Euler Systems and Arithmetic Geometry Barry Mazur and Tom...

Euler Systems and Arithmetic Geometry

Barry Mazur and Tom Weston

Contents

Chapter 1. Lecture 1 51. Galois Modules 52. Example : Quasi-Finite Fields 63. Local Fields 10

Chapter 2. Lecture 2 131. Discrete Valuation Rings 132. Local Fields 143. The Galois Theory of Local Fields 16

Chapter 3. Lecture 3 191. Ramification Groups 192. Witt Vectors 233. Projective Limits of Groups of Units of Finite Fields 24

Chapter 4. Lecture 4 271. The Absolute Galois Group of a Local Field 272. Galois Representations 31

Chapter 5. Lecture 5 351. Group Cohomology 352. Galois Cohomology 383. Tate Local Duality 42

Chapter 6. Lecture 6 451. Duality Preliminaries 452. Tate Local Duality 49

Chapter 7. Lecture 7 531. Finite/Singular Structures 532. Generalized Selmer Groups 573. Brauer Groups 60

Chapter 8. Lecture 8 651. Notation for Generalized Selmer Groups 652. A Global Pairing 673. The Finiteness of the Selmer Group 69

Chapter 9. Lecture 9 751. Abelian Varieties 752. Selmer Groups of Abelian Varieties 77

3

4 CONTENTS

Chapter 10. Lecture 10 811. Kummer Theory 812. Cohomology of Abelian Varieties 86

Chapter 11. Lecture 11 931. L/K Forms 932. Cohomological Interpretations 95

Chapter 12. Lecture 12 991. Torsors for Algebraic Groups 99

Chapter 13. Lecture 13 1031. The Picard Group of a Curve 1032. Direct Limits of Selmer Groups 1073. Conditions on Galois Representations 108

Chapter 14. Lecture 14 1151. Structures on Galois Representations 1152. Depth and Kolyvagin-Flach Systems 119

Chapter 15. Lecture 15 1211. The Main Theorem 1212. The Main Theorem for Rank 1 Modules 124

Chapter 16. Lecture 16 1271. Other Versions of the Main Theorem 127

Chapter 17. Lecture 17 1351. Modular Curves 1352. Operators on Modular Curves 138

Chapter 18. Lecture 18 1431. Representations on Torsion Points 1432. Heegner Points 146

Chapter 19. Lecture 19 1491. Hecke Operators on Heegner Points 149

Chapter 20. Lecture 20 1551. The Euler System for X0(11) 155

Chapter 21. Lecture 21 1591. Local Behavior of Cohomology Classes 159

Chapter 22. Lecture 22 1631. Completion of the Proofs 163

Bibliography 169

CHAPTER 1

Lecture 1

This lecture will be a summary of the material to be covered in the next fewlectures; many details will be left until then.

1. Galois Modules

Let K be a field, and fix a separable closure Ks. Set

GK = Gal(Ks/K) = lim←−K⊆K′⊆Ks

Gal(K ′/K)

where the projective limit is over all finite Galois extensions K ′ of K contained inKs. We give GK its usual profinite (Krull) topology.

Let M be a GK-module; that is, M is an abelian group with a left GK-action.We will say that M is a GK-Galois module if this action is continuous with respectto the profinite topology on GK and the discrete topology on M . (These are calledtopological GK-modules in [Se-LF].) Equivalently, for each m ∈ M there must bean open subgroup H of GK which fixes m; that is,

M =⋃

H open subgroup of GK

MH .

(Recall that MH is the subgroup of M of elements invariant under H.)In particular, if M is a finitely generated GK-Galois module, then there is an

open subgroup H of GK which fixes all of M . (H is just the intersection of the opensubgroups fixing each of the finitely many generators of M .) Since open subgroupsof G are just Gal(Ks/K

′) for finite extensions K ′ of K, replacing K ′ by its normalclosure we see that the GK-action on M factors through Gal(K ′/K) for some finiteGalois extension K ′ of K.

For future reference we will briefly clarify the relationship between GK-Galoismodules and abelian sheaves for the etale topology on SpecK. (See [Mi-EC, p.53] for more details.) In fact, there is an equivalence of categories between thetwo, although certain objects often belong more naturally in one category than theother. Specifically, set x = SpecK and x = SpecKs. Then given an abelian sheafF we associate to F the GK-Galois module

MF = F(x) = lim−→x′→xF(x′)

where the direct limit is over x′ = SpecK ′ for all finite Galois extensions K ′ of Kin Ks.

0Last modified September 4, 2003

5

6 1. LECTURE 1

In the other direction, we first recall that for any x-scheme U which is connectedand etale we can associate a GK-set (that is, a set with a GK-action) by

U 7→ Homx(x,U),

the Ks-valued points of U . Given a GK-Galois module M , we now associate to itthe etale sheaf FM such that

FM (U) = HomGK (Homx(x,U),M)

for any finite etale U over x. We leave it to the reader to check that these associa-tions give the asserted equivalence of categories. (In particular in the first directionone must check that MF really is a discrete GK-module. In the second direction,one must check that FM really is a sheaf and that Fm(x) = M .)

If M is a GK-Galois module the Galois cohomology groups

Hq(GK ,M) = lim−→H C GK ,H open

Hq(GK/H,MH)

are defined. (Note that if H is open and normal in GK then GK/H is finite. Forgeneralities on the cohomology of finite groups see [Se-LF, Chapters 7 and 8].) Inparticular,

H0(GK ,M) = MGK = HomGK (Z,M),

the GK-invariants of M . More generally, for any q ≥ 0

Hq(GK ,M) = ExtqGK (Z,M).

2. Example : Quasi-Finite Fields

2.1. Cohomology Groups. We now specialize the notions of the previoussection to the case of a quasi-finite field k. (See [Se-LF, Chapter 13, Section 2] forthe definition of a quasi-finite field.) When working with a quasi-finite field (whichfor us will almost always actually be finite), we will usually write gk or even just gfor its absolute Galois group Gal(ks/k). We then have an isomorphism

gk ∼= Z,

where Z is the profinite completion of Z with respect to its subgroups of finiteindex; that is,

Z = lim←−N

Z/NZ.

(If k is actually finite, then there is a canonical choice for this isomorphism; namely,we want 1 ∈ Z to correspond to the Frobenius automorphism. For general quasi-finite fields we do not have such a canonical choice.) Fix one such isomorphismand let ϕ ∈ gk correspond to 1. In this setting a gk-Galois module M is simply anabelian group M with an automorphism ϕ such that every orbit of ϕ is finite.

The cohomology theory in this case is quite simple, at least when M is torsion.It is determined by the exact sequence

0 −−−−→ Mgk −−−−→ M1−ϕ−−−−→ M −−−−→ Mgk −−−−→ 0

where Mgk is the gk-coinvariants M/(1− ϕ)M . (This is the largest quotient of Mon which gk acts trivially, just as Mgk is the largest submodule of M on which gkacts trivially.) In fact, these give the only non-zero cohomology groups.

2. EXAMPLE : QUASI-FINITE FIELDS 7

Theorem 2.1. If k is a quasi-finite field and M is a torsion gk-Galois module,then there are natural isomorphisms

Hq(gk,M) =

Mgk q = 0;Mgk q = 1;0 q ≥ 2.

In particular, given a gk-module map f : M → N the following diagrams commute:

H0(gk,M)f−−−−→ H0(gk, N)

∼=

y y∼=Mgk

f−−−−→ Ngk

H1(gk,M)f−−−−→ H1(gk, N)

∼=

y y∼=Mgk

f−−−−→ Ngk

where the horizontal maps in each case are the natural maps induced by f .

Proof. See [Se-LF, Chapter 13, Section 1].

In fact, somewhat more is true: suppose we are given an exact sequence

0→M → N → P → 0

of gk-Galois modules. Consider the exact commutative diagram

0

0

0

0 // Mgk //

Ngk //

P gk

0 // M //

1−ϕ

N //

1−ϕ

P //

1−ϕ

0

0 // M //

N //

P //

0

Mgk//

Ngk//

Pgk//

0.

0 0 0

The horizontal arrows in the first and last rows are either obvious or come fromfunctoriality, depending on which you prefer. The snake lemma now yields an exactsequence

0→Mgk → Ngk → P gk →Mgk → Ngk → Pgk → 0.

In particular, we obtain a boundary map P gk →Mgk .

8 1. LECTURE 1

Proposition 2.2. In the situation above, the diagram

H0(gk, P ) −−−−→ H1(gk,M)

∼=y y∼=P gk −−−−→ Mgk

commutes, where the top and bottom maps are the boundary map in cohomologyand the map constructed above respectively.

Proof. This follows easily from the explicit isomorphisms of [Se-LF, Chapter13, Section 1].

2.2. Cup Products. We continue to assume that k is a quasi-finite field (witha fixed topological generator ϕ of gk) and that M is a torsion gk-module. Sincemost of the cohomology groups of M are trivial, the cup product structure must beparticularly simple. (Recall that for any profinite group G and discrete G-modulesM and N , the cup product is a pairing

Hi(G,M)⊗Hj(G,N)→ Hi+j(G,M ⊗N)

for any i, j ≥ 0. See [AW, Section 7]. Recall that if M and N are G-modules,then M ⊗ N is made into a G-module by setting g(m ⊗ n) = gm ⊗ gn.) UsingTheorem 2.1 it is easy to describe the non-trivial cup product pairings for gk.

Proposition 2.3. For i = j = 0 the cup product is simply the natural map

Mgk ⊗Ngk → (M ⊗N)gk .

Similarly, for i = 1, j = 0 the cup product is the natural map

Mgk ⊗Ngk → (M ⊗N)gk .

Proof. This follows easily from the definition of cup product and the usualisomorphisms.

We can use the above proposition to determine the Pontrjagin dual of M .Recall that this is defined to be

M∨ = Hom(M,Q/Z).

(As always, if G is a group and M and N are G-modules, Hom(M,N) is givena G-module structure by (gf)(m) = gf(g−1m), f ∈ Hom(M,N), g ∈ G, m ∈M . In particular, the G-invariants of Hom(M,N) are precisely the G-equivarianthomomorphisms.)

Proposition 2.4. Let M and N be finite gk-Galois modules. Suppose thatthere is a perfect pairing

M ⊗N → Q/Z.

(This is equivalent to N ∼= M∨.) Then the induced cup product pairings

Hi(gk,M)⊗H1−i(gk, N)→ H1(gk,M ⊗N)→ H1(gk,Q/Z) = Q/Z

are also perfect pairings for i = 0, 1.

Proof. This follows easily from Proposition 2.3.

2. EXAMPLE : QUASI-FINITE FIELDS 9

2.3. Finite Fields. We now specialize to the case where k is a finite field. Setp = char k and q = pf = |k|. Since k is finite it is perfect and we have ks = k, thealgebraic closure of k. Furthermore, in this case we have a canonical isomorphism

gk → Z

given by sending the Frobenius automorphism ϕ to 1. (We normalize ϕ so that itsends x to xq.) For every n ≥ 1 there is a unique extension kn of k of degree n; wewill write gn for the Galois group Gal(kn/k). It is the unique (cyclic) quotient ofgk of order n.

We have |kn| = qn; set wn = |k∗n| = qn−1. For any m dividing n we have normmaps

k∗nN−→ k∗m;

these are in fact surjective. (See [Fr, Section 7, Corollary to Proposition 3].) Define∆k to be the projective limit of the k∗n under these maps. ∆k is a gk-module,although it is not a gk-Galois module in the sense of Section 1.

Proposition 2.5. ∆k is (very non-canonically) isomorphic to∏l 6=p Zl, where

the product is over all primes l not equal to p.

Proof. First note that the Chinese remainder theorem shows that the aboveproduct is simply the projective limit

lim←−N,(N,p)=1

Z/NZ,

the limit taken over only those N relatively prime to p. Further, note that for anyN relatively prime to p we can find a wn with N dividing wn. The propositionnow follows easily from the fact that ∆k is a projective limit of cyclic groups of allorders prime to p by surjective maps.

The proposition implies that for any finite extension k′ of k, ∆k′ and ∆k areabstractly isomorphic. In fact, a canonical isomorphism is given by the norm map.(This is one case where it is much more natural to think of things in terms of etaleabelian sheaves.) They are even isomorphic as gk′ -modules; since gk doesn’t acton ∆k′ , this is the best we could hope for. In any event, this leads one to consider∆Fp . Here one is able to use roots of unity to construct a canonical isomorphism

∆Fp ∼=∏l 6=p

Zl(1),

where Zl(1) is the Tate module of the l-power roots of unity. That is,

Zl(1) = lim←−n

µln ,

the maps

µln+1 → µln

being the lth-power map. Note that the Zl(1) notation is slightly misleading; Zl(1)is not canonically isomorphic to Zl as an abelian group. Zl(1) is called the Tatetwist of Zl; gk acts on it simply by the cyclotomic action.

10 1. LECTURE 1

3. Local Fields

We continue with the notation of the previous section: k is a finite field ofcharacteristic p and order q. Recall that to k one can associate the ring of Wittvectors

W (k) = (a0, a1, a2, . . .) | ai ∈ kwhich is made into a ring using certain universal polynomials. (See [Se-LF, Chapter2, Section 6]. With some effort they can be reconstructed by attempting to modelthe operations in Zp.) W (k) has two natural operators F and V given by

F : (a0, a1, a2, . . .) 7→ (ap0, ap1, a

p2, . . .);

V : (a0, a1, a2, . . .) 7→ (0, a0, a1, a2, . . .).

Clearly F and V commute, and in fact V F = FV is simply multiplication by p inW (k). It turns out that W (k) is a complete, absolutely unramified (that is, withuniformizer p) discrete valuation ring with residue field k.

Now, consider the units W (k)∗. They naturally sit in an exact sequence

0→ 1 + pW (k)→W (k)∗ → k∗ → 0,

where the map W (k)∗ → k∗ is reduction modulo p. Since 1 + pW (k) is pro-p andk∗ has order prime to p it follows that the sequence splits, so that k∗ lifts uniquelyto W (k)∗ and

W (k)∗ ∼= k∗ × (1 + pW (k)).

The splitting map k∗ → W (k)∗ is called the Teichmuller character and denotedω. (Its existence can also be seen more directly; for example, it follows easily fromHensel’s lemma. (See [Se-LF, Chapter 2, Section 2.2, Lemma] or [Cas, AppendixC].) Alternatively, one can check that k∗ is simply the roots of unity µ(W (k)).)

The same construction applies to kn; W (kn) is also an absolutely unramifiedcomplete discrete valuation ring. Letting K be the quotient field of W (k), Kn thequotient field of W (kn), W (k) the union of all of the W (kn)’s and K∞ its fractionfield, we obtain a diagram

K∞

::

:: _

gk

_

W (k)0

AA

::

::

Kn

gn

::::::::

k_

gk

_

::

:: W (kn)

0

BB

::::::: K

kngn

;;;;;;; W (k)0

AA

k

This construction can be shown to give all of the unramified extensions of K. (See[Se-LF, Chapter 3, Section 5]. Note that this approach only works for absolutelyunramified K.)

3. LOCAL FIELDS 11

The next step from K to K is the tamely ramified piece. For this we useKummer extensions. We adjoin to the field Kn the wth

n roots of p. More precisely,set

Ln = Kn[x]/(xwn − p).

Since Kn contains µwn this is indeed a Kummer extension. Also, xwn − p is anEisenstein polynomial, so Ln is totally ramified of degree wn over Kn; since p doesnot divide wn the ramification is tame. Furthermore, if we let V be the subgroupof K∗n/K

∗wnn generated by p, then Kummer theory (as in Lecture 8, Section 3.1)

identifies Gal(Ln/Kn) with Hom(V, µwn) as Gal(Kn/K)-modules. (Recall thatGal(Kn/K) acts on Gal(Ln/Kn) by extending an automorphism of Kn to Ln andthen using it to conjugate Gal(Ln/Kn); this is well-defined since Gal(Ln/Kn) isabelian.) In fact, since Kn is absolutely unramified and p ∈ K, V is isomorphicto Z/wnZ with trivial Galois action. Furthermore, we know that µwn ∼= k∗n, sofinally we can identify Gal(Ln/Kn) with k∗n as Gal(Kn/K)-modules. Thus, if welet Tn = Gal(Ln/K), we see that it sits in an exact sequence

1→ k∗n → Tn → gn → 1.

This sequence does not in general split; however, since p ∈ K we can take a singlewthn root of p, say α, and adjoin it to K:

Ln

K(α)

gnwwwwwwwww

Kn

k∗n

GGGGGGGGG

K

gn

wwwwwwwwww

GGGGGGGGG

This does allow us to identify a (non-canonical) lifting of gn to Tn, and thus toidentify Tn with the semi-direct product of k∗n and gn. We have shown above thatthe action of gn for this semi-direct product is simply its natural action on k∗n.

Now, define L∞ to be the union of all of the Ln, and let T = Gal(L∞/K).That L∞ is the maximal tamely ramified extension of K∞ can be seen easily fromKummer theory. Gal(L∞/K∞) is the projective limit of the k∗n’s; one can checkthat the maps (m divides n)

Gal(Ln/Kn)→ Gal(Lm/Km)

for this projective system are simply the norm maps

k∗nN−→ k∗m.

Thus Gal(L∞/K∞) ∼= ∆k.

12 1. LECTURE 1

We summarize our constructions to this point in the following diagram.

K _

IK

_

_

GK

_

L∞

Pwildly ramified

_

T

_

K∞

∆ktamely ramified

K

gkunramified

Here IK is the inertia group of K. Also, gk ∼= Z (canonically) and ∆k∼=∏l 6=p Zl

(non-canonically). P is a pro-p group which is quite complicated.

CHAPTER 2

Lecture 2

1. Discrete Valuation Rings

1.1. Definitions. Let A be a commutative ring with unit. We recall six equiv-alent characterizations of discrete valuation rings. (As a general convention, for anylocal ring A we let mA denote its maximal ideal.)

(a) A is a noetherian local ring and mA is principal, generated by a non-nilpotent.

(b) A is a noetherian, integrally closed integral domain with only one non-zeroprime ideal.

(c) A is a principal ideal domain with a unique non-zero prime ideal.(d) Let A be an integral domain with field of fractions K. Then K possesses

a surjective valuation homomorphism

v : K∗ Z,

for which, if we set v(0) = +∞, we have

v(x+ y) ≥ minv(x), v(y)

for all x, y ∈ K, and A can be recovered as

A = x ∈ K | v(x) ≥ 0.

(e) A has a non-nilpotent element π such that every non-zero element a ∈ Acan be written uniquely as a = uπn for some u ∈ A∗, n ≥ 0.

(f) A is a discrete valuation ring.

For proofs of the equivalences see [Se-LF, Chapter 1, Section 2]. These proofsare essentially a sort of “resolution of singularities” in Krull dimension 1; that is,the implication (b) implies (a) shows that taking the integral closure of a localdomain of dimension 1 gives a regular local ring of dimension 1. Geometrically,this corresponds to replacing a possibly non-regular point with a regular point.

Any element π ∈ A of valuation 1 is called a uniformizer or a local parameter;these are precisely the generators of mA. This name also has a geometric interpre-tation: if A is the local ring of a smooth k-rational point x of an algebraic curveX over k, then a “local parameter” in the above sense is simply a rational functionon X which provides a local parameter for X about the point x.

If A is a discrete valuation ring, we will write kA (or simply k if it is clear fromcontext) for its residue field A/mA.


13

14 2. LECTURE 2

1.2. Completions. There are two different, yet equivalent, points of view oncompletions. We begin with the algebraic approach. (See [Ma, Chapter 8] for thedetails.) For any noetherian local ring A, we define its mA-adic completion (or justcompletion) to be the ring

A = lim←−n

A/mnA.

A is again a local ring, with maximal ideal mA = mAA. There is a natural injection

A → A

since by Krull’s theorem ∩nmnA = 0. If A is a discrete valuation ring, then A is

also a discrete valuation ring, and any uniformizer of A is again a uniformizer of A(although there are of course many uniformizers of A which do not lie in A). We letK and K be the field of fractions of A and A respectively; we now have a diagramas below.

A //

>>>>>>>> A

>>>>>>>

K // K

The topological approach begins with the fraction field K; we define a metricon K by

‖ x ‖= e−v(x)

for x ∈ K. (The choice for the base of the exponent is irrelevant for our purposes.)The properties of v make K into a topological field with respect to this metric,and we define K to be the topological completion of K with respect to the inducedmetric topology. We recover the ring A simply as

A = x ∈ K; ‖ x ‖≤ 1.

For more details on this approach, see [Se-LF, Chapter 2]. For more details on thecomparison of the two methods, see [Wes-Ide, Sections 1 and 2].

One easily sees that the residue fields of A and A coincide; that is, the naturalmap A → A induces an isomorphism

kA∼−→ kA.

2. Local Fields

2.1. Definitions. Let A be a complete discrete valuation ring (that is, A = A)with fraction field K and residue field k. If K has characteristic 0 and k is finite,we will call such a K a local field. (This is not the usual definition of local field,but it will suffice for our present purposes.)

Since A is complete, we have

A = lim←−n

A/mnA.

By assumption A/mA = k is finite, and it follows easily that each A/mnA is finite,

since it has finite length over k. Specifically, if k has order q, then A/mnA has order

qn. In particular, each A is a profinite topological ring (since it is an inverse limitof finite rings) and thus is compact in its natural profinite (i.e., mA-adic) topology.

2. LOCAL FIELDS 15

The fraction field K can be written as

K =⋃n∈Z

π−nA

for any uniformizer π. Each π−nA is easily seen to be both an open and closedsubset of K (since the metric on K is discrete), and each π−nA is a homeomorphicimage of a compact set and thus compact. Thus K is a locally compact topologicalfield.

Given only the topology on K, we can recover A and its maximal ideal mA as

A = x ∈ K | xν∞ν=1 is contained in a compact set

andmA = x ∈ K | lim

ν→∞xν = 0.

From this we also recover A∗ = A − mA and the valuation homomorphism, sincewe know mA. Again, for more details on the topology of local fields see [Wes-Ide].

2.2. Construction of Local Fields. Let F be a number field with ring ofintegers OF . (Recall that this means that F is a finite extension of Q and OF isthe integral closure of Z in F .) Let p be a prime ideal of OF , and consider

OF,p = lim←−n

OF /pn.

Then OF,p is a complete discrete valuation ring (see [Se-LF, Chapter 1, Section3]). In fact, it can be shown that any local field (as we have defined them) arisesin this way.

There is another approach which is in some sense more self-contained. Thebasic result is the following theorem.

Theorem 2.1. Let K be a local field and let L be a finite extension of K. Thenthere is a unique local field structure on L extending that of K. (That is, there is aunique topology on L giving it the structure of a local field which restrictson K tothe given topology.)

Proof. (Sketch. For details, see [Se-LF, Chapter 2, Section 2].) Let B be theintegral closure of A in L.

B // L

A

OO

// K

OO

L is a finite dimensional vector space over K, so there is only one compatibletopology on L. (See [Cas, Section 10].) Now let p be a prime ideal of B. Supposep∩A = 0. Then one can use the integrality of B over A to show that p must be 0.Thus, for any non-zero prime p of B we must have p ∩A = mA. Again, integralityinsures that some such prime must exist. Furthermore, p is unique; for if it weren’t,each p could be used to define a different topology on L compatible with that onK (since p ∩ A = mA). It then follows from the equivalent characterizations ofdiscrete valuation rings that B is a discrete valuation ring, and thus that L is alocal field.

16 2. LECTURE 2

Now, choose a rational prime p. We start with the field Qp, which is a local fieldwith ring of integers Zp. Take K to be any finite extension of Qp; by the theoremit is a local field in a unique way, with ring of integers A the integral closure of Zpin K. Again, it can be shown that these are all of the possible local fields (as wehave defined them) with residue field of characteristic p.

3. The Galois Theory of Local Fields

Let L/K be a finite Galois extension of local fields. Set G = Gal(L/K). Notefirst that G stabilizes B; that is, if s ∈ G, then s(B) = B. This is essentiallybecause B is integral over A and s fixes A and thus integral equations over A aswell. Further, G stabilizes mB . (For this use the topological characterization of mB

and the fact that elements of G are homeomorphisms.)It follows that G induces an action on B/mB = kB . To unify notation, let

` = kB for the remainder of this section. Since `/k is a finite extension of finitefields it is automatically Galois. Set g = Gal(`/k). We now have a diagram

mB

G

// B

G

//

L

G

mA // A

//

K

`

g

B/mB

g

k A/mA

Since we have shown that G acts on `, and since it obviously fixes k, we haveconstructed a natural map

G→ g

which is obviously a group homomorphism. We define the inertia group I to be thekernel of this map; it is a subgroup of G.

Theorem 3.1. The natural map G → g is surjective; thus we have an exactsequence

0→ I → G→ g→ 0.

Proof. k is finite and thus perfect, so the primitive element theorem impliesthat we can write ` = k[b] for some b ∈ `. Consider the set of k-conjugates of b in `.g acts on this set, by definition, and it acts simply transitively since b is a primitiveelement. Now lift b to an element b of B. Let Pb(T ) be the minimal polynomial ofb over A; it has the form

Pb(T ) =∏

s∈S⊆G

(T − s(b)

)∈ A[T ]

where S is a subset of G such that S acts simply transitively on the K-conjugatesof b. Consider the reduction of Pb(T ), say Pb(T ) ∈ k[T ]. By construction b is aroot of Pb(T ). Since Pb(T ) has coefficients in k, it follows that any k-conjugate ofb is also a root of Pb(T ); that is, for any σ ∈ g there exists an s ∈ G such that

3. THE GALOIS THEORY OF LOCAL FIELDS 17

σ(b) = s(b). Since b generates the entire extension ` over k, it follows that s mapsto σ under the natural map G→ g, and thus that this map is surjective.

Let n, f and e be the number of elements in G, g and I respectively. n issimply the degree of L/K; we call f the residue degree and e the inertial degree orramification index of the extension L/K. By Theorem 3.1 we have n = ef . If weset (temporarily) Kur = LI , the subfield of L fixed by I, we have a diagram

L _

G

_

Kur

Itotally ramified

K

gunramified

CHAPTER 3

Lecture 3

1. Ramification Groups

1.1. Definitions. In an effort to further understand the ramification in Galoisextensions of local fields we will now define a certain filtration of the Galois group.As before let L/K be a finite Galois extension of local fields with ring of integersB and A respectively. Let ` and k be their residue fields, and let p be the residuecharacteristic. For any i ≥ 0 and s ∈ G, consider the following two equivalentconditions:

(a) s is the identity on B/mi+1B ;

(b) vL(s(b)− b) ≥ i+ 1 for all b ∈ B.Here vL is the valuation on L, normalized so that

vL : L∗ Z.

If in addition we have that B = A[β] for some β ∈ B, then we have a third conditionequivalent to the previous two.

(c) vL(s(β)− β) ≥ i+ 1.The proof of the equivalence is easy; see [Se-LF, Chapter 4, Section 1, Lemma

1]. We define Gi to be the subgroup of G of elements s satisfying any of the aboveconditions. Gi is called the ith ramification group of the extension L/K. It followseasily from condition (a) that each Gi is normal in G. By (b) we also see that forsufficiently large i we must have Gi = 1. Thus we have a filtration

1 C · · · C Gi C Gi−1 C · · · C G1 C G0 C G.

We note thatG0 = I by definition, where I is the inertia group for the extensionL/K. This means that the rest of the group G is irrelevant for the computation ofthe ramification groups; in particular, we may as well replace K by Kur (recall thatthis is the maximal unramified extension of K in L; that is, Kur = LI) and thusassume that L/K is totally ramified. We make this assumption for the remainderof our discussion of ramification groups.

We now investigate the quotients Gi/Gi+1. We begin with the special case ofi = 0. By definition G0 stabilizes mB and G1 stabilizes m2

B . Thus the naturalaction of G0 on mB/m

2B factors through G0/G1; in fact, G0/G1 acts faithfully on

mB/m2B , since any element of G0 which fixes all of mB/m

2B actually lies in G1.

Since mB is principal, mB/m2B is free of rank 1 over B/mB = `. Further, the action

of G0/G1 on mB/m2B is l-linear; this is easy to check using the fact that if b ∈ B

lifts an element of ` then s(b) ≡ b (mod mB) for any s ∈ G0. Combining all of thiswe have the following proposition.


19

20 3. LECTURE 3

Proposition 1.1. There is a natural injection

G0/G1 → Autl(mB/m2B) ∼= `∗.

Corollary 1.2. G0/G1 is cyclic of order dividing qB − 1, where qB is theorder of `; in particular, it is prime to p.

We now go off on a complete digression before considering the higher quotients.

1.2. Digression : Filtrations of Matrix Groups. Let R be an arbitrarycommutative, noetherian, complete local ring with finite residue field k and maximalideal mR. Thus

R = lim←−n

R/mnR

and we have an exact sequence

0→ mR → R→ k → 0.

R is profinite, and thus a compact topological ring. There is also a multiplicativeversion of the above exact sequence:

1→ 1 + mR → R∗ → k∗ → 1.

This is essentially the exact sequence comparing GL1(R) with GL1(k), where forany ring R, GLN (R) is the group of invertible N by N matrices with entries in R.(Recall that a necessary and sufficient condition for such a matrix to be invertibleis that its determinant lies in R∗.)

Fix some positive integer N . Consider the group GLN (R). It follows from thecompleteness of R that we have

GLN (R) = lim←−n

GLN (R/mnR);

thus GLN (R) is a compact topological group. For any i ≥ 1, we consider thesurjection

GLN (R) GLN (R/miR).

(This is easily seen to be a surjection; since R is local, any lifting to R of the entriesof a matrix in GLN (R/mi

R) will work.) Let Γi be the kernel; it is a normal subgroupof GLN (R) which is closed and of finite index, and thus open. Thus there is anexact sequence

1→ Γi → GLN (R)→ GLN (R/miR)→ 1,

and we have defined a filtration

· · ·Γi+1 C Γi C · · · C Γ1 C GLN (R)

of GLN (R).We now consider the quotients Γi/Γi+1. The first one, GLN (R)/Γ1, is just

GLN (k), which is an interesting group. Next consider the higher quotients. It iseasy to write down Γi explicitly; it is just the set of matrices of the form 1 + (Aαβ),with each entry Aαβ ∈ mi

R. (Here by 1 we mean the N ×N identity matrix and by(Aαβ) we mean the N ×N matrix with (α, β) entry Aαβ .) Now, define a map

Γi/Γi+1 → (Aαβ) | Aαβ ∈ miR/m

i+1R

1. RAMIFICATION GROUPS 21

by sending 1 + (Aαβ) to the reduction of (Aαβ) modulo mi+1R . (Here the second

set is made into a group by addition of matrices.) The fact that this is a grouphomomorphism comes from the calculation(

1 + (Aαβ))(

1 + (Bαβ))

= 1 + (Aαβ +Bαβ) + (Aαβ)(Bαβ) = 1 + (Aαβ +Bαβ)

since (Aαβ)(Bαβ) has all entries in m2iR , which vanishes modulo mi+1

R . It is easy tosee that this map is actually an isomorphism. The group structure of the secondgroup is quite simple; it is just isomorphic to

n2︷︸︸︷miR/m

i+1R × · · · ×mi

R/mi+1R ,

n2 copies of the additive group miR/m

i+1R . This in turn is an abelian group of

exponent p.We now look at the simplest case of this; take N = 1. Our Γi’s are thus defined

by the exact sequences

1→ Γi → R∗ → (R/miR)∗ → 1.

If we apply our above results with the ring B from earlier in this section, we obtaina filtration of the topological group B∗. The usual notation for the Γi in thissituation is Ui; B∗ is denoted by U . We thus have a filtration

· · · ⊆ Ui+1 ⊆ Ui ⊆ · · · ⊆ U1 ⊆ U .

Our above results show that each Ui/Ui+1 is a finite p-group, and thus that U1 is apro-p group; that is, U1 is an inverse limit of finite p-groups.

1.3. The Higher Ramification Groups. We now compare the two filtra-tions

1 C · · · C Gi C Gi−1 C · · · C G1 C G0

and· · · ⊆ Ui+1 ⊆ Ui ⊆ · · · ⊆ U1 ⊆ U

that we have constructed. As before, we assume that L/K is totally ramified, sothat G0 = I = Gal(L/K). Choose a uniformizer π = πB for B. One can easilycheck that B = A[π]. (See [Se-LF, Chapter 1, Section 6].) So in this case for anys ∈ G0 we have s ∈ Gi if and only if

s(π) ≡ π (mod mi+1B ).

This is in turn is equivalent to

s(π)π≡ 1 (mod mi

B),

since π has exact valuation 1. That is, s ∈ Gi if and only if

s(π)π∈ Ui.

Now consider the mapθi : Gi → Ui/Ui+1

defined by

s 7→ s(π)π

.

22 3. LECTURE 3

We claim first that θi is independent of the choice of π. To see this, let π′ beanother uniformizer for B; then π′ = uπ for some u ∈ U . We compute

s(π′)π′

=s(uπ)uπ

=s(u)u

s(π)π

=u+mi+1

u

s(π)π

for some mi+1 ∈ mi+1B , since s ∈ Gi. But

u+mi+1

u

s(π)π≡ s(π)

π(mod Ui+1)

since u is a unit. This shows that θi is well-defined independent of π. In fact, italso shows that θi is a group homomorphism. To see this, let s, t ∈ Gi. Note thatπ′ = t(π) is also a uniformizer. Thus

st(π)π

=st(π)t(π)

t(π)π

=s(π′)π′

t(π)π,

which shows that θi is a homomorphism, since in Ui/Ui+1 we have s(π′)/π′ =s(π)/π.

θi clearly vanishes on Gi+1, so we obtain a map

Gi/Gi+1 → Ui/Ui+1

which we shall hereafter refer to as θi. It is easy to see that θi is now injective, usingcriterion (c) for lying in Gi+1. Thus, by the determination of Ui/Ui+1 earlier wenow see that Gi/Gi+1 is abelian of exponent p, for i ≥ 1. This yields two importantcorollaries.

Corollary 1.3. The group G1 is a p-group.

Corollary 1.4. If L/K is tamely ramified, then G1 = 1. (Recall that anextension L/K is tamely ramified if its ramification index e is prime to p, theresidue characteristic.)

We now summarize our investigation of ramification. So assume L/K is anyfinite Galois extension of local fields (in particular, we no longer assume that it istotally ramified) with Galois group G and residue Galois group g. Set P = G1

and ∆ = G0/G1. Then P is a p-group (the wildly ramified inertia subgroup), ∆ iscyclic of order prime to p, and we have an exact sequence

1→ P → I → ∆→ 1.

Next, if we let T be the Galois group of the maximal tamely ramified extension ofK in L (that is, the extension LP), then we have exact sequences

1→ P → G→ T → 1

and

1→ ∆→ T → g→ 1.

∆ was previously identified with a subgroup of `∗, and one can check that theconjugation action of g on ∆ is just the usual action of g on `∗.

2. WITT VECTORS 23

1.4. The Upper Indexing. To this point we have been using the lower in-dexing of ramification groups. This is the simplest to define, but it unfortunatelybehaves very poorly with respect to extensions. To correct for this, one is led tointroduce the upper indexing as follows. (For more details, see [Se-LF, Chapter4].) First, we extend the definition of Gi to real indices by saying that s ∈ Gu ifand only if vL(s(b)− b) ≥ u+ 1 for all b ∈ B. Now define

ϕ(u) =∫ u

0

dt

(G0 : Gt).

(The integrand is piecewise constant, so this is really just a sum; the integralnotation is more convenient.) This can be checked to give a bijection from thepositive reals to itself. We define the upper indexing by

Gu = Gϕ(u).

The inverse function of ϕ can be determined to be

ψ(v) =∫ v

0

(G0 : Gw)dw.

ThusGv = Gψ(v).

It turns out that this upper indexing is compatible with extensions, in the sensethat if we have a diagram

M _

GM/K

_

L

K

GL/K

then under the natural surjection

GM/K GL/K

each GvM/K maps onto GvL/K .

2. Witt Vectors

Let R be a perfect ring of characteristic p. We define the Witt vectors over Rto be the set

W (R) = (a0, a1, . . .) | ai ∈ R,which is turned into a ring using certain universal polynomials. (See [Se-LF,Chapter 2, Section 6] for all references for this section.) The fundamental exampleis W (Fp) = Zp, which is really used to define the univeral polynomials in general.In this case the Witt vector of α ∈ Zp is just the p-adic expansion of α; that is,

(a0, a1, · · · )↔ α = a0 + a1p+ a2p2 + · · · .

In general the relationship is not this simple; it involves twisting by Frobenius.Now let k be a finite field of characteristic p. Then W (k) is is a complete

discrete valuation ring with residue field k. Further, W (k) is absolutely unramified;

24 3. LECTURE 3

that is, it has p as a uniformizer. The construction of Witt vectors is also functorial,so we get a map

Zp = W (Fp) →W (k).The Galois group g = Gal(k/Fp) acts on W (k), and one has a canonical isomor-phism

g ∼= Gal(K/Qp),where K is the fraction field of W (k). More generally, we have the following propo-sition.

Proposition 2.1. Let A be a complete discrete valuation ring with residue fieldk and fraction field K of characteristic 0. Then there is a unique ring homomor-phism

W (k) //

!!DDDDDDDD A

k

making this diagram commute, where both maps to k are the natural map.A is in fact a free W (k)-module of rank equal to the absolute ramification index

of K. Specifically, one can write A = W (k)[πA], where πA is any uniformizer forA.

3. Projective Limits of Groups of Units of Finite Fields

3.1. Topologically Cyclic Profinite Groups. Let Γ be a profinite groupwhich is topologically cyclic; that is, there is a γ ∈ Γ such that Γ is the closure ofthe group generated by γ. We further assume that γ has infinite order; this merelyexcludes the case where Γ is cyclic of finite order. Thus we have a map

γZ → Γ

with dense image. Morally speaking, any such Γ ought to be some sort of profinitecompletion of the infinite cyclic group Z. We will now attempt to make thatstatement more precise in certain special cases of interest to us.

First of all, we recall the standard examples. We have the full profinite com-pletion of Z,

Z = lim←−N

Z/NZ,

the inverse system being defined by divisibilities. We also have

Zp = lim←−n

Z/pnZ = lim←−N,(N,l)=1 for l 6= p

Z/NZ.

The first inverse system is with respect to the usual ordering of Z, the second withrespect to the multiplicative ordering. More generally, if L is any set of primenumbers of Z, we have∏

l∈L

Zl∼= lim←−N , N has all prime divisors in L

Z/NZ.

This isomorphism comes from the Chinese remainder theorem. All of these groupsare topologically cyclic, with canonical generator the image of 1 ∈ Z.

Conversely, suppose thatΓ = lim←−

N

CN ,

3. PROJECTIVE LIMITS OF GROUPS OF UNITS OF FINITE FIELDS 25

where each CN is cyclic of order wN and the maps of the inverse system are sur-jective. In particular Γ is topologically cyclic, as one can lift generators via thesurjective maps. Suppose further that every integer M prime to a fixed primenumber p divides some wN , and that all of the wN are prime to p. Then one caneasily show that

Γ ∼=∏l 6=p

Zl,

although the isomorphism is horribly non-canonical.

3.2. Example : Units of Finite Fields. Let k be a finite field of order qand characteristic p, let k be a fixed algebraic closure of k, and let kn be the uniquesubfield of k of degree n over k.

k

;;;;;;;

knn

<<<<<<<

k

Set Cn = k∗n and wn = |Cn| = qn − 1. This defines an inverse system via normmaps

kn′N−→ kn

for n dividing n′. These maps can be shown to be surjective (see [Fr, Section 7,Corollary to Proposition 3]), and one easily checks that the wn satisfy the conditionsat the end of the previous section. Thus, if we define

∆k = lim←−n

k∗n,

then∆k∼=∏l 6=p

Zl,

although the isomorphism is non-canonical. (Each factor Zl is canonical, but thechoice of generator and thus the identification with Zl is not.) This is actually ag-module, where g = Gal(k/k), and the action is continuous when ∆k is given theprofinite topology. (It is not continuous if ∆k is given the discrete topology; thus∆k is not a g-Galois module.)

Now let k′ be any finite extension of k in k. Then

∆k′∼= ∆k

since ∆k′ is defined to be an inverse limit of a cofinal subsystem of the inversesystem defining ∆k. In fact, this is a g′-isomorphism, where g′ = Gal(k/k′). Thissuggests that we should look at the Gal(k/Fp)-module ∆Fp . We use this module todefine Gal(k/Fp) modules Zl(1) by

∆Fp =∏l 6=p

Zl(1).

26 3. LECTURE 3

This notation is slightly misleading, in that Zl(1) is not canonically isomorphic toZl. It can actually be realized as the Tate module of the l-power roots of unity;that is,

Zl(1) = lim←−n

µln .

In fact, ∆k can also be realized as this same product, via our earlier isomorphism.Here g acts on each Zl(1) simply as it would as a subgroup of Gal(Fp/Fp).

CHAPTER 4

Lecture 4

1. The Absolute Galois Group of a Local Field

1.1. Finite Galois Extensions. We continue to let K be a local field, inthe sense defined previously, with ring of integers OK and residue field k = kA =OK/mA of characteristic p. For this section we will feel free to think of Galoisgroups as belonging to extensions of rings of integers; this will simplify notation,and no confusion should result.

Let g = Gal(k/Fp). Using our construction of Witt vectors, we get a diagram

A // k

W (k)

g

Aur //

g

k

g

W (Fp) Zp// Fp

In fact, using our study of ramification groups we know that we can further decom-pose the extension as

A

p-group

Atr

⊆k∗

Aur

g

A

where the notation above is supposed to indicate that we know that Gal(Atr/Aur)is isomorphic to the ramification quotient G0/G1, which canonically embeds in k∗.(See Lecture 3, Section 1.1.)

1.2. Passage to the Limit. We now generalize the above construction. Letkn be the unique extension of k of degree n, and let gn = Gal(kn/k). Let qn bethe order of kn and let wn = qn − 1 be the order of k∗n. Then Hensel’s lemma (see[Cas, Appendix C]) shows that the elements of k∗n lift to W (kn) as roots of unity;


27

28 4. LECTURE 4

more precisely, there is an isomorphism

ω : k∗n∼=−→ µ(W (kn)),

called the Teichmuller character, such that

ω(a) ≡ a (mod m)

for all a ∈ k∗n, where m is the maximal ideal of W (kn). Somewhat more explicitly,one can define ω(a) by

ω(a) = limν→∞

aqνn

where a is any choice of lifting of a to W (kn). (The basic idea here is that expo-nentiating by powers of qn will not affect the root of unity part of a, while it will“push” the non-root of unity part off to infinity. It is still congruent to a sinceexponentiating by qn is the identity on the residue field.)

Let us now construct the maximal totally tamely ramified extension of An =W (kn). This can be done using Kummer theory (for a review of Kummer theory,see Lecture 8, Section 3.1); simply set

Atrn = W (kn)[x]/(xwn − p).

As before, Kummer theory shows that the Galois group of Atrn over W (kn) is canon-

ically isomorphic to k∗n. (See Lecture 1, Section 3.) This further implies that Atrn

really is the maximal tamely ramified extension of An, since we have shown thatthe Galois group of any tamely ramified extension of An will inject into k∗n. (SeeLecture 3, Proposition 1.1. One must also use the fact that the residue field doesnot change under totally ramified extensions.) Choose a uniformizer πn of Atr

n .Letting Tn be the Galois group of Atr

n over W (Fp) = Zp, we have a diagram

Atrn

gnk∗n

IIIIIIIIII_

T

_

Zp[πn]

IIIIIIIIIIAn

gn

Zp

where the unlabeled extension isn’t even Galois, since Zp doesn’t contain the wthn -

roots of unity for n ≥ 2. Nevertheless, we do get a semi-direct product decomposi-tion

Tn = k∗n o gn

where the action of gn on k∗n is simply its natural action. (This is immediate, usingthe fact that the isomorphism of Kummer theory is Galois equivariant.)

1. THE ABSOLUTE GALOIS GROUP OF A LOCAL FIELD 29

Next, we take the limit as n goes to infinity of the above construction. Thisyields a diagram

Atr∞

g∆Fp

CCCCCCCC_

T

_

·

CCCCCCCCC A∞

g

Zp

where A∞ = W (Fp), g = Gal(Fp/Fp) ∼= Z, Atr∞ is the tamely ramified extension of

A∞ generated by all non p-power roots of p, T is the Galois group of the fractionfield of Atr

∞ over Qp, and the unnamed entry is some non-canonical non-Galoisextension of Zp. In particular, as above T = ∆Fp o g, where the action of g on ∆Fpis its natural action. We can also write this as an exact sequence

1→ ∆Fp → T → g→ 1

or, via our earlier canonical isomorphisms,

1→∏l 6=p

Zl(1)→ T → Z→ 1.

Both of the flanking terms in this exact sequence are topologically cyclic inthe sense of Lecture 3, Section 3. Let ϕ be any lifting of the canonical Frobeniusgenerator of g; we will refer to it as a Frobenius element of T . Let τ be anytopological generator of ∆Fp . No matter the choice of ϕ and τ , they are related by

ϕτϕ−1 = τp

since ϕ acts as Frobenius on ∆Fp , and the action in this setting is given by conju-gation. In fact, this is a presentation of T , in a sense that we make precise in thefollowing proposition.

Proposition 1.1. The group T is the profinite completion of the group on twogenerators τ and ϕ satisfying the single relation ϕτϕ−1 = τp.

Proof. We sketch the proof. Let F be the group generated by τ and ϕ withthe above relation. Every element can be written uniquely in the form τaϕb, and wehave the commutation relation ϕaτ b = τ bp

a

ϕa. One first shows that the profinitecompletion of the subgroup generated by ϕ is Z; this is because combining theabove relation with ϕn = 1 and τn = 1 doesn’t introduce any additional conditionssolely on ϕ. On the other hand, the same construction with the subgroup generatedby τ does result in the condition

τpn−1 = 1.

It follows from Lecture 3, Section 3 that the profinite completion of the subgroupgenerated by τ is isomorphic to ∏

l 6=p

Zl.

30 4. LECTURE 4

Also, the relation shows that this subgroup is normal. Thus we get an exact se-quence

1→∏l 6=p

Zl → F → Z→ 1,

where F is the profinite completion of F . Furthermore, we have a lifting of agenerator of Z to F : namely, ϕ. We also know how ϕ acts on the other factor, bythe defining relation. Thus

F ∼=∏l 6=p

Zl o Z,

with the same action of Z on∏l 6=p Zl as T . Thus T ∼= F .

It follows easily from this characterization that we also have a homeomorphism(although definitely not a homomorphism)∏

l 6=p

Zl × Z→ T

given by sending α× β to ταϕβ .Let us now consider a subgroup T ′ of T of finite index. T ′ must contain some

power of ϕ, since otherwise all of the powers of ϕ would lie in different cosets.Similarly, it must contain some power of τ , and we can take this power to be primeto p since p is “invertible” when acting on τ . In other words, there is some integere prime to p, some integer f and some lifting ϕ of 1 ∈ Z such that

τe, ϕf ⊆ T ′.

We will denote by Te,f the closed subgroup of T generated by τe and ϕf for e andf as above; the above argument shows that these are all of the subgroups of T offinite index. (In particular they are open.) They are not all normal, however; onecan check easily (by conjugating τe by ϕ and ϕf by τ) that this occurs if and onlyif e divides pf − 1.

1.3. Absolute Galois Groups. Since T is the maximal tamely ramified quo-tient of the absolute Galois group GQp = Gal(Qp/Qp), we have an exact sequence

1→ P → GQp → T → 1

where P is the wild ramification subgroup of GQp ; it is pro-p. Letting I be theinertia subgroup of GQp , we have an exact sequence

1→ I → GQp → g→ 1.

Restricting our attention to I alone, we have an exact sequence

1→ P → I → ∆Fp → 1.

2. GALOIS REPRESENTATIONS 31

This is all sumarized in the diagram

Qp _

I

_

_

GQp

_

Qtrp

Pwildly ramified

_

T

_

Qurp

∆Fptamely ramified

Qp

gunramified

Now let K be any finite extension of Qp. Let GK be its absolute Galois group,so that GK ⊆ GQp . We set IK = I ∩ GK and PK = P ∩ GK ; IK is simply theinertia group of K/K. We also let TK be the image of GK under the map GQp → T ;this is not the same as the tame inertia group of K in the usual sense if K has wildramification over Qp. One easily checks that TK = Te,f where e is the prime to ppart of the inertial degree of K and f is the residue degree of K. We now have anexact sequence

1→ PK → GK → TK → 1.

2. Galois Representations

2.1. Definitions and Reductions. Let K be a local field of residue char-acteristic p. Let F be a field of scalars; for our purposes we will take F to alsobe a local field, of residue characteristic l. We will usually, but not always, wantl 6= p. Let V be a vector space over F of finite dimension d. We consider GK-representations on V ; of course, we assume them to be continuous and F -linear.So, by a Galois representation we mean a continuous homomorphism

ρ : GK → AutF (V ) ∼= GLd(F ),

the isomorphism being given by any choice of an F -basis of V .We begin by showing that one may actually assume that the image of GK lands

in GLd(OF ), where OF is the ring of integers in F . More generally, let Γ be anyprofinite group which acts continuously and F -linearly on V :

Γ→ AutF (V ).

Recall that a lattice Ω ⊆ V is a free OF -submodule of V of rank d, such that

Ω⊗OF F∼=−→ V.

We claim that there is some lattice Ω which is stabilized by Γ. Let Ω0 beany lattice of V . Then if we choose the basis for V to also be an OF -basis ofΩ0, we can realize GLd(OF ) as the compact open subgroup of GLd(V ) preservingΩ0. The cosets GLd(F )/GLd(OF ) correspond to the set of all lattices of V , thecorrespondence given by sending g ∈ GLd(F ) to gΩ0. Now, for our purposes wemay replace Γ by its image in GLd(F ), so that Γ is now a compact subgroup ofGLd(F ). Consider the set

Γ0 = Γ ∩GLd(OF ).

32 4. LECTURE 4

This is open in Γ, and thus has finite index since Γ is profinite and thus compact.Let γ1, . . . , γν be a set of coset representatives for Γ/Γ0. Take

Ω = (γ1 + · · ·+ γν)Ω0.

One can check that Ω is a lattice, and it is clearly stabilized by Γ since Ω0 isstabilized by Γ0. (Use the easily checked fact that given a lattice of rank n ina vector space of dimension n over a local field, adding any vector to the latticeresults again in a lattice of rank n.)

The fact that Γ stabilizes the lattice Ω implies that, by choosing the basis of Fto be an OF -basis of Ω, the action of Γ on GLd(F ) factors through GLd(OF ). Westate the Galois case as a proposition.

Proposition 2.1. Let V be an F -vector space of dimension d, and let

ρ : GK → AutF (V )

be a Galois representation. Then there is a basis of V such that with respect to thisbasis the image of ρ is in GLd(OF ).

2.2. Potentially Semistable Representations. By our presentation of theGalois group T in Section 1.2, we know that a representation of T on V is simplygiven by specifying two invertible F -linear operators τ and ϕ satisfying the singlerelation

ϕτϕ−1 = τp.

By our above reductions we can further assume that τ and ϕ preserve OF .Let v ∈ V be an eigenvector of τ with eigenvalue λ. Set vn = ϕ−nv for all

n ≥ 0. Then using the above relation one easily checks that vn is an eigenvector ofτ with eigenvalue λp

n

. (Each time you “commute” a ϕ−1 with τ the τ is replacedby a τp.) But V is finite dimensional, so τ has only finitely many eigenvalues. Thusfor some m < n,

λpm

= λpn

.

In particular, λ is a (pn−m − 1)pm-root of unity. In fact, we can take the above mto be zero, for we are free to replace λ by λp

m

(by replacing v with vm), in whichcase the above equality becomes

λ = λpn−m

.

Thus λ is a (pν − 1)th root of unity for some ν. Since all of the eigenvaluesof τp

ν−1 are thus 1, we see that τpν−1 becomes unipotent (that is, the sum of

the identity matrix and a nilpotent matrix) over some finite extension of F whichcontains all of its eigenvalues. unipotent; that is, it is the sum of the identity matrixand a nilpotent matrix.

We are now in a position to prove a fundamental result of Grothendieck. Wefirst need a definition.

Definition 1. Assume l 6= p. A Galois representation

ρ : GK → GLd(F )

is said to be semistable if the image of IK is unipotent. ρ is said to be potentiallysemistable if there is some finite extension L/K such that ρ|GL is semistable.

Proposition 2.2 (Grothendieck). Let ρ : GK → GLd(F ) be a Galois repre-sentation. Suppose that F and K have different residue characteristics l and prespectively. Then ρ is potentially semistable.

2. GALOIS REPRESENTATIONS 33

Proof. As before, we can conjugate ρ so as to assume that its image is inGLd(OF ); this is permitted because conjugation does not affect the property ofbeing unipotent. Consider the diagram

0

ker

GK //

$$IIIIIIIII GLd(OF )

GLd(kF )

where kF is the residue field of OF and ker is the kernel of the above map. SinceGLd(kF ) is finite, ker has finite index in GLd(OF ). Thus, since we are allowingfinite extensions, we can replace K by the fixed field of ker and thus assume that

GK → GLd(kF )

is the trivial representation.It follows that ρ actually maps GK into the above kernel. But we know that

this kernel is pro-l (see Lecture 3, Section 2). In particular, since PK ⊆ GK is pro-p,it must vanish under ρ. Thus we have shown that ρ factors through GK/PK = TK .

We can also extend K so that all of the eigenvalues of τ (acting on V ) lie inK. The image of IK in TK is just ∆k, where k is the residue field of OK . ∆k

is (topologically) generated by τe, where τ is a topological generator of ∆Fp ande is the tame inertial degree of K/Qp. We saw above that τn is unipotent for itsaction on V for some n prime to p; let L be a the unique totally tamely ramifiedextension of K of degree n. L is the fixed field of τne and ϕ, and it follows that theimage of IL acting on V is unipotent. Thus ρ|GL is semistable, which completesthe proof.

The above proof rested very heavily on the fact that l was distinct from p; thecase where l = p is very far from settled. Indeed, the above definition of semistableisn’t even appropriate in this case.

2.3. Further Reductions. Suppose for now that l 6= p. We will now usethe methods of the proof of Proposition 2.2 to get additional reductions on ourrepresentations.

Let K/Qp be a finite extension and let k be the residue field of K. We have acommutative diagram with exact rows relating the groups T and TK .

1 // ∆k//

_

TK // _

gk // _

1

1 // ∆Fp // T // g // 1

Here gk is the absolute Galois group of k. The index of ∆k in ∆Fp is the tameinertial degree and the index of gk in g is the residue degree [k : Fp].

34 4. LECTURE 4

We want to consider only the pro-l part of ∆k. More precisely, let Tl,K be thequotient of TK by the maximal non-pro-l subgroup of ∆k. This reduces the exactsequence

1→∏r 6=p

Zr(1)→ TK → Z→ 1

to an exact sequence1→ Zl(1)→ Tl,K → Z→ 1.

The methods of Proposition 2.2 now give the following result.Proposition 2.3. Let ρ : GK → GLd(F ) be a Galois representation with p 6= l.

Then there exists a finite extension L/K such that ρ|GL factors through Tl,L andthe image of the inertia group of L acts unipotently.

Proof. The proof is virtually the same as that of Proposition 1.1.

We make one final comment on reducing the domain of Galois representations.Let ρ : GK → GLd(OF ) be a Galois representation, and consider the residualrepresentation ρ : GK → GLd(kF ). Let G0

K be the kernel of ρ; G0K is the set of

elements of GK such that ρ(g) is in the kernel of the natural map

GLd(OF )→ GLd(kF ).

G0K is an open, normal subgroup of GK of finite index since GLd(kF ) is finite. We

let G0,lK be its pro-l completion; that is, G0,l

K is the projective limit of all finitequotients of G0

K of order a power of l. Since G0K is already profinite, there is a

natural surjection G0K G0,l

K ; let H be the kernel:

1→ H → G0K → G0,l

K → 1.

ρ must vanish on H, since H is completely non-pro-l and ρ maps it into thepro-l kernel of GLd(OF )→ GLd(kF ). Thus ρ factors as a map

GK/H → GLd(OF ).

But GK/H is simply an extension of the pro-l group GL0,l by the finite groupGK/G

0K . In this way one can often gain more control over the Galois representation

ρ.

CHAPTER 5

Lecture 5

1. Group Cohomology

1.1. Basic Facts. Let G be a profinite group and let M be a discrete sub-module of G in the usual sense that

M =⋃

H open subgroup of G

MH .

Equivalently we could take the union over all open normal subgroups. This condi-tion merely means that G acts on each element of M through a finite quotient. Werecall that the cohomology groups of G with coefficients in M are abelian groupsdefined by

Hj(G,M) = lim−→HCG

Hj(G/H,MH)

where the limit is over all open normal subgroups H of G and the maps of the di-rected system are the inflation maps. (Note that open subgroups of G automaticallyhave finite index, so that all of the groups G/H are finite. See [Se-LF, Chapter 7]for the definitions of the standard maps of cohomology groups.) There are variousother equivalent definitions of these cohomology groups; for example,

Hj(G,M) = ExtjZ[G](Z,M) = lim−→

HCG

ExtjZ[G/H](Z,M

H)

where here one can simply consider Ext as it is usually defined for rings. Theequivalence of these definitions stems from the fact that (for profinite G and discreteM ; things are more complicated in general) these Hj(G,M) are the right derivedfunctors of the left exact functor

M 7→MG,

where MG is the submodule of G-invariants of M . That is,

MG = m ∈M | gm = m for all g ∈ G.

In particular, this shows that

H0(G,M) = MG,

which we note is independent of all topological considerations.


35

36 5. LECTURE 5

1.2. The First Cohomology Group. We now turn to the first cohomologygroup, which is the one which we will primarily be concerned with. In this case,inflation is injective (see below), so we can actually realize H1(G,M) as a union

H1(G,M) =⋃HCG

H1(G/H,MH),

the union as usual over open, normal subgroups of G.There are other somewhat more concrete descriptions of H1(G,M) as well. The

most standard is to realize it as equivalence classes of crossed homomorphisms. Wedefine a crossed homomorphism to be a continuous map

h : G→M

such thath(g1g2) = h(g1) + g1h(g2).

We define CrossHom(G,M) to be the set of all (continuous) crossed homomor-phisms from G to M . CrossHom(G,M) is made into an abelian group through theabelian group structure of M . We say that a crossed homomorphism h : G → Mis principal if it has the form

h(g) = gm−mfor some fixed m ∈ M and all g ∈ G. (Such a map is indeed continuous, thanksto our assumption that M is discrete.) It is easy to check that every such map isa crossed homomorphism. We let PrincCrossHom(G,M) denote the subgroup ofCrossHom(G,M) of principal crossed homomorphisms. With these definitions, itcan be shown that

H1(G,M) ∼= CrossHom(G,M)/PrincCrossHom(G,M);

see [Se-LF, Chapter 7, Section 3]. (In fact, if one is working with general G and M(not necessarily profinite and discrete, respectively), one usually takes this as thedefinition of H1(G,M); it is a theorem that in our case this agrees with the derivedfunctor and direct limit definitions.) An element of CrossHom(G,M) is sometimesalso called a 1-cocycle; an element of PrincCrossHom(G,M) is a 1-coboundary.

As an important special case of this, note that if G acts trivially on M , then

CrossHom(G,M) = HomZ(G,M)

andPrincCrossHom(G,M) = 0,

so thatH1(G,M) ∼= HomZ(G,M).

Using this description, we can see directly that inflation is injective. In thissetting, for N closed and normal in G, inflation is induced by the natural map

CrossHom(G/N,MN )→ CrossHom(G,M)

defined by composing crossed homomorphisms with the map G G/N . (It is clearthat PrincCrossHom(G/N,MN ) maps to PrincCrossHom(G,M), so that we get awell-defined map on H1’s.) To show injectivity we must check that if h : G/N →MN is a crossed homomorphism, and if it becomes principal as a map G→M , sayby h(g) = gm−m for some m ∈ M , then we can actually write h(g) = gm0 −m0

for some m0 ∈MN . But this is easy; since h vanishes on N , we must have nm = mfor all n ∈ N . Thus we can take m = m0, so inflation is indeed injective on H1’s.

1. GROUP COHOMOLOGY 37

1.3. The Basic Exact Sequence. We can actually extend our injection

H1(G/N,MN ) inf // H1(G,M)

to a longer exact sequence. We first recall how G/N acts on the cohomology groupH1(N,M). Most concretely, if h : N → M is a crossed homomorphism, we defineγh : N →M for γ ∈ G/N as follows : first choose a lifting γ ∈ G of γ. Now define

γh(x) = γh(γ−1xγ)

for x ∈ N . This makes sense since N is normal, and one can check easily that it isstill a crossed homomorphism. Furthermore, it is not hard to show that a differentchoice of γ gives a crossed homomorphism which differs from γh by a principalcrossed homomorphism, so that we really do get a well-defined action of G/N onH1(N,M).

There is another way to define this action which is more abstract but somewhatcleaner. Recall that Hj(G,M) is contravariant in G and covariant in M . For anyγ ∈ G, we can define maps

N ← N, M →M

by sending n ∈ N to γ−1nγ ∈ N and m ∈ M to γm ∈ M . These maps arecompatible in the sense of [Se-LF, Chapter 7, Section 5] (this just means that themap M →M is a map of N -modules with the normal G-action on the second factorand the twisted action on the first factor) so that we get induced homomorphisms

Hj(N,M)γ→ Hj(N,M)

for all j. This defines our G-action. However, one can check easily that the actionof any γ ∈ N is in fact trivial; this is done by looking at the 0th level and theneither using the universal property of derived functors or else a dimension shiftingargument. (See [Se-LF, Chapter 7, Section 5, Proposition 3].) Thus we get anaction of G/N on all of the Hj(N,M).

We are now in a position to state the extension of our above injection. In it’ssimplest form it is an exact sequence

0 −−−−→ H1(G/N,MN ) inf−−−−→ H1(G,M) res−−−−→ H1(N,M)G/N .

This can be proved directly, working with crossed homomorphisms; see [Se-LF,Chapter 7, Section 6]. We can get a more general result by using the full Hochschild-Serre spectral sequence: for N a closed normal subgroup of G and M a discreteG-module this is a second stage, first quadrant, cohomological spectral sequence

Epq2 = Hp(G/N,Hq(N,M))⇒ Hp+q(G,M).

The exact sequence of low degree terms takes the form

0 −−−−→ H1(G/N ;MN ) inf−−−−→ H1(G;M) res−−−−→ H1(N ;M)G/N

−−−−→ H2(G/N ;MN ) inf−−−−→ H2(G;M).

For the construction of the Hochschild-Serre spectral sequence, see [Wei, Chapter5]; for a proof of the above exact sequence and some generalizations, see [Wes-IR].

38 5. LECTURE 5

1.4. Cup Products. We briefly recall the notion of cup products; for a state-ment of the main properties, see [Se-LF, Chapter 8, Section 3]. For proofs of theseproperties, see [AW, Section 7].

Let M1 and M2 be A-modules, where A is some commutative ring, which arealso discrete G-modules for some profinite group G. We further suppose that theaction of G is A-linear; this is all equivalent to saying that M1 and M2 are A[G]-modules rather than just Z[G]-modules. We endow the tensor product M1 ⊗AM2

with a structure of G-module by setting g(m1 ⊗m2) = gm1 ⊗ gm2. It is also, ofcourse, still an A-module. The cup product is then a pairing

Hi(G,M1)⊗A Hj(G,M2)→ Hi+j(G,M1 ⊗AM2).

We will usually write m1 ∪m2 for the cup product of m1 ∈ Hi(G,M1) and m2 ∈Hj(G,M2).

Often one takes cup products in a situation where there is also a natural map

M1 ⊗AM2 →M3

for some A[G]-module M3, in which case we can consider the natural map

Hi(G,M1)⊗A Hj(G,M2)→ Hi+j(G,M3)

induced by cup product and the above map.

2. Galois Cohomology

2.1. Torsors for Finite GK-modules. For this section we fix a field K andlet GK = Gal(Ks/K) be its absolute Galois group. We further assume that M isa finite GK-module. (M is also assumed to be discrete and the action continuous,of course.) We will refer to finite sets on which GK acts as GK-sets. What we saybelow is actually true for general groups, but we will confine ourself to the case ofGK .

Definition 2. An M -torsor is a GK-set T together with a GK-equivariantprincipal homogeneous action of M ,

α : M × T → T.

We will write α(m, t) as m + t, and by GK-equivariance we mean that g(m +t) = gm + gt. Recall that a principal homogeneous action is one which is simplytransitive; that is, for any fixed t0 ∈ T , the map

M → T

given by sending m to m+ t0 is a bijection.Thus an M -torsor is a GK-set which looks exactly like M , except that it isn’t

itself an abelian group; by choosing a t0 to act as the identity it can be made intoone. So an M -torsor is a “copy” of M without a distinguished identity element.

There are other ways to describe M -torsors which can be somewhat more illu-minating. For each t0 ∈ T we get a commutative diagram

M × T α // T

M ×M //

∼=

OO

M

∼=

OO

2. GALOIS COHOMOLOGY 39

where the bottom map is just the group law on M , the first vertical isomorphismsends (m1,m2) to (m1,m2 + t0) and the second vertical isomorphism sends m tom+ t0. This diagram expresses both the fact that α is a group action and that itis principal homogeneous.

Combining all of these diagrams over T , we get a diagram

(M × T )× T α×1// (T )× T

(M ×M)× T //

∼=

OO

(M)× T

∼=

OO

where all of the maps on the last T -factor are the identity, and the vertical mapsnow send (m1,m2, t) to (m1,m2 + t, t) and (m, t) to (m+ t, t).

Considering the last T factor as a sort of parameter space for the other commu-tative diagrams, it might make somewhat more sense to write the previous diagramas

(M × T )Tα // TT

(M ×M)T //

∼=

OO

MT

∼=

OO

2.2. Connetions between Torsors and H1(G,M). We first recall that itis possible to put a group structure on the set of M -torsors. Let T1 and T2 betwo M -torsors. Consider the finite set T1×T2 with its natural product GK-action.M ×M acts naturally on T1×T2, and we define an equivalence relation on T1×T2

by decreeing that(t1, t2) ∼ (t1 +m, t2 −m)

for all m ∈M . We define the Baer sum T1 T2 of T1 and T2 to be the quotient ofT1 ⊗ T2 by this equivalence relation. One can easily check that T1 T2 is actuallyan M -torsor, with its M -action defined by having M act on the first factor but notthe second.

We are now in a position to state our main theorem on torsors. We say thattwo M -torsors T1 and T2 are isomorphic as M -torsors if there is na isomorphism ofGK-sets between them commuting with the action of M .

Theorem 2.1. H1(GK ,M) is naturally isomorphic to the set of M -torsors upto isomorphism of M -torsors. It is in fact an isomorphism of groups where theM -torsors are made into a group by the Baer sum.

Proof. This is actually fairly easy. One uses the identification

H1(GK ,M) = lim−→HCG

Ext1Z[GK/H](Z,M

H),

together with the description of this group as isomorphism classes of extensions

0→M → ε→ Z→ 0.

Given such an extension, the corresponding torsor is simply the inverse image of1 ∈ Z in ε. We leave the details to the reader.

40 5. LECTURE 5

Recall that the category of GK-sets is equivalent to the category of etale alge-bras over K. (An etale algebra is one which is flat and unramified; see [Mi-EC,Chapter 1, Section 3, p. 21] for the definition of an unramified algebra.) In thecase that GK acts trivially on M , we have that

H1(GK ,M) ∼= Homcts(GK ,M)

classifies etale algebras over K with M -action. If the homomorphism GK → M isactually surjective, then it corresponds to a field extension L/K with Galois groupM .

2.3. Torsors for Algebraic Groups. There is a similar theory of torsors foralgebraic groups. We replace M by an algebraic group Γ defined over K (recallthat an algebraic group is an algebraic variety with a group structure such thatthe multiplication and inversion maps are actually morphisms of algebraic varietiesand the identity is K-rational), and for simplicity we assume that Γ is smooth asan algebraic variety over K. With appropriate definitions everything we say canbe generalized to the case where Γ is non-commutative, but we will concentrate onthe commutative case. (Note that if Γ is non-commutative we haven’t even definedthe cohomology set (it isn’t a group in this generality) H1(GK ,Γ(Ks)).)

In this setting, a Γ-torsor T is an algebraic variety over K together with amorphism

Γ× T α−→ T

satisfying the same conditions as our map α above. Specifically, we have a diagram

(Γ× T )× T α×1// (T )× T

(Γ× Γ)× T //

u ∼=

OO

(Γ)× T

v∼=

OO

where the bottom map is the multiplication morphism on Γ and the identity on T ,

u = id×α× id

(where α operates on the last two factors Γ× T ) and

v = α× id .

On points,u(γ1, γ2, t) = (γ1, γ2 + t, t)

andv(γ, t) = (γ + t, t).

It follows from this definition that T is isomorphic to K over K (it might not beisomorphic over K since it might not even have any points defined over K), so Tis necessarily smooth. Also, we can define a Baer sum in an analagous way to thatused previously.

We have in this setting the following analogue of Theorem 2.1.Theorem 2.2 (Grothendieck). If Γ is a smooth algebraic group over K, then

H1(GK ,Γ(Ks)) is naturally isomorphic to the group of Γ-torsors (defined over K)up to isomorphism.

2. GALOIS COHOMOLOGY 41

Proof. The proof of this theorem is quite difficult, in contrast to the fairlyimmediate proof of Theorem 2.1. The fact that Γ is smooth is absolutely essential; ifΓ isn’t smooth, then we must allow flat coverings rather than merely etale coverings.For a proof in the much easier case that Γ is an elliptic curve, see [Si-AEC, Chapter10, Theorem 3.6].

2.4. The Local Invariant. We now consider the algebraic group Gm over alocal field K (in our usual sense). More accurately, we will fix an algebraic closureK of K and consider the attached GK-module K∗. We will denote its torsionsubgroup by µ(K); it is just the group of roots of unity of K∗.

In this setting we have the following fundamental theorem of local class fieldtheory.

Theorem 2.3. Let K be a local field. Then there is a canonical isomorphisminv, called the invariant map,

H2(GK , µ(K∗))∼=−→ Q/Z.

Proof. See [Se-LF, Chapter 13, Section 3]. This theorem is often stated as

H2(GK , K∗)∼=−→ Q/Z,

but either form can easily be obtained from the other.

In this section we will use our knowledge of the structure of GK to compute thel-adic component of H2(GK , µ(K∗)); that is, we will show that H2(GK , µl∞(K∗))is canonically isomorphic to Ql/Zl, at least in the case l 6= p, where p is thecharacteristic of the residue field of K. Here µl∞(K∗) is the group of l-power rootsof unity in K∗.

Our main tools will be the Hochschild-Serre spectral sequence and the followinglemma.

Lemma 2.4. Let G be a pro-p group and let M be a discrete G-module in whicheach element is killed by a power of l. If p 6= l, then for all j ≥ 1,

Hj(G,M) = 0.

Proof. Recall that

Hj(G,M) = lim−→HCG

Hj(G/H,MH),

the direct limit being over the open, normal subgroups of G. Choose an elementx ∈ Hj(G/H,MH) and let c : (G/H)j → MH be a j-cocyle representing x. (See[Se-LF, Chapter 7, Section 3].) Since G/H is finite, c takes on only finitely manyvalues in MH . Thus there must be some n ≥ 0 such that lnc = 0, and thereforelnx = 0. But G/H has order pm for some m, so pmx = 0 by [Se-LF, Chapter 8,Section 2, Corollary 1]. Since l 6= p, we thus must have x = 0. So every group inthe direct limit is 0, and Hj(G,M) = 0 as well.

We will use various exact sequences from Lecture 4, Sections 1.3 and 2.3. First,we have an exact sequence

1→ PK → GK → TK → 1

42 5. LECTURE 5

where PK is a p-group. The Hochschild-Serre spectral sequence for this exactsequence collapses to the bottom row by Lemma 2.4, since PK is pro-p. Thus weget an isomorphism

H2(GK , µl∞(K∗)) ∼= H2(TK , µl∞(K∗)).

Next we recall that we have an exact sequence

1→∏r 6=p,l

Zr(1)→ TK → Tl,K → 1.

Since∏r 6=p,l Zr(1) has no pro-l component, the same argument as above shows that

H2(TK , µl∞(K∗)) ∼= H2(Tl,K , µl∞(K∗)),

and thus thatH2(GK , µl∞(K∗)) ∼= H2(Tl,K , µl∞(K∗)).

Next we use the exact sequence

1→ Zl(1)→ Tl,K → Z→ 1.

This time the Hochschild-Serre spectral sequence is entirely zero outside of thebottom left 2× 2 square, since Z and Zl(1) cohomology vanish on torsion modulesin all dimensions greater than or equal to 2. (See [Se-LF, Chapter 13, Section 1,Proposition 2].) From this we get an isomorphism

H2(GK , µl∞(K∗)) ∼= H1(Z,H1(Zl(1), µl∞(K∗))

).

Since l 6= p, µl∞(K) ⊆ Kur, so Zl(1) (which at the moment lies in the inertiagroup) acts trivially on it. Thus,

H1(Zl(1), µl∞(K∗)) = Hom(Zl(1), µl∞(K∗)) = Hom(Zl,Ql/Zl) ∼= Ql/Zl,

since as abelian groups Zl(1) is just Zl and µl∞(K∗) is just Ql/Zl. (All Hom-groupsare continuous homomorphisms, of course.) In fact, H1(Zl(1), µl∞(K∗)) even hastrivial Z = Gal(Kur/K)-action, since Z acts via the l-cyclotomic character on eachand the actions cancel. Thus,

H1(Z,H1(Zl(1), µl∞(K∗))

)= Hom(Z,Ql/Zl) ∼= Ql/Zl.

Thus, as promised, we have shown that

H2(GK , µl∞(K∗)) ∼= Ql/Zl.

Of course, this is not yet the full determination of the invariant map. Most impor-tantly, we have not computed the p-part; our methods above would be completelyineffective in that case.

3. Tate Local Duality

Let K be a local field and let M be a finite discrete GK-module. We define thePontrjagin dual M∨ of M by

M∨ = Hom(M,Q/Z),

with the usual adjoint GK-action. We further define the Cartier dual M∗ by

M∗ = Hom(M,µ(K∗)),

again with the adjoint GK-action. M∨ and M∗ are isomorphic as abelian groups,but not as GK-modules, and in both cases the double dual is simply M .

3. TATE LOCAL DUALITY 43

Pontrjagin duality is simply the obvious assertion that there is a natural, GK-equivariant, perfect pairing

M ⊗M∨ → Q/Z.

Similarly, Cartier duality is the assertion that there is a natural GK-equivariantperfect pairing

M ⊗M∗ → µ(K∗).Tate local duality is essentially the extension of Cartier duality to cohomologygroups.

Theorem 3.1 (Tate Local Duality). Let K and M be as above. Then, for alli, Hi(GK ,M) is finite, and there are perfect pairings

Hi(GK ,M)⊗H2−i(GK ,M∗)→ Q/Z

induced by cup product, Cartier duality and the invariant map.

Proof. See [Mi-ADT, Chapter 1, Section 2].

CHAPTER 6

Lecture 6

1. Duality Preliminaries

1.1. Pontrjagin Duality. As always, let K be a local field with residue fieldk of characteristic p. Let GK and gk be the absolute Galois groups Gal(K/K) andGal(k/k) respectively. We let M be a finite GK-module (where the GK-action iscontinuous with respect to the discrete topology on M , of course).

We define two duals of M . The first is the Pontrjagin dual

M∨ = HomZ(M,Q/Z),

the Hom being as abelian groups. M∨ receives the usual adjoint GK-action; in thiscase, since GK acts trivially on Q/Z, this means that for any f : M → Q/Z andσ ∈ GK , σf : M → Q/Z is given by

(σf)(m) = f(σ−1m)

for all m ∈M . There is a natural perfect pairing of abelian groups

M ⊗ZM∨ → Q/Z,

given bym⊗ f 7→ f(m)

for m ∈ M and f ∈ M∨. (Recall that for a pairing M ⊗A N → P to be a perfectpairing of A-modules means that the induced maps

M → HomA(N,P )

andN → HomA(M,P )

are isomorphisms.) The pairing is perfect in this case because M is a finite abeliangroup, and thus (loosely speaking) every element of M can find something in Q/Zto map to. In fact, if M ⊗M∨ is given the usual induced GK-action, one easilychecks that this pairing, and thus the induced maps as well, are GK-equivariant.Note also that (M∨)∨ is canonically isomorphic to M as GK-modules.

Now, let H be an arbitrary subgroup of GK . Then M is an H-module in the ob-vious way, and one easily checks directly from the definitions that (MH)∨ = (M∨)Hand (MH)∨ = (M∨)H , where MH and MH are the H-invariants and coinvariantsrespectively. (These two equalities are actually formally equivalent; simply replaceM by M∨ in either to get the other one.) Put differently, our above perfect pairing“restricts” to perfect pairings

MH ⊗ (M∨)H → Q/Z


45

46 6. LECTURE 6

and

MH ⊗ (M∨)H → Q/Z.

1.2. Cartier Duality. The Cartier dual of M is a GK-module M∗ which isisomorphic to M∨ as abelian groups, but with a different GK-action. Precisely, wedefine

M∗ = HomZ(M,µ(K)),

where µ(K) is the group of roots of unity in K. The fact that M∨ ∼= M∗ (non-canonically) as abelian groups follows from the fact that there are non-canonicalisomorphims Q/Z ∼= µ(K). The difference is that µ(K) is a non-trivial GK-module,so the induced adjoint GK-action on M∗ is different from that on M∨: for anyf : M → µ(K) and σ ∈ GK , we have

(σf)(m) = σf(σ−1m)

for all m ∈ M , the outside action being that of GK on µ(K). We still have(M∗)∗ = M as GK-modules.

Just as in the Pontrjagin case, we have a canonical perfect pairing

M ⊗ZM∗ → µ(K)

given by evaluation. It is again easily checked to be GK-equivariant. In this case,however, the pairing does not always restrict to subgroups. Specifically, let H beany subgroup of GK . Then the direct argument from the definitions shows onlythat (MH)∗ = (M∗)H and (MH)∗ = (M∗)H if H acts trivially on µ(K). However,with a little more work we can recover these results in the following special case.

Lemma 1.1. Suppose that M is a finite GK-module of prime to p order, wherep is the residue characteristic of k. Then

(MIK )∗ = (M∗)IK

and

(MIK )∗ = (M∗)IK ,

where IK is the inertia subgroup of GK .

Proof. Let µ′(K) be the subgroup of µ(K) of roots of unity of prime to porder. Using the decomposition

µ(K) = µ′(K)× µp∞(K)

we get a decomposition

M∗ = Hom(M,µ(K)) = Hom(M,µ′(K))×Hom(M,µp∞(K)),

and the last of these groups is zero, since M has prime to p order. Thus

M∗ = Hom(M,µ′(K)).

Now, since K(µ′(K)) is unramified over K, IK acts trivially on µ′(K). The usualargument now extends to this case to finish the proof.

1. DUALITY PRELIMINARIES 47

1.3. Tate Modules. In order to clarify the relationship between M∨ and M∗

we will now present formally a subject which we have touched on before. Let l bea prime number (possibly equal to p), and let W be a l-divisible, l-torsion abeliangroup. Thus

W =⋃n

W [ln],

where W [ln] is the ln-torsion in W . We define a functor Tal from the category ofsuch groups to the category of Zl-modules by defining

Tal(W ) = lim←−n

W [ln],

the inverse limit being with respect to the maps

W [ln+1] l−→W [ln]

given by multiplication by l. These are surjective since W is l-divisible. Tal(W )is a Zl-module since each W [ln] is a Z/lnZ-module. In fact, if W [l] is finite, thenit follows easily that Tal(W ) is a free Zl-module of finite rank equal to |W [l]|/l.(The relevant structure of W is actually determined by W [l], as can be seen justby staring at it for a moment.) It is also easy to see that if W is a GK-module,then Tal(W ) is also a GK-module. However, it is not true that Tal(W ) need bediscrete, even if W was.

The canonical example is, of course,

Tal(Ql/Zl) = Zl;

it follows immediately that

Tal((Ql/Zl)r

)= Z

rl ,

the free Zl-module of rank r. Since µl∞(K) is isomorphic to Ql/Zl as abeliangroups, Tal(µl∞(K)) must be a free Zl-module of rank 1. In fact, one checks easily(since all of the GK-actions involved essentially come from µl∞(K)) that

Tal(µl∞(K)) ∼= Zl(1).

(See Lecture 3, Section 3.2.) In the case l = p we use this to define the moduleZp(1), which we had not previously defined.

If M is any pro-l GK-module, we define its (first) Tate twist by

M(1) = M ⊗Zl Zl(1).

More generally, we define inductively

M(n) = M(n− 1)⊗Zl Zl(1)

for any positive integer n. We also define

Zl(−1) = HomZl(Zl(1),Zl),

where the Zl in the range has trivial GK-action and correspondingly

M(−1) = M ⊗Zl Zl(−1) = HomZl(Zl(1),M)

andM(n) = M(n+ 1)⊗Zl Zl(−1)

for any negative integer n. One checks immediately that

Zl(n1)⊗Zl Zl(n2) = Zl(n1 + n2)

48 6. LECTURE 6

for any integers n1 and n2.This entire theory generalizes to the case where one considers all primes l at

once. Specifically, we define a functor Ta from torsion, divisible groups to Z-modules(which are just profinite abelian groups) by

Ta(W ) = lim←−N

W [N ],

where the maps of the inverse system are the multiplication by m maps

W [mN ] m−→W [N ];

these are surjective since W is divisible. As above, Ta(W ) inherits a GK-actionfrom W if W is a GK-module.

As an example, we findTa(Q/Z) = Z

canonically, andTa(µ(K)) =

∏l prime

Zl(1)

which we will denote simply Z(1). Suppressing the p-part, we find that we haveessentially recovered the Galois group ∆k.

If M is any profinite GK-module, we define its (first) Tate twist to be

M(1) = M ⊗ZZ(1).

Also definingM(−1) = Hom

Z(Z(1),M),

we proceed as above to get GK-modules M(n) for all integers n. In the case thatM is pro-l it is easy to check that these definitions agree with our earlier ones.

It is immediate from these definitions that if M is any finite GK-module, thePontrjagin and Cartier duals are related by

M∗ = M∨(1).

1.4. Cohomology of Topologically Cyclic Profinite Groups. Let C bea profinite group which is (not necessarily canonically) isomorphic to∏

l∈L

Zl

for some set of prime numbers L. Let M be a finite discrete C-module, withcontinuous C-action. If C were actually canonically isomorphic to the product ofZl’s, meaning that it has a chosen generator γ, then there is a canonical isomorphism

H1(C,M) = MC = M/(γ − 1)M,

given by sending a cocycle ϕ : C → M to ϕ(γ). (This adapts easily from [Se-LF,Chapter 13, Section 1].) In the case where C has no chosen generator, we can stillobtain a canonical isomorphism via the map

C ⊗ZH1(C,M)→MC

given by sending a pair c⊗ ϕ of an element of C and a cocycle to the class of ϕ(c)in MC . That this is an isomorphism follows easily either from the fact that thenon-canonical version is, or else it can also be seen easily from the finite cyclic case.


One can actually “solve” for H1(C,M) in the above equation in the case whereM has order divisible only by primes in L. Simply take the tensor product of bothsides with Hom

Z(C,∏l∈L Zl). This yields an isomorphism

H1(C,M) = HomZ(C,MC),

since HomZ(C,∏l∈L Zl)⊗Z C =

∏l∈L Zl.

2. Tate Local Duality

2.1. An Example : Local Class Field Theory. We keep the notation ofthe last section. Recall that Tate local duality states that the cohomology groupsH1(GK ,M) and H1(GK ,M∗) are finite, and that there is a perfect pairing

H1(GK ,M)⊗H1(GK ,M∗)→ Q/Z

given by cup product, Cartier duality and the local invariant map. (See Lecture 5,Section 3.)

We begin by working through the implications in an example. Let N be apositive integer and set M = Z/NZ, with trivial GK-action. Then

M∗ = Hom(Z/NZ, µ(K)) = Hom(Z/NZ, µN (K)) = µN (K),

where µN (K) is the group of N th roots of unity in K. Thus in this case Tate localduality gives a perfect pairing

H1(GK ,Z/NZ)⊗H1(GK , µN (K))→ Q/Z;

in fact, the image must land in 1NZ/Z, since both groups are N -torsion. Since

Z/NZ has trivial GK-action, the first of these cohomology groups is given by

H1(GK ,Z/NZ) = Hom(GK ,Z/NZ) = Hom(GabK ,Z/NZ).

We can compute H1(GK , µN (K)) using Kummer theory. We begin with theexact sequence of GK-modules

0→ µN (K)→ K∗N→ K∗ → 0.

The long exact sequence of cohomology gives us an exact sequence

0→ K∗/(K∗)N → H1(GK , µN (K))→ H1(GK , K∗)[N ]→ 0.

But H1(GK , K∗) = 0 by Hilbert’s Theorem 90. (See [Se-LF, Chapter 10, Section1, Proposition 2]. Alternatively, if one thinks of H1(GK , K∗) as classifying isomor-phism classes of one-dimensional K-vector spaces which become isomorphic overK, the result is clear, since isomorphism classes of vector spaces are completelydetermined by dimension.) Thus

K∗/(K∗)N ∼−→ H1(GK , µN (K∗)).

Combining these results, we have a perfect pairing

Hom(GabK ,Z/NZ)⊗K∗/(K∗)N → Q/Z.

Equivalently, we have an isomorphism

Hom(GabK ,Z/NZ) ∼= Hom(K∗/(K∗)N ,Q/Z).

Taking the direct limit over N yields

Hom(GabK ,Q/Z) ∼= Hom(K∗,Q/Z),

50 6. LECTURE 6

where K∗ is the profinite completion of K∗. (K∗ is not already complete; it isisomorphic to U × Z, where U is the group of units. U is complete, but Z is not.)Taking Pontrjagin duals finally yields the isomorphism

GabK∼= K∗.

Thus we have obtained the main isomorphism of local class field theory fromTate local duality. However, there are two flaws with this approach. First, localclass field theory is an essential ingredient in most proofs of Tate local duality, sothe argument is circular. (In fact, in spirit a good deal of the content of the proofof Tate local duality consists in compiling the known reciprocity isomorphisms oflocal class field theory.) Second, it would require a lot of effort to relate the abovecomputation to the usual reciprocity isomorphism, which is a key ingredient of thetheory.

2.2. Restrictions of Tate Local Duality : First Attempt. We now maketwo assumptions on the finite GK-module M . First, we assume that it is of prime top order. Second, we assume that the GK-action on M is at worst tamely ramified;that is, we assume that the wild ramification group PK acts trivially on M . ThusGK acts on M through GK/PK = TK , in our usual notation.

We will refer to the inflation-restriction exact sequence

0→ H1(gk,MIK )→ H1(GK ,M)→ H1(IK ,M)gk → 0

as the basic exact sequence. (It is exact on the right since H2(gk,MIK ) = 0, asM is torsion.) We are interested in relating this exact sequence with Tate localduality.

Recall that we have an exact sequence

1→ ∆k → TK → gk → 1,

where ∆k is the image of IK in TK . We claim that we have an isomorphism

H1(∆k,M) ∼= H1(IK ,M).

To see this, we simply write down a portion of the inflation-restriction sequenceassociated to the exact sequence

1→ PK → IK → ∆k → 1;

specifically, consider

0→ H1(∆k,MPK )→ H1(IK ,M)→ H1(PK ,M).

By Lecture 5, Lemma 2.4, H1(PK ,M) = 0 since PK is pro-p and M has primeto p order. Since by hypothesis PK acts trivially on M , we obtain the desiredisomorphism.

Thus we can rewrite our basic exact sequence as

0→ H1(gk,M∆k)→ H1(GK ,M)→ H1(∆k,M)gk → 0.

We now use the results of Section 1.4 to compute the outside terms. Since gk ∼= Z

canonically, we have a canonical isomorphism

H1(gk,M∆k) ∼= (M∆k)gk .


Since ∆k∼=∏l 6=p Zl(1) and M has prime to p order, we have a canonical isomor-

phism

H1(∆k,M)gk ∼= Hom(∆k,M∆k)gk ∼= Hom

∏l 6=p

Zl(1),M∆k

gk

∼=(M∆k

(−1))gk .

Our basic exact sequence now takes the form

0→ (M∆k)gk → H1(GK ,M)→(M∆k

(−1))gk → 0.

We now rewrite the Pontrjagin dual of our basic exact sequence together withthe basic exact sequence for M∗.

0 //((M∆k

(−1))gk)∨ // H1(GK ,M)∨ //

((M∆k)gk

)∨ // 0

0 //((M∗)∆k

)gk

// H1(GK ,M∗) //

∼=

OO

((M∗)∆k

(−1))gk // 0

The vertical isomorphism is induced by Tate duality. Now, we claim that the othertwo vertical pairs of groups are isomorphic. For example,(

(M∗)∆k)gk

=((M∨(1)

)∆k)

gk

=((

(M(−1))∨)∆k

)gk

=((M(−1)∆k

)∨)gk

=((M(−1)∆k

)gk)∨=((M∆k

(−1))gk)∨.

Here we have used some easy isomorphisms together with some already mentionedin Section 1.1. This shows that the first two terms are isomorphic, and the last twoare done in a similar way.

One should note, however, that there is absolutely no apparent reason whythese isomorphisms should make the above diagram commutative. We will statewhat will be useful to us as a proposition.

Proposition 2.1. With K, k and M as above, the groups H1(gk,MIK ) andH1(IK ,M∗)gk have the same order. Similarly, H1(IK ,M)gk and H1(gk, (M∗)IK )have the same order.

Proof. This follows immediately from the isomorphisms constructed above,after translating back into the language of cohomology. Note that the fact thatthese groups are all finite follows from the statement of Tate duality.

2.3. Restrictions of Tate Duality : Second Attempt. We keep the sameassumptions on M . We can now use the results we obtained above to relate Tatelocal duality to our basic exact sequences


and0→ H1(gk, (M∗)IK )→ H1(GK ,M∗)→ H1(IK ,M∗)gk → 0.

52 6. LECTURE 6

We claim that under the Tate pairing

H1(GK ,M)⊗H1(GK ,M∗)→ Q/Z,

H1(gk,MIK ) and H1(gk, (M∗)IK ) are orthogonal. To see this consider the com-mutative diagram

H1(gk,MIK )⊗H1(gk, (M∗)IK )

inf ⊗ inf// H1(GK ,M)⊗H1(GK ,M∗)

H2(gk,MIK ⊗ (M∗)IK )

H2(GK ,M ⊗M∗)

H2(gk, µ(k))inf // H2(GK , µ(K))

Q/Z

where the first vertical map in the second column is induced by the inclusionsMIK →M and (M∗)IK →M∗ and the rest of that column gives the Tate pairing.Note that since M has prime to p order we really do only need to consider µ(k).But H2(gk, µ(k)) = 0, so for H1(gk,MIK ) and H1(gk, (M∗)IK ) the pairing factorsthrough 0; thus they are orthogonal under the Tate pairing, as claimed.

This orthogonality together with Proposition 2.1 implies the following funda-mental theorem.

Theorem 2.2. Let K be a local field with residue field k. Let M be a finitetamely ramified GK-module of prime to p order. Then under Tate local duality,H1(gk,M) and H1(IK ,M∗)gk are orthogonal complements. Similarly, H1(IK ,M)gk

and H1(gk,M∗) are orthogonal complements. Thus Tate local duality induces per-fect pairings

H1(gk,M)⊗H1(IK ,M∗)gk → Q/Z

andH1(IK ,M)gk ⊗H1(gk,M∗)→ Q/Z.

In particular, we have an exact commutative diagram

0 //(H1(IK ,M)gk

)∨ //(H1(GK ,M)

)∨ //(H1(gk,M)

)∨ // 0

0 // H1(gk,M∗) //

∼=

OO

H1(GK ,M∗) //

∼=

OO

H1(IK ,M∗)gk //

∼=

OO

0

Proof. The injectivity of the first vertical maps and the surjectivity of thesecond vertical map follow from the the orthogonality constructions above. Thefact that they are isomorphisms then follows from Proposition 2.1. The rest of thetheorem is just a restatement of the diagram.

CHAPTER 7

Lecture 7

1. Finite/Singular Structures

1.1. Local Structures. Let K be a local field with residue field k of charac-teristic p, and let M be a finite GK-module of prime to p order. In any case wehave exact sequences


and0→ H1(gk, (M∗)IK )→ H1(GK ,M∗)→ H1(IK ,M∗)gk → 0.

If in addition M is only tamely ramified, then Theorem 2.2 of Lecture 6 applies.In this case we define the finite and singular parts of H1(GK ,M) by

H1f (GK ,M) = H1(gk,MIK )

andH1s (GK ,M) = H1(IK ,M)gk .

The definition also applies to M∗. We will refer to these choices as the standardfinite/singular structure on M . (We could of course make these definitions withoutany conditions on M , but they would not in general have the properties that wewant.) Thus there is an exact commutative diagram

0 // H1s (GK ,M)∨ // H1(GK ,M)∨ // H1

f (GK ,M)∨ // 0

0 // H1f (GK ,M∗) //

∼=

OO

H1(GK ,M∗) //

∼=

OO

H1s (GK ,M∗) //

∼=

OO

0

with the vertical isomorphisms coming from Tate local duality. Put differently,Tate duality identifies H1

f (GK ,M) and H1s (GK ,M∗) as duals; it also identifies

H1s (GK ,M) and H1

f (GK ,M∗) as duals.Note that in the case that M is unramified as a GK-module (meaning that IK

acts trivially on M), we have the slightly simpler definitions

H1f (GK ,M) = H1(gk,M)

andH1s (GK ,M) = Hom(IK ,M)gk .

We will now briefly describe an interpretation of H1f (GK ,M) that helps to

explain the terminology “finite”. We do this using the interpretation of H1(GK ,M)in terms of torsors. (See Lecture 5, Section 2.) So let T be an M -torsor. Inparticular, T is a finite GK-set. There is an equivalence of categories between the


53

54 7. LECTURE 7

Figure 1. Extending finite etale schemes over SpecK

category of finite GK-sets and the category of finite, etale schemes over SpecK.(See [Mi-EC, Chapter 1, Section 5, Theorem 5.3].) Let T be the finite, etale schemeover SpecK associated with T .

Let OK be the ring of integers of K. SpecOK consists of the generic pointSpecK and the closed point Spec k. A natural question is when the SpecK schemeT can be extended to a finite, etale scheme T ′ over SpecOK . It is not difficult to seethat any finite, etale T over SpecK can be extended to a quasi-finite, etale schemeover SpecOK ; the difficult part is getting it to be finite and etale . It turns outthat, at least in the case where M is unramified, T ′ can be taken to be finite if andonly if the cohomology class associated to T lies in H1

f (GK ,M). Thus the “finite”terminology is appropriate, at least in this case.

1.2. Global Structures. We now consider a global field F . For us we willsimply take global field to mean number field; that is, a finite extension of Q. Wewill always consider F to come along with a choice of algebraic closure F , and we setGF = Gal(F /F ) as usual. For each place v of F (archimedean or non-archimedean)we let Fv be the completion of F at v; we fix an algebraic closure Fv of Fv. Wealso choose an embedding of F into Fv; this is equivalent to choosing a place of Fabove v, which in turn is equivalent to choosing a decomposition group of v in GF .We will write GFv or just Gv for the Galois group Gal(Fv/Fv); this is (because ofour choices) identified with the decomposition group of v in GF . In the case wherev is non-archimedean, we will write gv for the absolute Galois group of the residuefield of Fv.

Fv

GvF

GF

Fv

F

Let M be a finite GF -module, with the GF -action continuous for the discretetopology on M . Note that for every place v, M is also a Gv-module by restriction.Thus our choice of embedding

Gv → GF

induces maps on cohomology

Hi(GF ,M) resv−→ Hi(Gv,M)

for all i ≥ 0. Here a remarkable thing happens: the maps resv are independent ofour choice of embedding Gv → GF . This follows from the fact that Gv is deter-mined as a subgroup of GF up to conjugation by elements of GF ; since conjugationinduces the identity automorphism on cohomology ([Se-LF, Chapter 7, Section 5,Proposition 3]), resv is thus independent of the choice of Gv in GF .

1. FINITE/SINGULAR STRUCTURES 55

We want to impose local structures on the first global cohomology group ofM , H1(GF ,M); we essentially want to use our finite/singular structures definedpreviously. However, we have only defined these local structures in the case that vis non-archimedean, M is at worst tamely ramified as a Gv-module, and the residuecharacteristic of Fv does not divide the order of M . (From now on we will say thatthe non-archimedean place v divides the order of M if its residue characteristicdoes.) We do not yet want to impose any particular structure at the other placesof F , so we make the following definition.

Definition 3. A structured GF -module is a finite, discrete GF -module Mtogether with a choice of finite/singular structure at each place v of F ; by this wesimply mean a choice of an exact sequence

0→ H1f (Gv,M)→ H1(Gv,M)→ H1

s (Gv,M)→ 0

for each v. (That is, simply pick either a subgroup or a quotient group of H1(Gv,M)and define the other group via the exact sequence.) We further require that thereis a finite set S of places of F such that for all v /∈ S,

H1f (Gv,M) = H1(gv,MIv )

andH1s (Gv,M) = H1(Iv,M)gv .

That is, we require that our choice of finite/singular structure agree with thatdefined previously, at least for v /∈ S.

We will always take S to contain all archimedean places, all places where Mhas wild ramification, and all places dividing the order of M , since we have not evendefined finite/singular structures in these cases. It is easy to check that since M isfinite and discrete there are only finitely many such places, so the definition makessense. We can in fact improve this statement slightly; M is actually unramified asa Gv-module for all but finitely many v.

In order to exploit Tate local duality we want to be somewhat careful as tohow the structure on M relates to that on its Cartier dual M∗. In order to do thiswe must carefully consider the archimedean places for the first time. In fact, theybehave much like the non-archimedean places. Since the case where v is complexis trivial (everything vanishes), the next proposition is all that we will need.

Proposition 1.1. Let M be a finite G = Gal(C/R)-module. Then the pairing

H1(G,M)⊗H1(G,M∗)→ H2(G,M ⊗M∗)→ H2(G,µ(C∗)) ∼→ 12Z/Z

given by cup product, Cartier duality and an easy computation, is perfect.

Proof. Write G = 1, c, where c is complex conjugation. c acts on µ(C∗) byinversion, so we can choose an identification of µ(C∗) with Q/Z on which c acts bynegation. Cartier duality between M and M∗ now becomes a sort of Pontrjaginduality

M ⊗M∗ → Q/Z,

where ϕ ∈M∗ is considered as a map ϕ : M → Q/Z such that

(cϕ)(m) = cϕ(c−1m)

56 7. LECTURE 7

for all m ∈ M . Given our action of c on Q/Z and the fact that c = c−1, thisbecomes

(cϕ)(m) = −ϕ(cm).

Recall that we have an identification

H1(G,M) = CrossHom(G,M)/PrincCrossHom(G,M).

A crossed homomorphism h must send 1 ∈ G to 0 ∈ M , so it is determinedcompletely by where it sends c. The cocycle condition is just

c(h(c)) + h(c) = 0,

so we can identify CrossHom(G,M) with the subgroup of M on which c acts bynegation; we write this as M−. Under this identification the principal crossedhomomorphisms are just (1− c)M , so we have an identification

H1(G,M) = M−/(1− c)M.

We have a similar expression for M∗.Now, we return to our “Pontrjagin” duality

M ⊗M∗ → Q/Z.

The annihilator of M∗− in M is easily seen to be (1− c)M , since ϕ ∈M∗− if andonly if

ϕ(cm) = ϕ(m)

for all m ∈M . Similarly, the annihilator of M− is just (1− c)M∗. Thus the abovepairing restricts to a perfect pairing

M−/(1− c)M ⊗M∗−/(1− c)M∗ → Q/Z.

Now our above observations and an easy computation using the definition of cupproducts (see [AW, Section 7]) identify this pairing with the cup product followedby Cartier duality, so the proof is complete.

We will refer to the above pairing simply as a local pairing.

Definition 4. Let M be a structured GF -module. We define the Cartier dualof M as a structured GF -module to be the GF -module

M∗ = HomZ(M,µ(F ))

together with choices of local finite/singular structures

0→ H1f (Gv,M∗)→ H1(Gv,M∗)→ H1

s (Gv,M∗)→ 0

such that H1f (Gv,M∗) is the exact dual of H1

s (Gv,M) under the local pairing forall v. Since the local pairings are perfect, it follows that H1

s (Gv,M∗) is the exactdual of H1

f (Gv,M).

Note that Theorem 2.2 of Lecture 6 insures that the above definition reallydoes give a finite/singular structure on M∗.

2. GENERALIZED SELMER GROUPS 57

2. Generalized Selmer Groups

2.1. Definitions. We continue to let F be a number field and we let M be a fi-nite, structured GF -module. We now define finite and singular parts of H1(GF ,M)using local conditions. First note that under the maps

H1(GF ,M) resv−→ H1(Gv,M),

each element h ∈ H1(GF ,M) maps to H1f (Gv,M) for all but finitely many v. To

see this, recall that H1(GF ,M) is defined as a direct limit, so we can consider h asan element of H1(Gal(L/F ),MGL) for some finite, Galois extension L of F . Letv be a place of F and let w be the place of L above v determined by our choiceGv → GF . Further suppose that v is non-archimedean and that L/F is unramifiedat v; this is true for almost all v. But in this situation, letting `w and fv be theresidue fields of Lw and Fv respectively, the inflation-restriction sequence gives anisomorphism

H1(Gal(`w/fv),MGL)∼=−→ H1(Gal(Lw/Fv),MGL),

since the inertia group is trivial. If also M has the standard structure at v, thecommutative diagram

H1(Gal(L/F ),MGL)resv //

inf

H1(Gal(`w/fv),MGL)

inf

H1f (Gv,M)

_

inf

H1(GF ,M)resv // H1(Gv,M)

shows that resv(h) ∈ H1f (Gv,M); thus this is true almost always, as claimed.

We want to define H1f (GF ,M) to be the set of global cohomology classes which

are always locally finite. Precisely we define

H1f (GF ,M) = ker

(H1(GF ,M)→ ⊕vH1

s (Gv,M)),

where the direct sum is over all places v of F . (Our argument above shows thatthis map really does land in the direct sum.) That is,

H1f (GF ,M) = h ∈ H1(GF ,M) | resv(h) ∈ H1

f (Gv,M) for all v.

We will call H1f (GF ,M) the Selmer group of the structured GF -module M .

We defineH1s (GF ,M) = H1(GF ,M)/H1

f (GF ,M),

so that there is an exact sequence

0→ H1f (GF ,M)→ H1(GF ,M)→ H1

s (GF ,M)→ 0.

We will call H1s (GF ,M) the Kolyvagin group of the structured GF -module M .

We should note that our choice of notation is not intended to imply that thereis any sort of duality between the Selmer groups and Kolyvagin groups of M andM∗. The statement of Poitou-Tate global duality is significantly more complicatedthan that of Tate local duality.

58 7. LECTURE 7

2.2. A First Example. In order to illuminate the meaning of a Selmer groupslightly, we now consider a simple example. Let M be a finite abelian group con-sidered as a GF -module with trivial GF -action. We define a structure on M asfollows: Let S be a set of places of F containing all of the bad places. That is, Scontains the archimedean places and the set of places dividing the order of M . Forv ∈ S, we simply take

H1f (Gv,M) = H1(Gv,M)

so thatH1s (Gv,M) = 0.

For v /∈ S, we take the standard finite/singular structure. Since M is unramifiedeverywhere, this takes the simple form

0→ H1(gv,M)→ H1(Gv,M)→ Hom(Iv,M)gv → 0.

Now, consider H1(GF ,M). Since GF acts trivially on M , we have

H1(GF ,M) = Hom(GF ,M) = Hom(GabF ,M).

(The homomorphisms are continuous, of course.) A cocycle h ∈ H1(GF ,M) lies inthe Selmer group H1

f (GF ,M) if it maps to 0 in H1s (Gv,M) for all places v. If v ∈ S,

this is automatic. If v /∈ S, this simply asks that h vanishes on the inertia groupIv. (We get these simple interpretations since h is just a homomorphism.) ThusH1f (GF ,M) is simply the subgroup of Hom(Gab

F ,M) of homomorphisms vanishingon all inertia groups of v ∈ S. In other words, letting F ab,S be the maximal abelianextension of F unramified outside of S, we have

H1f (GF ,M) = Hom(Gal(F ab,S/F ),M).

(Hom(Gal(F ab,S/F ),M) is considered as a subgroup of Hom(GabF ,M) via the nat-

ural surjection GabF Gal(F ab,S/F ).) Thus knowledge of H1

f (GF ,M) should yieldinformation about F ab,S .

If instead we had imposed the finite/singular structure with

H1f (Gv,M) = 0

andH1s (Gv,M) = H1(Gv,M)

for v ∈ S, one can easily check in the same way as above that we would haveobtained

H1f (GF ,M) = Hom(Gal(F ′/F ),M),

where F ′ is the maximal abelian extension of F which is unramified everywhereand in which all places in S split completely. (The key fact here is that all of thelocal Galois groups for a place v vanish if and only if v splits completely.)

2.3. Interpretation in terms of Etale Cohomology. We now give an in-terpretation of (some) Selmer groups in terms of etale cohomology. We let M beany finite GF -module, and we again take S to be a set of places containing all of thebad places. (This time it also contains those places where M is wildly ramified.)We take the standard finite/singular structure at v /∈ S, and we set

H1f (Gv,M) = H1(Gv,M)

for v ∈ S. We let S′ be the subset of S of non-archimedean places.

2. GENERALIZED SELMER GROUPS 59

Recall (see Lecture 1, Section 1) that to M we can associate an etale sheaf Mover SpecF . Let OF be the ring of integers of F , and consider the natural maps

SpecFj→ SpecOF − S′

i→ SpecOF .

Here SpecF maps to the generic point. Note that SpecOF−S′ is an open subschemeof SpecOF and can be realized as SpecOF [1/N ] where N is any element of OFdivisible by every place of S′ and no other places. In this situation it can be shownthat

H1f (GF ,M) = H1

et(SpecOF − S′, j∗M).We sketch a proof. Set X = SpecOF − S′. The Leray spectral sequence

([Mi-EC, Chapter 3, Theorem 1.18]) for j : SpecF → X takes the form

Hpet(X,R

qj∗M)⇒ Hp+qet (SpecF,M).

The exact sequence of low degree terms begins

0→ H1et(X, j∗M)→ H1

et(SpecF,M)→ R1j∗M(X).

But H1et(SpecF,M) is just H1(GF ,M) (see [Mi-EC, Chapter 3, Example 1.7]),

so we have identified H1et(X, j∗M) with the subgroup of elements of H1(GF ,M)

which vanish in R1j∗M(X).To check that an element of this last group vanishes it is enough to check on

the stalks. So let v → X be a geometric point sitting over some prime v of F ,v /∈ S′. The stalk of OX at v is just the strict Henselization of the local ring ofOX at v (recall that the strict Henselization of a local ring A is the smallest localring containing A which has algebraically closed residue field and satisfies Hensel’slemma); in this case, this is just the ring of integers of the maximal unramifiedextension of Fv. The base change of SpecOX,x up to SpecF is then SpecF ur

v , soby [Mi-EC, Chapter 3, Theorem 1.15] we have

R1j∗(M)v = H1(SpecF urv ,M) = H1(Iv,M),

where Iv is the inertia group at v. A similar calculation shows that the stalk atthe generic point is 0.

Combining this with our above exact sequence, we see that an element h ∈H1(F,M) lies in H1

et(X, j∗M) if and only if h maps to 0 in each H1(Iv,M) forv /∈ S. But this is exactly the definition of H1

f (GF ,M), so

H1f (GF ,M) = H1

et(SpecOF − S′, j∗M),

as claimed.

2.4. Other Notation. Our notation differs somewhat from that in [Ru-ES].The translation is easy; for convenience, and so the reader can go through [Ru-ES,Chapter 1, Section 6], we record it here.

First, Rubin uses the standard notations

Hi(K,M) = Hi(Gal(Ks/K),M)

for any field K, and

Hi(L/K,M) = Hi(Gal(L/K),M)

for any Galois extension L/K. He defines standard finite/singular structures forarchimedean v and for all v not dividing the order of M ; they agree with ours whenboth are defined.

60 7. LECTURE 7

Our Selmer group H1f (GF ,M), assuming our finite/singular structures agree

with his for v not dividing the order of M and given our choices for those v which dodivide the order of M , would be written as S(F,M). Rubin also defines “partial”Selmer groups where not all local conditions are enforced, and “strict” Selmergroups where some H1

f (GF ,M) are set to be 0 (in our terminology). For details,see [Ru-ES, Chapter 1, Section 5].

In [Ru-ES, Chapter 1, Section 6] he gives several interesting interpretations ofSelmer groups, relating them to ideal class groups, units and abelian varieties. Thereader is strongly encouraged to look at that section.

We should also note that Rubin’s Galois modules are rarely finite; they tendto be p-adic. This distinction can probably be safely ignored.

3. Brauer Groups

3.1. Definitions. We begin by recalling the various interpretations of theBrauer group. For proofs of the claims in this section, see [FD, Chapter 4]. Forany field L, we take as our definition

BrL = H2(GL, Ls).

As a first alternate definition, BrL can be considered as the set of finite dimen-sional central simple algebras over L modulo the equivalence relation where twocentral simple algebras are considered to be equivalent if they become isomorphicas L-algebras after tensoring each with (possibly different) total matrix algebrasMn(L). BrL is made into a group by tensor product. It can be shown that everycentral simple algebra over L is a matrix algebra over a central divison algebraover L, and that central simple algebras are equivalent in our sense if and only ifthey come from the same division algebra. Thus BrL also classifies central divisionalgebras over L. Lastly, it is also true that an L-algebra is central simple over L ifand only if it becomes isomorphic toMn(L), for some n, after tensoring with L. Inparticular, the dimension of any central simple algebra over L is a perfect square.

To recall a few examples, if k is a finite field we have Wedderburn’s theorem(see [FD, Chapter 3, Theorem 3.18])

Brk = 0,

and for the real numbers we have

BrR = Z/2Z

with the non-trivial division algebra being the quaternions H. (See [FD, Chapter3, Theorem 3.20]) We also note that

BrL = 0

for any algebraically closed field L; the proof of this is easy.

3.2. Brauer Group of a Local Field. Recall that if K is a local field, thenlocal class field theory gives an isomorphism

BrK = H2(GK , K∗)invK−→ Q/Z.

(See [Se-LF, Chapter 13, Section 3].) For completeness, we include here the proofthat this implies that

H2(GK , µ(K∗)) ∼= Q/Z.

3. BRAUER GROUPS 61

We use the Kummer exact sequence

1→ µN (K∗)→ K∗N−→ K∗ → 1

for every N . The long exact sequence in cohomology yields

H1(GK , K∗)→ H2(GK , µN (K∗))→ H2(GK , K∗)[N ]→ 0,

where by H2(GK , K∗)[N ] we mean the N -torsion. But by Hilbert’s Theorem 90the first term is zero, and by the invariant map the last term is 1

NZ/Z. Taking thedirect limit over N now completes the proof.

We recall the explicit description of division algebras over K. Let m/n ∈ Q/Zbe in lowest terms, with 0 < m < n. (0 ∈ Q/Z corresponds to K, of course.) We willconstruct the central division algebra A mapping to m/n under the invariant map.Let L be the unique unramified extension of K of degree n. A has dimension n2

over K, and contains L as a maximal commutative subalgebra. In fact, it requiresonly one more generator: A = L[γ], where γ normalizes L (that is, γLγ−1 = L),and for any x ∈ L, γxγ−1 = Frm x, where Fr is the Frobenius automorphism forK. See [Pi, Chapter 17, Section 10] for details.

3.3. Global Brauer Groups. We now consider the Brauer group of a numberfield F . Recall that we have canonical maps

H2(GF , F ∗)resv−→ H2(Gv, F ∗v )

for every place v of F . We claim that for any h ∈ H2(GF , F ∗), the image of h inH2(Gv, F ∗v ) is zero for almost all v.

Proposition 3.1. For any h ∈ H2(GF , F ∗), resv(h) = 0 for all but finitelymany places v of F .

Proof. Note that as usual, since H2(GF , F ∗) is a direct limit, we can assumethat h lies in H2(Gal(L/F ), L∗) for some finite Galois extension L/F . Now, let cbe a cocycle representing h. Since Gal(L/F ) is finite, c takes on only finitely manyvalues; in particular, its image is divisible by only finitely many primes. So let Tbe the set of all such places, together with the archimedean places and the placeswhere L/K is ramified; T is finite.

Now consider resv(h) ∈ H2(Gal(Lw/Fv), L∗w) for any v /∈ T , where w is theplace of L above v induced by our choice of embedding F → Fv. By the definitionof the restriction map, resv(h) is represented by the restriction of c. In particular, ctakes values in O∗w, the units of the ring of integers of Lw. This means that resv(h)lies in the image of the natural map

H2(Gal(Lw/Fv),O∗w)→ H2(Gal(Lw/Fv), L∗w).

Consider the exact sequence

0→ U1 → O∗w → `∗ → 0,

where U1 are the units congruent to 1 modulo mw, and `∗ is the residue field of Lw.This is an exact sequence of Gal(Lw/Fv)-modules, and using the fact (see [Se-LF,Chapter 12, Section 3, Lemma 2]) that Hi(Gal(Lw/Fv),U1) = 0 for all i ≥ 1, weget an isomorphism

H2(Gal(Lw/Fv),O∗w) ∼= H2(Gal(Lw/Fv), `∗).

62 7. LECTURE 7

Finally, since Lw/Fv is unramified, the cohomology of a finite cyclic group([Se-LF, Chapter 8, Section 4]) and the surjectivity of the norm on a finite fieldshow that

H2(Gal(Lw/Fv), `∗) = 0,which implies that the image of H2(Gal(Lw/Fv),O∗w) in H2(Gal(Lw/Fv), L∗w) iszero, and thus that resv(h) = 0 for all places not in T . This proves the claim.

For a proof of this fact using the cohomology of the ideles, see [Ta, Section7, Proposition 7.3 and Section 9.6]. (Note that he doesn’t actually prove that themaps he is considering are the restriction map, but it is not too difficult to see this.)For a proof in terms of divison algebras, see [Pi, Chapter 18, Section 5].

It follows that the restriction maps combine to give a map

H2(GF , F ∗)res−→ ⊕vH2(Gv, F ∗v ),

the direct sum being over all places v of F . One of the fundamental theorems ofglobal class field theory is the determination of the kernel and cokernel of this map.Before we state it, we consider

⊕vH2(Gv, F ∗v ).

This can be evaluated as

⊕v non-archQ/Z⊕⊕v real12Z/Z,

given the results of the previous two sections.Theorem 3.2. There is an exact sequence

0→ BrFres−→ ⊕v BrFv → Q/Z→ 0,

where the last map is summation of elements of Q/Z in the identification above.

Proof. This is proved along with the fundamental results of global class fieldtheory; see [Ta, Section 9.6] for the statement.

CHAPTER 8

Lecture 8

1. Notation for Generalized Selmer Groups

1.1. Structures. For convenience we will set some notation for our general-ized Selmer groups. We continue to let F be a number field with the conventionsof Lecture 7, Section 1.2. Let M be a finite GF -module. We recall that a structureσ on M is a choice of exact sequences

σv : 0→ H1f (Gv,M)→ H1(Gv,M)→ H1

s (Gv,M)→ 0

for all places v of M , such that the exact sequence σv is the standard local fi-nite/singular exact sequence for almost all v. We will write Mσ for the GF -moduleM with the structure σ; if there is no chance of confusion we will just write M .

Given two structures σ1 and σ2 on M , we will say that σ1 is more stringentthan σ2, written

σ1 < σ2,

if

H1f (Gv,Mσ1) ⊆ H1

f (Gv,Mσ2)

for all places v. In particular, this implies that

H1f (GF ,Mσ1) ⊆ H1

f (GF ,Mσ2),

since the conditions to lie in H1f (GF ,Mσ1) are more restrictive than those to lie in

H1f (GF ,Mσ2). This relation gives a partial ordering on the set of all structures.

Given a structured module Mσ we will write σ∗ for the structure defined on M∗

in Definition 4 of Lecture 7. It follows easily from the definition of this structureon M∗ that if σ1 < σ2, then σ∗2 < σ∗1 .

Let S be a finite set of places of F containing all the archimedean places, all theplaces dividing the order of M and all places where M is ramified. We define the S-structure on M , written as MS , to be the structure with the standard finite/singularstructures for all v /∈ S and with the choices

H1f (Gv,MS) = H1(Gv,M)

for all v ∈ S. (We considered this structure in Lecture 7, Sections 2.2 and 2.3.)So an S-structure is the least stringent structure on M with the standard choicesoutside of S. In particular, if σ is any structure, then there is some S-structurewhich is less stringent than σ; simply take S to contain all places where σ is notstandard.


63

64 8. LECTURE 8

1.2. The Singular Restriction Map. Recall that for every place v we havea canonical restriction map

H1(GF ,M) resv−→ H1(Gv,M).

We define the local singular restriction map at v

ress,v : H1(GF ,M)→ H1s (Gv,M)

to be the composition of resv with the quotient map

H1(Gv,M) H1s (Gv,M).

We saw in Lecture 7, Section 2.1 that for any h ∈ H1(GF ,M) we have resv(h) ∈H1f (Gv,M) for almost all v; in our new notation this says that ress,v(h) = 0 for

almost all v. Thus we obtain a map, which we call the singular restriction map,

ress : H1(GF ,M)→ ⊕vH1s (Gv,M),

the direct sum being over all places of F . With this notation we have

H1f (GF ,M) = ker ress,

so there is an exact sequence

0→ H1f (GF ,M)→ H1(GF ,M)→ ⊕vH1

s (Gv,M).

Recall that by definition of the Kolyvagin group H1s (GF ,M) we have an exact

sequence

0→ H1f (GF ,M)→ H1(GF ,M)→ H1

s (GF ,M)→ 0.

Comparing this to the exact sequence above we see that there is a natural injection

H1s (GF ,M) → ⊕vH1

s (Gv,M).

Given any c ∈ H1s (GF ,M), this map is simply the map

c 7→ (ress,v(c))v,

where c is any lifting of c to H1(GF ,M). We will write cv for ress,v(c) ∈ H1s (Gv,M),

so that the above map is simply

c 7→ (cv)v.

We conclude with a few more definitions. Given any

c = (cv)v ∈ ⊕vH1s (Gv,M),

define the support of c, Supp(c), to be the set of places v for which cv 6= 0; ourprevious discussion shows that Supp(c) is finite. If c is actually in H1

s (GF ,M),we again write Supp(c) for the set of places for which cv 6= 0. Also, for anyh ∈ H1(GF ,M) we will write hv for resv(h) ∈ H1(Gv,M). In particular, if h ∈H1f (GF ,M), then hv ∈ H1

f (Gv,M) for all v

2. A GLOBAL PAIRING 65

2. A Global Pairing

We continue with the notation of the previous section. In particular, M is afinite structured GF -module with (structured) Cartier dual M∗. For any place vwe will write 〈·, ·〉v for the pairing

H1(Gv,M)⊗H1(Gv,M∗)→ H2(Gv,M ⊗M∗)→ H2(Gv, µ(F ∗v ))→ Q/Z

given by cup product, Cartier duality and the invariant map. Thus for non-archimedean v, 〈·, ·〉v is just the Tate pairing; for real v it is the pairing

H1(Gv,M)⊗H1(Gv,M∗)→12Z/Z

of Lecture 7, Proposition 1.1; and for complex v all of the groups are 0, sinceGv = 0. For any cv ∈ H1(Gv,M∗), we will also think of the pairing as giving amap

cv : H1(Gv,M)→ Q/Z,

given by cv(hv) = 〈hv, cv〉.We now define a global pairing

H1f (G,M)⊗⊕vH1

s (Gv,M∗)→ Q/Z

by

h⊗ (cv)v 7→∑v

〈hv, cv〉v .

(Recall the definition of hv in the previous section.) This is actually well-definedsince hv ∈ H1

f (Gv,M) for all v and by our definition of the structure on M∗ theperfect pairings

H1(Gv,M)⊗H1(Gv,M∗)→ Q/Z

yield perfect pairings

H1f (Gv,M)⊗H1

s (Gv,M∗)→ Q/Z

for all v. Also, the sum is simply a finite sum∑v

〈hv, cv〉v =∑

v∈Supp(c)

〈hv, cv〉v

since (cv)v is an element of the direct sum.We will write the pairing as 〈h, c〉; alternately, for any c ∈ ⊕vH1

s (Gv,M∗), wewill also write c for the map

c : H1f (GF ,M)→ Q/Z

given by c(h) = 〈h, c〉.This pairing, together with the next proposition, will be our principal tool for

bounding the order of Selmer groups.Proposition 2.1. The subgroup H1

s (GF ,M∗) of ⊕vH1s (Gv,M∗) is orthogonal

to all of H1f (GF ,M) under the above pairing; that is, the composition

H1f (GF ,M)⊗H1

s (GF ,M∗)→ H1f (GF ,M)⊗⊕vH1

s (Gv,M∗)→ Q/Z

is zero.

66 8. LECTURE 8

Proof. Take any h ∈ H1f (GF ,M) and c ∈ H1

s (GF ,M∗), and let c be a liftingof c to H1(GF ,M∗). Consider the commutative diagram

H1(GF ,M)⊗H1(GF ,M∗) //

∏vH

1(Gv,M)⊗H1(Gv,M∗)

H2(GF ,M ⊗M∗) //

∏vH

2(Gv,M ⊗M∗)

H2(GF , µ(F ∗)) //∏vH

2(Gv, µ(F ∗v ))

BrF //∏v BrFv

where all horizontal maps are restriction and the second column is the product ofall of the local pairings 〈·, ·〉v. Note also that by Lecture 7, Proposition 3.1, theimage of BrF actually lands in ⊕v BrFv . Further, by Lecture 7, Theorem 3.2, BrFis the kernel of the summation map

⊕v BrFv → Q/Z.

Now, following h⊗ c clockwise around the diagram and then mapping by sum-mation to Q/Z gives the global pairing 〈h, c〉, by definition. But going counter-clockwise shows that the image of h ⊗ c in ⊕v BrFv lies in the image of BrF , andthus maps to 0 under summation. This completes the proof.

The above result is extremely useful in bounding the order of Selmer groups.We quote Mazur:

Any Kolyvagin element c ∈ H1s (G,M∗) imposes some local condi-

tion (at places v in its support) that must be satisfied by all Selmerelements h ∈ H1

f (G,M).For example: If the support of c 6= 0 is concentrated at ex-

actly one place vo, then any Selmer element h must have the prop-erty that hv0 ∈ ker(cv0) ⊂ H1

f (Gv0 ,M). Here is a way of thinkingabout the local condition detected by this Kolyvagin element c:It tells us that we may strengthen the f/s structure on the Galoismodule M , without changing the finite part of its (1-dimensional)cohomology. That is, let us define M ′ to be the structured modulewhose underlying Galois module is M again, whose f/s structurefor all places v 6= v0 is equal to the f/s structure on M , but on v0

we place a more stringent f/s structure on the Gv0-module M bysetting:

H1f (Gv0 ,M

′) := kercv0 : H1f (Gv0 ,M)→ Q/Z ⊂ H1

f (Gv0 ,M).

The above discussion gives us that the finite part of the 1-dimensional cohomology of the more stringent structureM ′ is equalto the finite part of the 1-dimensional cohomology of M . Now thisis already a curiously favorable state of affairs, because if we are

3. THE FINITENESS OF THE SELMER GROUP 67

trying to to show that H1f (G,M) is small, we have (by constructing

the Kolyvagin element c) that

H1f (G,M ′) = H1

f (G,M),

where M ′ is more stringent than M , and there is nothing to stop usfrom trying to do this again inductively, to force the finite cohomol-ogy to satisfy more and more stringent local conditions by findingfurther Kolyvagin elements for these increasingly stringent struc-tures. In the most extreme cases one then shows that H1

f (G,M)satisfies such stringent conditions that it is forced to vanish, or elseit is generated by specifically constructed cohomology classes. Theefficacy of this method is determined by our ability to manufactureKolyvagin elements with good control over their local components.This approach, combined with the following two further ideas isthe underlying philosophy behind Kolyvagin’s method:(1) It is possible, at times, to judiciously enlarge the base field F

without losing too much of the Selmer group, but this largerbase field gives us an even better chance of constructing Koly-vagin elements.

(2) In various different contexts, one can find well-behaved col-lections of objects (these are the Euler systems of algebraiccycles, Heegner points, circular units, elements in algebraic K-groups, etc.) which produce the desired Kolyvagin elementsin cohomology.

3. The Finiteness of the Selmer Group

3.1. Kummer Theory : Perfect Fields. Later in this section we will showthat the Selmer group H1

f (GF ,M) is finite; we review first some Kummer theorywhich we will need. For the time being, let L be any field; for simplicity we assumethat L is perfect. Fix an integer n and consider the exact Kummer sequence

1→ µn(L∗)→ L∗n−→ L∗ → 1.

The long exact sequence of GL-cohomology begins

1→ µn(L)→ L∗n−→ L∗ → H1(GL, µn(L))→ H1(GL, L∗).

By Hilbert’s Theorem 90, H1(GL, L∗) = 0, so we obtain an isomorphism

L∗/L∗n∼−→ H1(GL, µn(L)).

Recalling the definition of the boundary map in cohomology, this map simply sendsany α ∈ L∗ to the cocycle

c : GL → µn(L∗)

given by

c(σ) =σ(α1/n)α1/n

;

it is independent of the choice of nth root of α and is constant on the equivalenceclass of α in L∗/L∗n (up to coboundaries, of course).

68 8. LECTURE 8

Let us now assume that µn = µn(L∗) ⊆ L; this is the usual situation of Kummertheory. Our above isomorphism becomes

L∗/L∗n∼−→ Hom(GL, µn).

Since µn is abelian of exponent n, we can rewrite our isomorphism as

L∗/L∗n∼−→ Hom(Gal(L′/L), µn),

where L′ is the maximal abelian extension of L of exponent n. Taking duals givesa canonical isomorphism

Gal(L′/L) ∼= Hom(L∗/L∗n, µn).

Let V be a subgroup of L∗/L∗n. Then Hom(V, µn) is a quotient of Hom(L∗/L∗n, µn),so it corresponds under our above isomorphism to a quotient of Gal(L′/L); this inturn corresponds to an intermediate field L ⊆ E ⊆ L′. This correspondance sets upa bijection between subgroups of L∗/L∗n and abelian extensions of L of exponentn; if V corresponds to E, then there is an isomorphism

Gal(E/L) ∼= Hom(V, µn).

Let us attempt to make this correspondance more explicit. To do this, notethat Gal(L′/E) is precisely the subgroup of Gal(L′/L) mapping to 0 in Hom(V, µn);that is, σ ∈ Gal(L′/E) if and only if

σ(α1/n)α1/n

= 1

for all α ∈ L∗ lifting elements of V . It now follows immediately from Galois theorythat

E = L(V 1/n),the notation meaning that E is obtained by adjoining nth roots of lifts of everythingin V to L. We state our results of this section as a theorem.

Theorem 3.1. Let L be a perfect field containing µn(L∗), and let L′ be themaximal abelian extension of L of exponent n. Then

Gal(L′/L) ∼= Hom(L∗/L∗n, µn).

Furthermore, there is a bijective correspondance between subgroups of L∗/L∗n andintermediate fields between L and L′; if V corresponds to E, then

Gal(E/L) ∼= Hom(V, µn)

andE = L(V 1/n).

3.2. Kummer Theory : Local and Global Fields. We now take K to bea local field with uniformizer π. We continue to assume that K contains the nth

roots of unity. Let V ⊆ K∗/K∗n and E/K correspond under the correspondanceof Theorem 3.1. We wish to relate the ramification in E/K to V .

This is fairly easy given that E = K(V 1/n). Let α be a lift of some element ofV and let β be an nth root of α. E is the compositum of K(β) for all such β, so itwill be ramified over K if and only if K(β)/K is ramified for some α.

Write α = uπm for some unit u. Suppose first that n divides m. Then K(β) =K(β′), where β′ is an nth root of u. β′ satisfies the polynomial

xn − u,


although it may not be minimal. In any event, a standard theorem of algebraicnumber theory (see [Se-LF, Chapter 3, Section 5, Corollary 1 and Section 6, Corol-lary 2]) says that K(β′)/K is ramified only if nun−1 is not a unit in K(β′); thisoccurs only if π divides n. (However, even if π divides n we can not be sure thatthe extension is ramified.)

On the other hand, if n does not divide m, then K(β) must contain somenon-trivial root of a uniformizer of K; this immediately implies that K(β)/K isramified. We rephrase our results so far as a proposition.

Proposition 3.2. Let K be a local field with uniformizer π. Suppose that Kcontains the nth roots of unity. Let E/K be an abelian extension of exponent ncorresponding to a subgroup V ⊆ K∗/K∗n. If there is an α ∈ V such that

v(α) 6= 0 (mod n),

where v is the normalized valuation on K, then E/K is ramified. If instead

v(α) ≡ 0 (mod n)

for all α ∈ V and v(n) = 0, then E/K is unramified.Now let F be a global field. Considering ramification locally immediately yields

the following theorem.Theorem 3.3. Let F be a number field containing the nth roots of unity.

Let E/F be an abelian extension of exponent n corresponding to a subgroup V ⊆F ∗/F ∗n. Then E/F is ramified at a non-archimedean place v if there is an α ∈ Vwith

v(α) 6= 0 (mod n);

E/F is unramified at all other v except possibly those dividing n.Corollary 3.4. Let F be a number field and let S be a finite set of places of

F . Let L be the maximal abelian extension of F of exponent n, unramified outsideof S. Then L/F is finite.

Proof. Let F ′ = F (µn(F ∗)). Then the corresponding extension of F ′ containsthat of F , so we can assume that F contains the nth roots of unity. In the sameway we can assume that F is totally imaginary, so that the archimedean places areirrelevant.

There is a natural map

F ∗ → ⊕v non-archZ,

given by the valuation maps at all non-archimedean places, with kernel U , the unitsof the ring of integers of F . This map descends to an exact sequence

0→ U/Un → F ∗/F ∗n → ⊕vZ/nZ.

(The kernel is a priori U/(F ∗n∩U), but any element of F ∗ which has a power whichis a unit is itself a unit.) If V is the subgroup of F ∗/F ∗n corresponding to L, thenby Theorem 3.3 the image of V in each Z/nZ factor for v /∈ S is zero; thus we havean exact sequence

0→ V ∩ U/Un → V → ⊕v∈SZ/nZ.By Dirichlet’s Theorem, U is finitely generated, so U/Un is finite. Since S is alsofinite, V must be finite. This implies that L/F is finite.

70 8. LECTURE 8

3.3. The Finiteness of Selmer Groups.

Theorem 3.5. Let F be a number field and let M be a finite structured GF -module. Then H1

f (GF ,M) is finite.

Proof. Let S be a finite set of places of F containing all places where Mdoes not have the standard finite/singular structure; in particular, S contains allarchimedean places, all places dividing the order of M , and all places where M iswildly ramified. Let us also take S to contain all places where M is at all ramified.Let MS be the same GF -module as M with the S-structure (see Section 1.1). Thenwe have

H1f (GF ,M) ⊆ H1

f (GF ,MS),

so it will be enough to prove finiteness for MS .Next, let L be a finite Galois extension of F which splits M ; that is, such that

GL acts trivially on M . (It is easy to see that such an L exists; see Lecture 1,Section 1.1.) If L/F is ramified outside of S, then enlarge S to contain all placeswhere L/F is ramified. Let T be the set of places of L lying above places of F inS, and let MT be M considered as a GL-module with the T -structure. Consider,for all places v of F and all places w of L lying over v, the maps

H1(GFv ,MS) res−→ H1(GLw ,MT ).

We claim that under these maps the image of H1f (GFv ,MS) lies in H1

f (GLw ,MT ). Ifv ∈ S (so w ∈ T ) this is immediate; if v /∈ S, then this follows from the commutativediagram

H1(gv,MS) res //

inf

H1(gw,MT )

inf

H1(GFv ,MS) res // H1(GLw ,MT )

This implies that we have a natural map

⊕vH1s (GFv ,MS)→ ⊕wH1

s (GLw ,MT )

sitting in a commutative diagram

H1(GF ,MS) //

⊕vH1s (GFv ,MS)

H1(GL,MT ) // ⊕wH1s (GLw ,MT )

We now get induced maps on the kernels of the horizontal maps, which are justthe Selmer groups:

H1f (GF ,MS)→ H1

f (GL,MT ).


Let ker be the kernel of this map; it sits in an exact commutative diagram

0

0

0 // ker //

H1(Gal(L/F ),M)

0 // H1f (GF ,MS) //

H1(GF ,MS) //

⊕vH1s (GFv ,MS)

0 // H1f (GL,MT ) // H1(GL,MT ) // ⊕wH1

s (GLw ,MT )

It is clear from the cocycle interpretation that H1(Gal(L/F ),M) is finite; thus keris also finite. It follows that it will be enough to prove that H1

f (GL,MT ) is finite.We are now in the situation of Lecture 7, Section 2.2: we have

H1f (GL,MT ) = Hom(Gab

L,T ,M)

where GabL,T is the Galois group of the maximal abelian extension of L unramified

outside of T . Decomposing M as a product of cyclic groups, we see that to showthis it will be enough to show that

Hom(GabL,T ,Z/nZ)

is finite for any n.We have

Hom(GabL,T ,Z/nZ) = Hom(Gal(L′/L),Z/nZ),

where L′ is the maximal abelian extension of L of exponent n, unramified outsideof T . By Corollary 3.4 we know that L′ is a finite extension of L, and thus that

Hom(Gal(L′/L),Z/nZ)

is finite. This completes the proof.

CHAPTER 9

Lecture 9

1. Abelian Varieties

We will refer to the two articles [Ro] and [Mi-AV] in [CS] for most of theresults in this section. Both articles refer to [Mu] for most proofs, but since we feelthat attempting to read [Mu] is a much more demanding activity than reading thesummaries in [CS] we will simply refer to those and let them refer the reader tothe appropriate part of [Mu].

1.1. Complex Tori. A complex torus T is a compact, connected commutativeLie group over C. Let g be its dimension, and let t be the tangent space to T atthe origin; we have t ∼= C

g. The exponential map

exp : t→ T

in this situation is surjective, and since T is compact and connected the kernel Λmust be a a discrete subgroup of t ∼= C

g of maximal rank; that is, Λ is a lattice. Inthis way we get a complex analytic group isomorphism

T ∼= t/Λ ∼= Cg/Λ.

(See [Ro, Section 1] for details.)Let

m : T → T

be the multiplication by m map; it is a homomorphism since T is commutative. Infact, the identification T ∼= C

g/Λ makes it clear that it is surjective, so we have anexact sequence

0→ T [m]→ Tm−→ T → 0,

where T [m] is the m-torsion in T . Considering T once again as Cg/Λ, we find that

T [m] ∼=1m

Λ/Λ ∼= (Z/mZ)2g,

the last isomorphism being non-canonical.

1.2. Abelian Varieties. An abelian variety over a field L is a proper, geo-metrically connected variety A defined over L together with a point O ∈ A(L), amultiplication morphism

µ : A×A→ A

and an inversion morphismι : A→ A,

both defined over L, making A into a group object in the category of proper varietiesover L, with O as the identity element. (All this means is that the appropriate


73

74 9. LECTURE 9

diagrams expressing associativity, inverses and the identity all commute. See [Sh,Section 2] for the diagrams.)

There are some rather remarkable consequences of this definition. First, A isnonsingular, since any variety is nonsingular on a non-empty open set and µ canbe used to translate this set over all of A. Second, A is actually projective; see[Mi-AV, Section 7]. Lastly, the group law must be commutative; see [Mi-AV,Section 2]. Thus for any field L′ ⊇ L the L′-valued points A(L′) form an abeliangroup.

Now, suppose that L is a subfield of C. Then A(C) is a compact, connected(since we assumed that A was geometrically connected), commutative Lie groupover C; that is, A(C) is a complex torus. (Not every complex torus can be giventhe structure of an algebraic variety; see [Ro, Section 3].) In particular, we havean exact sequence

0→ A(C)[m]→ A(C) m−→ A(C)→ 0as above. Recall that A(C)[m] ∼= (Z/mZ)2g, where g is the dimension of A.

Lemma 1.1. Every point of A(C)[m] actually lies in A(L′) for some finiteextension L′ of L.

Proof. Let P be an m-torsion point over C. Since µ and thus m : A → A isdefined over L,

m(Pσ) = m(P )σ = 0for any σ ∈ Aut(C/L). In particular, all of the conjugate points to P are also inA(C)[m]. Thus P has only finitely many conjugates over L, which implies that itis defined over a finite extension of L.

In particular, A(C)[m] = A(L)[m]. This is not exactly what one would like;the optimal situation would be to define a projective algebraic group A[m] over Lby

A[m] = A×A 0,where the map A → A is multiplication by m and 0 is the trivial abelian varietyover L. (Intuitively this is the correct notion, since a point in A×A 0 is an orderedpair (a, 0) where a maps to 0 under multiplication by m.) One then would wantto show that the morphism A

m−→ A is surjective with finite kernel (such a mapis called an isogeny), so that A[m] is a finite projective algebraic group over L.(Recall that a map f : A→ B of abelian varieties over L is surjective if it satisfiesany of the following equivalent conditions:

(1) f is surjective as a map of topological spaces;(2) f : A(L)→ B(L) is surjective as a map of abelian groups;(3) the induced map between the etale sheaves defined by A and B is surjec-

tive.)This program can actually be carried out over an arbitrary field L; one finds

that multiplication by m is an isogeny of degree m2g and that it is etale if and onlyif the characteristic of L does not divide m. In particular, A[m] has all points inL, and

A[m](L) ∼= (Z/mZ)2g,

assuming that the characteristic of L does not divide m. See [Mi-AV, Section 8]for more details. (There may be fewer points in A[m](L) when the characteristicof L divides m.)

2. SELMER GROUPS OF ABELIAN VARIETIES 75

The usefulness of our above results is that we have obtained a well-behavedGal(L/L)-module A[m](L); in fact, by an easy variant of Lemma 1.1 it is actuallydiscrete.

2. Selmer Groups of Abelian Varieties

2.1. Definitions. Let F be a global field, let A be an abelian variety definedover F and let m be an integer. Since multiplication by m is surjective, we have anexact sequence

0→ A[m](F )→ A(F ) m−→ A(F )→ 0.

We saw in the previous section that A[m](F ) is a finite, discrete GF -module, so wecan apply all of the theory we have developed to it. We will write it (only slightlyambiguously) just as A[m].

Consider the portion of the long exact cohomology sequence

A(F ) m−→ A(F )→ H1(GF , A[m])→ H1(GF , A) m−→ H1(GF , A)

We can reduce this to a short exact sequence

0→ A(F )/mA(F )→ H1(GF , A[m])→ H1(GF , A)[m]→ 0.

If we are interested in the arithmetic of A(F ), this exact sequence is a good start;we have expressed a quotient of A(F ) as a subgroup of a cohomology group. Un-fortunately, both of the cohomology groups in the exact sequence are quite large.We will use local considerations to replace them by much smaller subgroups.

To do this, we consider the composition

H1(GF , A[m])→ H1(GF , A(F ))→∏v

H1(Gv, A(Fv)),

where the last map is the product of the restriction maps over all places v of F .The groups we are looking for are the kernel of the second map and the kernel ofthe composition.

Definition 5. The mth Selmer group Sm(A/F ) is defined by

Sm(A/F ) = ker

(H1(GF , A[m])→

∏v

H1(Gv, A(Fv))

),

the map being the composition above. The Shafarevich-Tate group X(A/F ) isdefined by

X(A/F ) = ker

(H1(GF , A(F ))→

∏v

H1(Gv, A(Fv))

).

76 9. LECTURE 9

By their definitions, Sm(A/F ) and X(A/F ) fit into an exact commutativediagram

0

0

0

0 // A(F )/mA(F ) // Sm(A/F ) //

X(A/F ) m //

X(A/F )

0 // A(F )/mA(F ) // H1(GF , A[m]) // H1(GF , A(F )) m //

H1(GF , A(F ))

∏vH

1(Gv, A(Fv))m //

∏vH

1(Gv, A(F ))

Exactness of the top row follows from a simple diagram chase. Extracting that row,we obtain the fundamental exact sequence

0→ A(F )/mA(F )→ Sm(A/F )→X(A/F )[m]→ 0.

2.2. The Abelian Variety Structure on A[m]. The next step is to endowA[m] with a suitable finite/singular structure (in the sense of Lecture 7, Section1.2) so that we can apply our general theory. In fact, by choosing all of the lo-cal structures correctly we will identify H1

f (GF , A[m]) with the mth Selmer groupSm(A/F ); Theorem 3.3 of Lecture 8 will then have immediate important arithmeticconsequences.

To define this structure, we want to extend the bottom row of our above com-mutative diagram by using the local versions

0→ A(Fv)/mA(Fv)→ H1(Gv, A[m])→ H1(Gv, A)[m]→ 0

of our original exact sequence

0→ A(F )/mA(F )→ H1(GF , A[m])→ H1(GF , A)[m]→ 0.

(Note that the A[m] notation is not really ambiguous here, since we have chosenan embedding F ⊆ Fv.) We obtain a commutative diagram

0

0

0

0 // A(F )/mA(F ) // Sm(A/F ) //

X(A/F ) m //

X(A/F )

0 // A(F )/mA(F ) //

H1(GF , A[m]) //

H1(GF , A(F )) m //

H1(GF , A(F ))

0 //∏v A(Fv)/mA(Fv) //

∏vH

1(Gv, A[m]) //∏vH

1(Gv, A(Fv))m //

∏vH

1(Gv, A(Fv))

with exact rows, although at the moment we only know that the last two columnsare exact. After a little bit of comtemplation of the above diagram and the definitionof Sm(A/F ) it becomes clear what the appropriate finite/singular structure shouldbe.

2. SELMER GROUPS OF ABELIAN VARIETIES 77

Definition 6. We define the abelian variety structure on A[m] to be the fi-nite/singular structure σ defined by the local exact sequences

σv : 0→ A(Fv)/mA(Fv)→ H1(Gv, A[m])→ H1(Gv, A)[m]→ 0;

that is, we define

H1f (Gv, A[m]) = im

(A(Fv)/mA(Fv) → H1(Gv, A[m])

)and

H1s (Gv, A[m]) = im

(H1(Gv, A[m])→ H1(Gv, A)

).

The exact sequence

0→ A(Fv)/mA(Fv)→ H1(Gv, A[m])→ H1(Gv, A)

shows that these definitions are compatible. It is not immediately clear, however,that σ is really a finite/singular structure. We will prove this in the next lecture;for the time being we derive some consequences.

Proposition 2.1. If F is a number field, A/F is an abelian variety and A[m]is endowed with the abelian variety structure σ, then

H1f (GF ,Mσ) = Sm(A/F ).

Proof. This is just an easy diagram/definition chase in the above diagram.

Corollary 2.2. The mth Selmer group Sm(A/F ) is finite.

Proof. This follows immediately from Proposition 2.1 and Theorem 3.3 ofLecture 8.

Corollary 2.3 (The Weak Mordell-Weil Theorem). If A is abelian varietyand F is a number field, then A(F )/mA(F ) is finite for any integer m.

The full Mordell-Weil theorem states that A(F ) is a finitely generated abeliangroup; to derive this from Corollary 2.3 one must use the theory of height functions.(See [Si-AEC, Chapter 8, Sections 5 and 6] or [Si-H].)

Corollary 2.4. The m-torsion in X(A/F ) is finite for any m.Sm(A/F ) itself is effectively computable in many cases; see [Si-AEC, Chapter

10, Section 1] for the case of elliptic curves with 2-torsion. More recent work ofSchaffer and Grant has given good algorithms for computing it for abelian varietiesof dimension 1 or 2, independent of any other hypotheses.

To relate Sm(A/F ) to A(F )/mA(F ) requires knowledge of X(A/F ), however,and this group has proved much more resistant to computation. Note that Corol-lary 2.4 does not imply that X(A/F ) itself is finite. That X(A/F ) is finite is astandard conjecture; it was not known for a single abelian variety over a number fielduntil the work of Karl Rubin in 1986. He showed that it was finite for elliptic curvesover Q with complex multiplication and analytic rank 0 or 1. (See [Ru-TS].) It isnow known, using the work of Kolyvagin, Gross-Zagier, Murty-Murty and Bump-Friedberg-Hoffstein, for all modular elliptic curves over Q of analytic rank 0 or 1.(See [Ko], [GZ], [MM], [BFH].)

Note that knowledge of A(F )tors and A(F )/mA(F ) for a single m > 1 deter-mines the abelian group structure of A(F ). Alternately, knowledge of A(F )/mA(F )for a few values of m would suffice. Nevertheless, we know very little about the

78 9. LECTURE 9

rank of A(F ) in general. For example, although it is relatively easy to constructinfinite families of abelian varieties over Q of rank r and dimension g such that

r

g≥ 1

for all sufficiently large g, it is not known how to extend these constructions to giver

g≥ 1 + ε

for any ε.

CHAPTER 10

Lecture 10

1. Kummer Theory

1.1. The Ring of S-Integers. We present in this section an alternative (andmore precise) approach to Kummer theory over global fields. Let F be a numberfield and let S be a finite set of primes of F . (Later, when we assume that Scontains all of the archimedean places, there will be times when only the non-archimedean places are relevant and our notation may not reflect this. We leavethese issues to the reader to sort out. Also, we will use both p and v to denoteprimes, depending on the context.) Let OF be the ring of integers of F . We definethe ring of S-integers, OF,S , by

OF,S = x ∈ F | ordp(x) ≥ 0 for all p /∈ S.

That is, OF,S consists of those elements of F which are integral away from S andarbitrary in S.

Intuitively, then, OF,S is just the localization of OF at an element of F divisibleexactly by the primes in S. However, since OF need not be a principal ideal domain,it is not immediately clear that such an element exists. That they do in fact existfollows from the finiteness of the ideal class group. (See [La-ANT, Chapter 5,Section, pp. 100-101].) Specifically, let p be a prime in S. Then, since the idealclass group of OF is finite, pn is principal for some n, say pn = αpOF . Setting

α =∏p∈S

αp,

we see thatOF,S = OF [α].

This description gives an immediate determination of the primes of OF,S : theyare simply the primes of OF which are not in S. It also now follows immediatelythat the localizations and completions of OF and OF,S at primes p /∈ S agree.Essentially, then, OF,S has none of the information of OF about p ∈ S, but has allof it for p /∈ S.

The following theorem is one of two fundamental finiteness results for OF,S .Theorem 1.1 (Dirichlet Unit Theorem). O∗F,S is a finitely generated abelian

group. More precisely, O∗F,S/µ(F ∗) is free abelian of rank r1 + r2 + |S| − 1, wherer1 is the number of real embeddings of F and r2 is the number of pairs of complexconjugate embeddings.

Proof. See [La-ANT, Chapter 5, Section 1, Corollary of p. 105].


79

80 10. LECTURE 10

In order to state the other finiteness result, which is just a slight generalizationof the finiteness of the ideal class group, we first must define locally free modulesover OF,S . We will say that an OF,S-module V is locally free if

Vv = V ⊗OF,S (OF,S)v = V ⊗OF,S OFvis free as an OFv -module for all v /∈ S. (This is equivalent to V being free over allof the localizations of OF,S , rather than going all the way to the completions.) Wedefine the Picard group Pic(OF,S) of OF,S to be the set of isomorphism classes offinitely generated OF,S-modules which are locally free of rank 1. Pic(OF,S) is madeinto a group by tensor product; it is easy to see that the tensor product of twofinitely generated locally free of rank 1 OF,S-modules is of the same form. Inversesare given simply by duals,

V −1 = HomOF,S (V,OF,S),

as is easy to check. (For a more geometric treatment of the Picard group, see [Ha,Chapter 2, Section 6, esp. Remark 6.3.2 and pp. 143-146].)

Pic(OF,S) also has an interpretation in terms of ideals. Recall that a fractionalideal of OF,S is a non-zero finitely-generated sub-OF,S-module of F . The fractionalideals are made into a group by multiplication. This group is simply free abelianon the primes of F not in S. A fractional ideal is said to be principal if it is of theform xOF,S for some x ∈ F ∗.

Proposition 1.2. There is a canonical isomorphism between Pic(OF,S) andthe group of fractional ideals of OF,S modulo principal fractional ideals.

Proof. For a proof in the context of divisors and invertible sheaves, see [Ha,Chapter 2, Section 6, Proposition 6.15]. We give a purely algebraic proof.

There is a natural map from the group of fractional ideals to Pic(OF,S), simplyconsidering a fractional ideal as a locally free OF,S-module. The kernel of the mapis easily seen to be the principal fractional ideals, so we are left to show that themap is surjective.

Let V be a locally-free OF,S-module of rank 1. We claim that

V ⊗OF,S F ∼= F.

To see this, choose any prime p not in S and let R be the localization of OF,S at p.Then

V ⊗OF,S F ∼= (V ⊗OF,S R)⊗R F ∼= R⊗R F ∼= F,

since V is locally free. (Note that here we only used the fact that V is locally freeat one prime.)

Given this isomorphism, we tensor the injection

OF,S → F

with the flat (since it is locally free) OF,S-module V to get an injection

V → V ⊗OF,S F ∼= F,

which realizes V as a submodule of F , and thus, since it is finitely generated, as afractional ideal.

Corollary 1.3. Pic(OF,S) is finite.

1. KUMMER THEORY 81

Proof. It is a standard fact that the ideal class group of OF is finite; thus,Pic(OF ) is finite. The finiteness for Pic(OF,S) follows immediately, since under theisomorphism of Proposition 1.2 Pic(OF,S) can be identified with a subquotient ofPic(OF ).

In fact, one sees easily from the proof of the corollary that by taking S largeenough to kill everything in Pic(OF ) we get Pic(OF,S) = 0. More precisely, oneonly has to choose representative ideals of Pic(OF ) and let S contain every primedivisor of each representative.

1.2. The Kummer map. Let N ≥ 1 be an integer and let F be a numberfield such that µN (F ) ⊆ F ∗; we will just write µN for µN (F ). Let S be a set ofplaces of F containing all of the archimedean places and all places of F dividingN . We wish to define a natural map

O∗F,S/O∗NF,S → Hom(GabF,S , µN ).

Here GabF,S is the Galois group of the maximal abelian extension of F unramified

outside of S.Let u be in O∗F,S . Fix a ξ ∈ F such that ξN = u. (Normally making such a

choice is a very bad idea, but we will see that since µN ⊆ F ∗ the choice will notmatter.) Note that ξ is actually in the maximal abelian extension of F unramifiedoutside of S, since F (ξ)/F is cyclic of order dividing N with discriminant dividingNξN−1. Since ξ is a unit away from S, F (ξ)/F can only be ramified at places inS.

Define ϕu ∈ Hom(GabF,S , µN ) by

ϕu(g) =g(ξ)ξ.

Note first that since the conjugates of ξ are just ζξ for various ζ ∈ µN , ϕu reallydoes have image in µN . In fact, the map ϕu is independent of the choice of N th

root of u: the fact that µN ⊆ F ∗ implies that any g ∈ GabF,S acts trivially on µN , so

g(ζξ)ζξ

=ζg(ξ)ζξ

=g(ξ)ξ.

Thus we have a canonical map

O∗F,S → Hom(GabF,S , µN ),

and it is easily seen to be a homomorphism (using the fact that we can use any N th

root of u to define ϕu). Furthermore, u ∈ O∗F,S defines the trivial homomorphismon Gab

F,S if any only if

g(ξ) = ξ

for all g ∈ GabF,S , where ξN = u. By Galois theory this happens if and only if ξ ∈ F ;

thus we have defined a natural injection

O∗F,S/O∗NF,S → Hom(GabF,S , µN ).

The determination of the cokernel, however, is somewhat more difficult.

82 10. LECTURE 10

1.3. Preparation for Descent. In order to determine the cokernel of themap of the previous section we must introduce the notion of twisting locally freemodules. We begin with some easier examples.

Let K be any field and let L be a finite Galois extension, with Galois group Gof order d. Then L is a K-vector space of dimension d on which G acts K-linearly.Thus L is a K-representation space for G; in other words L is a K[G]-module. Infact, L is just the regular representation; that is, L is a free K[G]-module of rank1. This follows immediately from the normal basis theorem ([La-Al]).

In particular, suppose G is abelian and let ϕ : G→ K∗ be a character. Definethe ϕ-representation space of L by

Lϕ = x ∈ L | g(x) = ϕ(g)x for all g ∈ G.

Identifying L with K[G], we find that

Lϕ ∼=K[G]ϕ

=

∑g∈G

agg ∈ K[G] |∑

aghg =∑

ϕ(h)agg for all h ∈ G

=∑

agg ∈ K[G] | ag = ϕ(g−1)a1 for all g ∈ G

=a∑

ϕ(g−1)g,

so Lϕ is one dimensional over K.Restrict now to the case where K is a local field. Let OK and OL be the rings

of integers of K and L respectively. Then OL is a free OK-module of rank d as wellas a OK [G]-module. However, it is not always free of rank 1 over OK [G], since thenormal basis theorem does not apply to rings of integers. In fact, it can be shownthat if L/K is unramified, then there is a form of the normal basis theorem for thering of integers and OL ∼= OK [G] as OK [G]-modules.

Suppose that L/K is an arbitrary finite abelian extension of local fields andthat V is an OK-module which is free of rank 1. Let ϕ : G → O∗K be a characterof G. (Any character of G to K∗ must land in O∗K since its image is just roots ofunity, which are integral.) We define the ϕ-representation space of V by

V ϕ = (V ⊗OK OL)ϕ

= v ⊗ x ∈ V ⊗OK OL | v ⊗ g(x) = ϕ(g)(v ⊗ x) for all g ∈ G.

We know that V ϕ is torsion-free, since it is a submodule of V ⊗OK OL, which isfree of rank n over OK . Thus, since OK is a principal ideal domain, V ϕ is free. Infact, it is easy to show that V ϕ ⊗OK K ∼= Kϕ, so that the rank must be 1.

1.4. Descent for Global Fields. We now return to the case where F is aglobal field containing µN and S is a finite set of places containing the archimedeanplaces and the places dividing N . We will define a map

Hom(GabF,S , µN )→ Pic(OF,S).

We first work in slightly more generality. Let V be a locally free OF,S-moduleand let ϕ be in Hom(Gab

F,S , µN ). Since µN is finite, ϕ factors through some finite

1. KUMMER THEORY 83

extension L/F , which we can take to be abelian and unramified outside of S:

GabF,S

ϕ

// // Gal(L/F )

ϕyyssssssssss

µN

Let T be the set of places of L lying above places of S. Then OL,T is an OF,S-module. We define the ϕ-representation space of V by

V ϕ = (V ⊗OF,S OL,T )ϕ

= v ⊗ x ∈ V ⊗OF,S OL,T | v ⊗ g(x) = ϕ(g)(v ⊗ x) for all g ∈ G.For example,

OϕF,S = x ∈ OL,T | g(x) = ϕ(g)x for all g ∈ G.For general V , V ϕ is just

V ϕ = V ⊗OF,S OϕF,S .

Note first that the OF,S-modules V ϕ defined by different choices of splittingfields are canonically isomorphic, so that the above definition makes sense. Notealso that if V is locally free of rank 1, then so is V ϕ; this follows from the localcomputations of the previous section.

Thus, we can define the desired map

Hom(GabF,S , µN )→ Pic(OF,S)

by sending ϕ to OϕF,S . It is easy to check from the definitions that this is a homo-morphism, using the fact that the definition is independent of the choice of splittingfield.

Theorem 1.4. There is a natural exact sequence

1→ O∗F,S/O∗NF,S → Hom(GabF,S , µN )→ Pic(OF,S)[N ]→ 1

where the maps are as defined above.

Proof. We already know that the first map is injective. Next we show thatthe composition of the maps is zero. So let u be in O∗F,S and let ξ be an N th

root. We wish to show that OϕuF,S is isomorphic to OF,S . Set L = F (ξ) (whichis indeed abelian and unramified outside of S since µn ⊆ F ∗ and S contains theplaces dividing N) and note that

OϕuF,S = x ∈ OL,T | g(x) = ϕu(g)x for all g ∈ G

= x ∈ OL,T | g(x) =g(ξ)ξx for all g ∈ G

= x ∈ OL,T | g(x

ξ

)=x

ξfor all g ∈ G.

From this one sees easily that the map

OϕF,S → OF,Sgiven by multiplication by ξ−1 is an isomorphism.

To show that the last map is surjective we use the interpretation of Pic(OF,S)in terms of fractional ideals. So let I be a fractional ideal such that IN is trivial.Thus IN = αOF,S for some α ∈ F ∗. Choose an N th root ξ of α and set L = F (ξ).

84 10. LECTURE 10

Then L/F is unramified outside of S since α is the N th power of a fractional idealof OF,S . (This criterion for ramification becomes quite transparent when viewedlocally.) Define

ϕ : GabF,S → µN

by

ϕ(g) =g(ξ)ξ.

Then ϕ factors through Gal(L/K). We claim that OϕF,S ∼= I. To see this, first doa computation as above to show that

OϕF,S = ξOF,S .

To check that ξOF,S and I are isomorphic it is enough to check that they agree asfractional ideals of L, since L/K is unramified outside of S. But this is clear, whichestablishes surjectivity.

Finally, let ϕ : GabF,S → µN be such that OϕF,S is trivial in Pic(OF,S)[N ]. Thus

there is an isomorphismOϕF,S ∼= OF,S .

Let ξ be the element of OϕF,S corresponding to 1 ∈ OF,S under this isomorphism.One easily checks that ϕ is given simply by

ϕ(g) =g(ξ)ξ,

and that OϕF,S = ξOF,S . ξ may not be a unit, but we note that the fractional ideal ofOF,S it defines is trivial by hypothesis. Thus we can modify ξ by an element of OF,Sto get a unit ξ′ of OL,T . Now (ξ′)N ∈ O∗F,S and ξ′ yields the same homomorphismϕ, which completes the proof of exactness.

2. Cohomology of Abelian Varieties

2.1. Lang’s Theorem. Recall that if A is an abelian variety over a field K,we have an exact sequence of GK-modules

0→ A[N ]→ A(K) N−→ A(K)→ 0,

where N is any positive integer prime to the characteristic of K and A[N ] is theN -torsion in A(K). The long exact sequence in GK-cohomology yields an exactsequence

0→ A(K)/NA(K)→ H1(GK , A[N ])→ H1(GK , A(K))[N ]→ 0.

In order to understand this exact sequence it is important to gain more control overH1(GK , A(K)). An important result in this direction is Lang’s theorem.

Theorem 2.1 (Lang). Let A be a smooth, connected, commutative algebraicgroup over a finite field k. Then

H1(gk, A(k)) = 0.

Proof. We sketch the proof, the underlying idea being to “think geometri-cally”. For details see [Car, p. 32] or [PR, Chapter 6, Section 2].

Recall that we have evaluated the cohomology group H1(gk, A(k)) as

H1(gk, A(k)) ∼= A(k)/(Fr−1)A(k),

2. COHOMOLOGY OF ABELIAN VARIETIES 85

where Fr is the Frobenius automorphism x 7→ xq, q = |k|. So we want to show that

Fr−1 : A(k)→ A(k)

is surjective. To do this we want to interpret Fr geometrically and then see thatFr−1 is surjective.

We begin by considering the category of commutative k-algebras. If R is suchan algebra, then there is a natural k-linear automorphism of R

Fr : R→ R

given by x 7→ xq. Under the natural equivalence of categories between k-algebrasand affine schemes over Spec k this yields a Spec k-automorphism Fr for SpecR.

Let X now be an arbitrary scheme over Spec k. Let U = SpecR and V = SpecSbe two affine open subsets of X. In order to show that the automorphisms Frglue to an automorphism of X it will be enough to show that they agree for anyW = SpecRr = SpecSs, r ∈ R, s ∈ S. But this is clear, since Fr is given by thesame p-power map in both instances. Thus we obtain a natural automorphism Fr onany Spec k-scheme X. (See [Ha, Chapter 4, Section 2, p. 301] for a slightly differentapproach to defining Fr. Note that in our situation we simply have Xp = X.)

The first key point is to observe that Fr induces the zero map on differentials.The essential reason for this is that on the level of differentials, the map induced byFr involves multiplication by q, which kills any k-algebra; see [Si-AEC, Chapter2].

Now, let A be an algebraic group. Consider the morphism A→ A which “sendsx ∈ A to x−1 Fr(x).” (We leave it to the reader to write this down algebraically interms of the multiplication, inversion and Frobenius morphisms.) We will simplywrite this morphism as Fr−1. One now shows that the induced map on differentialsis just the sum of the maps induced by Fr and by −1; since Fr induces the zeromap and −1 induces isomorphisms, Fr−1 induces isomorphisms on differentials.Since A is smooth, [Ha, Chapter 3, Section 10, Proposition 10.4] now implies thatFr−1 is etale . Thus, by [Mi-EC, Chapter 1, Section 2, Theorem 2.12], it has openimage.

Further suppose that A is connected. Choose any a ∈ A(k) and consider themap ψa sending x ∈ A(k) to x−1aFr(x). This is just the translation of Fr−1 bya, and thus also has open image. In particular, the image of ψa intersects that ofFr−1, since any two opens on a connected variety intersect. Thus there must bex1, x2 ∈ A(k) such that

(Fr−1)(x1) = ψa(x2)

and a little algebra now shows that

(Fr−1)(x1x−12 ) = a.

Thus Fr−1 is surjective, as claimed.

Corollary 2.2. If A is an abelian variety over a finite field k, then

A(k)/NA(k) ∼= H1(gk, A[N ])

for any integer N .

Proof. This follows immediately from Theorem 2.1 and the Kummer se-quence.

86 10. LECTURE 10

2.2. Abelian Schemes over Local Fields. Let S be a scheme. An abelianscheme over S is a proper, smooth group scheme over S with connected geometricfibers. (As usual, see [Sh, Section 2] for the relevant diagrams.) In particular, forevery S scheme X, the set

A(X) def= HomS(X,A)

of X-valued points of A is a group; in fact, it turns out that the group structuremust be commutative. (See [Mi-AV, Section 20, Corollary 20.2]. If T = SpecR isaffine we will often just write A(R) for A(T ).) Note first that if T is an S-scheme,then

A(T ) = (A×S T )(T ),

as follows immediately from the universal property of the fiber product. Also, in thecase that S is the spectrum of a field K, then an abelian scheme over S is the sameas an abelian variety over S; thus any such abelian scheme is necessarily projective,and under any projective embedding it is easy to check that the above notionof SpecK-valued points corresponds with the usual notion of points in projectivespace.

Let K be a local field with ring of integers OK and residue field k. Let A beabelian variety over SpecK. We will say that A has good reduction if it extends toan abelian scheme over SpecOK ; that is, if there is some abelian scheme A′ overSpecOK such that

A′ ×SpecOK SpecK ∼= A.

A fundamental result for abelian varieties with good reduction is that all SpecOK-valued points extend to SpecK-valued points. Intuitively the idea is that any pointin projective space over K can be “scaled” to have at least one coordinate integral.We give a “coordinate-free” proof instead.

Proposition 2.3. Let A be an abelian scheme over OK . Then the natural mapA(OK)→ A(K) is a bijection.

Proof. The valuative criterion for properness ([Ha, Chapter 2, Theorem 4.7])tells us that given any diagram

SpecK //

A

SpecOK // SpecOK

there is a unique map SpecOK → A making the diagram commute. Phraseddifferently, the natural map A(OK)→ A(K) is an isomorphism, as required.

We can use the above result to gain more control over the structure of theK-valued points of an abelian variety. So let A be an abelian scheme over SpecOKand let m be the maximal ideal of OK . Then we have the equality

A(OK) = lim←−ν

A(OK/mν).

(In fact, this is true for any SpecOK-scheme X. In the affine case it follows im-mediately from the universal property of inverse limits and the general case followsfrom the affine case.) Thus A(OK) is a profinite group; in fact, it is a p-adic Lie


group. Define a subgroup A1 of A(OK) to be the kernel of the natural reductionmap

A(OK)→ A(k),so that there is an exact sequences

0→ A1 → A(OK)→ A(k)→ 0.

(The last map is surjective since OK → k is formally smooth.) It is a standard factfor abelian varieties that A1 can be expressed as the m-valued points of a certainformal group. (See [Si-AEC, Chapter 4, Section 1] for the case of elliptic curvesand [Co2] for the general case.)

Now, suppose that N is prime to the characteristic of k. Then the map

A1N−→ A1

is an isomorphism by [Si-AEC, Chapter 4, Proposition 2.3], so there is an exactcommutative diagram

0 // A1//

N

A(OK) //

N

A(k) //

N

0

0 // A1//

A(OK) //

A(k) //

0

0 // A(OK)/NA(OK) // A(k)/NA(k) // 0

Combined with Theorem 2.1 and Proposition 2.3 this gives us isomorphisms

A(K)/NA(K) ∼= A(OK)/NA(OK) ∼= A(k)/NA(k) ∼= H1(gk, A[N ]).

Now consider the above commutative diagram with K replaced by arbitraryfinite extensions L. If L is taken large enough to contain all of A[N ](K), then thekernels of the diagram yield an isomorphism

A[N ](K) ∼= A[N ](k).

In fact, this isomorphism is GK-equivariant, as is easy to check. In particular,A[N ](K) must be unramified as a GK-module.

2.3. Local Abelian Variety Structures. Let A be an abelian variety overa local field K and suppose that A has good reduction. Let N be an integer primeto the residue characteristic of K. We wish to compare the two exact sequences

0→ A(K)/NA(K)→ H1(GK , A[N ])→ H1(GK , A(K))[N ]→ 0

and0→ H1(gk, A[N ])→ H1(GK , A[N ])→ H1(IK , A[N ])gk → 0.

In the previous section we defined an isomorphism between the first terms in theexact sequences. We claim that this isomorphism lies in a commutative diagram

0 // A(K)/NA(K) //

H1(GK , A[N ]) // H1(GK , A(K))[N ] //

0

0 // H1(gk, A[N ]) // H1(GK , A[N ]) // H1(IK , A[N ])gk // 0

88 10. LECTURE 10

The proof of this is quite easy if A has been embedded in projective space overKin such a way that it “reduces” to an abelian variety over k; the main step becomesidentifying the map A(K) → A(k) with the usual “scaling and reduction” map.(See [Si-AEC, Chapter 7, Section 2].) We instead again present a “coordinate-free” proof.

Since A has good reduction it extends to an abelian scheme over SpecOK whichwe will also denote by A. Note that the group of K-valued points A(K) is naturallya GK-module even when thought of as HomSpecOK (Spec K, A), since any g ∈ GKinduces a SpecK-automorphism

Spec Kg−→ Spec K.

Any choice of projective embedding of A will identify this GK-module with theusual GK-module of K-valued points. Similarly A(k) has a natural structure ofgk-module. Considering A[N ] as a group scheme over SpecOK we can realize theGK-module A[N ](K) as the K-valued points of the group scheme A[N ].

The key to the coordinate-free proof of the commutativity of the above diagramis the following commutative diagram which comes immediately from the fact thatA is a group scheme over SpecOK (here L is any finite extension of K, OL is itsring of integers and ` is its residue field):

A(L)×A(L) // A(L)

A(OL)×A(OL) //

∼=

OO

A(OL)

∼=

OO

A(`)×A(`) // A(`)

Here the vertical maps are the natural maps induced by SpecL → SpecOL andSpec ` → SpecOL and the horizontal maps are multiplication; the indicated mapsare isomorphisms by Proposition 2.3. Passing to the limit over L and the inducedmultiplication-by-N diagram, we obtain a commutative diagram of “reduction”maps

A(K) N //

A(K)

A(k)N // A(k)

Now, note that in order to establish the desired commutativity it will be enoughto establish it for the diagram

A(K) //

H1(GK , A[N ](K))

A(k)

H1(gk, A[N ](k))inf // H1(GK , A[N ](K))


since the map of cokernels is simply the one induced by the first two vertical maps.(Here the map A(K) → A(k) is the reduction map and the other two unlabeledmaps are the boundary maps from the Kummer exact sequence.) We recall thedefinition of the Kummer maps: given P ∈ A(K), pick a Q ∈ A(K) such thatN ·Q = P . Then the image of P in H1(GK , A[N ]) is the cocycle which sends anyg ∈ GK to Qg −Q ∈ A[N ]. The definition is similar for A(k).

So let P be in A(K) and pick Q ∈ A(K) as above. By the commutative diagramfor multiplication by N , we find that N · Q = P , where P and Q are the images ofthe respective points in A(k). Thus P maps to the GK-cocycle

g 7→ Qg −Qand P maps to the gk-cocycle

g 7→ Qg − Q = Qg −Q.To complete the proof we now simply have to use the fact that A[N ](K) ∼= A[N ](k)as GK-modules. This shows that the two cocycles become the same after inflation,so the diagram really does commute.

2.4. Global Abelian Variety Structures. Let A be an abelian variety de-fined over a global field F . Let N be an integer and let M = A[N ] be the GF -module of N -torsion points. A fundamental fact about such an A is that it hasgood reduction at almost all places v of F ; that is, the base change

Av = A×SpecF SpecFvhas good reduction for almost all v.

We sketch the proof. For a few more details see [Mi-AV, Section 20, Remark20.9], or for more many more details and a “coordinate-free” approach see [Co].We need to show that there is some finite set S of places such that A extends toan abelian scheme over SpecOF,S . Recall that any abelian variety is projective.Fix an embedding of A into projective space. Here A is simply the zero set ofsome finite set of homogeneous polynomials. Since all that matters are the zerosof the polynomials, we can scale them so as to assume that all of the coefficientsare integral. Let A′ be the projective scheme the equations define over SpecOF .A′ is automatically proper and is easily seen to be flat, and its generic fiber isA. However, A′ need not yet be smooth, a group scheme or have geometricallyconnected fibers.

By [Ha, Chapter 3, Section 10, Theorem 10.2], to get smoothness it is enoughto remove all primes which have nonsingular fibers lying over them. A fiber is non-singular if and only if the Jacobian determinant vanishes modulo the correspondingprime. Since the fibers are all defined by the same equations and by assumptionthe Jacobian does not vanish in K, it can only vanish for finitely many fibers. Thuswe need only remove a finite number of primes to get smoothness.

Next consider the multiplication and inversion morphisms of A. These aregiven by ratios of homogeneous polynomials with coefficients which we can take tobe in OK , and thus will be regular for almost all primes. Thus by removing a finitenumber of primes we can get multiplication and inversion morphisms for A′ overOF,S , where S is the (finite set) of discarded primes.

Lastly, then, we must consider the geometrically connected fiber condition.Here one shows that the existence of the (group) identity section shows that con-nected implies geometrically connected, so that it will be enough to establish that

90 10. LECTURE 10

the fibers are connected. Here one again uses the fact that the generic fiber isconnected to show that almost all of the fibers are connected. See [ST, Section 1,Lemma 3] for the details.

Finally, then, we have constructed an abelian scheme A′ over SpecOF,S (forsome finite set of primes S) with generic fiber A. Base changing A′ to the appro-priate completion for v /∈ S shows that A has good reduction for v /∈ S, and thusthat A has good reduction almost everywhere.

Recall now that we defined the abelian variety structure on M by setting

H1f (Gv,M) = im

(A(Fv)/NA(Fv) → H1(Gv, A[N ])

)under the Kummer map. We now immediately see that this really is a finite/singularstructure; this follows from the computations of the previous section, which applywhenever A has good reduction at v and v does not divide N . In particular, thiscompletes the proofs of all of the results of Lecture 9, Section 2.2.

Note, however, that the abelian variety structure on M depends on the partic-ular abelian variety A and the isomorphism A[N ] ∼= M and thus is not an invariantof M as a GK-module.

CHAPTER 11

Lecture 11

1. L/K Forms

1.1. Definitions and Examples. Let L/K be a finite extension of fields. Weassume that L/K is Galois, although that will not really be necessary until later.Let V be an algebraic variety defined over K. We will say that another algebraicvariety W , defined over K, is an L/K form of V if

W ×SpecK SpecL ∼= V ×SpecK SpecL.

(Note that the specific isomorphism is not part of the data attached to an L/Kform; we merely assume that one exists.) An L/K form W of V , then, is simplyan algebraic variety which becomes isomorphic to K after base change to L; this,of course, does not imply that V and W are isomorphic over K. Such a W is oftencalled a twist of V . We define E(L/K, V ) to be the set of K-isomorphism classesof varieties W , defined over K, which are L/K forms of V . In fact, it is a pointedset, with the class of V itself being the distinguished element.

This notion of L/K form is not in any way restricted to the case of algebraicvarieties. Grothendieck was able to extend the key ideas to a much more generalsituation, replacing L/K with any faithfully flat quasi-compact map of schemesY → X, and replacing the above twists with certain fibered categories.

We will content ourselves with a few more examples. First, suppose that insteadof simply considering a variety V over K, we consider a base-pointed variety V overK; that is, we consider pairs (V, v), where v is a K-valued point of V . L/K forms ofV are now pairs (W,w) of a variety W defined over K and a K-valued point w of Wsuch that v maps to w under some isomorphism of VL and WL. There is an obviousnotion of isomorphism of base-pointed schemes, and we can define E(L/K, (V, v))to be the set of K-isomorphism classes of base-pointed varieties, defined over K,which are L/K forms of (V, v). In the exact same way one can define L/K formsfor algebraic groups defined over K, isomorphisms now required to be isomorphismas algebraic groups; or even L/K forms for algebraic varieties with chosen groupsof automorphisms.

Moving away from varieties, we note that a central simple algebra over K ofrank n2 which splits over L is nothing more than an L/K form of the central simpleK-algebra Mn(K). Thus E(L/K,Mn(K)) is simply the set of central simples K-algebras which split over L. As another example, let F be a number field and recallthe notion of locally free rank 1 OF,S-module introduced in Lecture 10, Section 1.3.If V is such a module, then for some N we have V N ∼= OF,S since Pic(OF,S) isfinite. Interpreting V as a fractional ideal, we have V N = αOF,S . It follows easilythat V becomes trivial over F (α1/N ), so that it is a F (α1/N )/F form for OF,S .


91

92 11. LECTURE 11

1.2. The Automorphism Group. We return now to the case of algebraicvarieties over K; almost everything we will say remains true in the other situationswe mentioned, but this case is the simplest to explain. So let V be a fixed varieityover K. If L is an extension of K, we define the automorphism group for V overL, AutL(V ), to be the group of automorphisms of V ×SpecK SpecL as an algebraicvariety over L. Thus AutL(V ) is just the set of isomorphisms

ϕ : V ×SpecK SpecL→ V ×SpecK SpecL

of SpecL-schemes. (Viewing Aut(V ) as a function of L one obtains a presheaf forthe etale topology on SpecK; in fact, it is actually a sheaf. See [Mi-EC, Chapter3, Section 4, pp. 134-135]. We will not really need this interpretation, however.)

The (not necessarily abelian) group AutL(V ) has a natural left Gal(L/K)-action. Specifically, given ϕ ∈ AutL(V ) and g ∈ Gal(L/K), we define gϕ ∈AutL(V ) by the following commutative diagram of isomorphisms:

V ×SpecK SpecL

1×g

ϕ// V ×SpecK SpecL

1×g

V ×SpecK SpecLgϕ//___ V ×SpecK SpecL

That is, gϕ is the inverse of the isomorphism 1× g followed by ϕ followed by 1× g.Here by 1× g we mean the map on V ×SpecK SpecL making the following diagramcommute:

V1 // V

%%KKKKKKKKKK

V ×SpecK SpecL1×g //___

πV

55kkkkkkkkkkkkkkkkk

πL

))SSSSSSSSSSSSSSSV ×SpecK SpecL

πV

77nnnnnnnnnnnnn

πL

''PPPPPPPPPPPPSpecK

SpecLg // SpecL

99ttttttttt

It is easy to check that this is really a group action, so that we have realizedGal(L/K) as a group of automorphisms of AutL(V ). Note that with this defini-tion the identity is Gal(L/K)-invariant. More generally, an important fact is thatthe subgroup of AutL(V ) of automorphisms fixed by every g ∈ Gal(L/K) can becanonically identified with AutK(V ). If one is willing to work simply with equa-tions this is just standard Galois theory, but the full statement is really that theetale presheaf defined above is actually a sheaf.

If AutL(V ) were abelian it would thus be a Gal(L/K)-module in our usualsense. For the next few sections we will amend the definition of Gal(L/K)-moduleto include modules whose underlying groups are non-abelian, so that any AutL(V )is a Gal(L/K)-module. However, to make the context in which AutL(V ) is beingused clear, we will write AutL(V ) for it when we want to regard it as a Gal(L/K)-module.

Let us now generalize the above situation somewhat. We continue to let V bean algebraic variety over K, and we now fix a finite extension L/K and an L/Kform W of V . We define IsomL(W,V ) to be the set of SpecL-isomorphisms from

2. COHOMOLOGICAL INTERPRETATIONS 93

W to V defined over L; that is, the set of isomorphisms

W ×SpecK SpecL→ V ×SpecK SpecL.

First note that IsomL(W,V ) is non-empty, since W is an L/K form for V . Ithas a natural left AutL(V ) action given by postcomposition: if ψ ∈ IsomL(W,V )and ϕ ∈ AutL(V ), then we define ϕψ to simply be the composition

W ×SpecK SpecLψ→ V ×SpecK SpecL

ϕ→ V ×SpecK SpecL.

It is easy to see that this action exhibits IsomL(W,V ) as a principal homogeneousspace for AutL(V ); that is, it is an AutL(V )-torsor.

IsomL(W,V ) in fact has even more structure. It has a natural Gal(L/K)-action: if ϕ ∈ IsomL(W,V ) and g ∈ Gal(L/K), we define gϕ ∈ IsomL(W,V ) to bethe composition

W ×SpecK SpecL

1×g

ϕ// V ×SpecK SpecL

1×g

W ×SpecK SpecLgϕ//___ V ×SpecK SpecL

We will write IsomL(W,V ) when we want to emphasize that IsomL(W,V ) should bethought of with its Gal(L/K)-action. In fact, the Gal(L/K)-actions on IsomL(W,V )and AutL(V ) are compatible with the action of AutL(V ) on IsomL(W,V ). We willmake this notion precise in the next section.

2. Cohomological Interpretations

2.1. G-action Torsors. In this section we wish to formalize the sort of situ-ation studied in the previous section. Let G be a group and let A be a finite groupon which G acts. Let X be a set on which both G and A act. We will say that X isa G-action torsor for A if it is a torsor for A in the usual sense and if the A-actionis compatible with the G-structures. That is, there is a map

A×X α−→ X

exhibiting X as a principal homogeneous space for A and such that

α(ga, gx) = gα(a, x)

for any a ∈ A, x ∈ X and g ∈ G.As a first example (with varieties rather than sets), it is easy to see that the

constructions of the previous section exhibit IsomL(W,V ) as a Gal(L/K)-actiontorsor for AutL(V ), where V is an algebraic variety over K and W is an L/K formfor V .

If G is a group and A is another group on which G acts, we define T (G,A) tobe the set of isomorphism classes of G-action torsors for A. T (G,A) is actually apointed set, with the trivial torsor A as the distinguished element.

The fundamental classification theorem for G-action torsors is the followingcohomological expression, which we have essentially seen before.

Theorem 2.1. If G is a group and A is a finite group with a G-action, thenthere is a canonical isomorphism of pointed sets

T (G,A) ∼= H1(G,A).

94 11. LECTURE 11

We state this for non-abelian A, even though we have not yet defined H1(G,A)when A is non-abelian, because it seems unnecessarily restrictive to assume thatA is abelian. In fact, the definition of H1 for non-abelian modules is essentiallyprecisely what makes the proof of the theorem work. We recall it briefly here; formore details on non-abelian cohomology, see [Se-LF, Appendix to Chapter 7] or[Se-GC, Chapter 1, Section 5] (although the reader should be forewarned that inthe second source Serre appears to change his conventions halfway through).

To define non-abelian cohomology we work with cocycles and we must be carefulas to what order operations occur in. All actions are left actions, and we will writeall G-actions as exponentials on the left. We define the set CrossHom(G,A) to bethe pointed set of maps

ϕ : G→ A

such thatϕ(g1g2) = ϕ(g1) g1ϕ(g2)

for all g1, g2 ∈ G. (We write A multiplicatively, of course.) The distinguishedelement is the trivial map sending all of G to 1 ∈ A.

CrossHom(G,A) has a natural leftA-action: for any a ∈ A and ϕ ∈ CrossHom(G,A)we define a ∗ ϕ ∈ CrossHom(G,A) by

a ∗ ϕ(g) = aϕ(g) g(a−1).

(Of course one must check that a ∗ ϕ is still a crossed homomorphism; this is aneasy computation. Note also that this action does not preserve the base point ofCrossHom(G,A).) We define H1(G,A) to be the set of orbits of CrossHom(G,A)under this action; it is a pointed set with the distinguished element given by theclass of the trivial crossed homomorphism. One checks immediately that if A isabelian this yields the usual definition of H1(G,A).

We now give the proof of Theorem 2.1.

Proof. Let (X,α) be a G-action torsor for A. Choose any x ∈ X. We willuse x to define a cocycle

cx : G→ A.

Specifically, we define cx(g) to be the unique element of A such that

α(cx(g), gx) = x.

cx(g) exists and is unique since α is a principal homogeneous action.We check that cx really is a cocycle. For g1, g2 ∈ G,

α(cx(g2), g2x) = xg1α(cx(g2), g2x) = g1x

α(g1cx(g2), g1g2x) = g2x

α(cx(g1), α(g1cx(g2), g1g2x)) = α(cx(g1), g2x)

α(cx(g1) g1cx(g2), g1g2x) = x.

On the other hand, by definition

α(cx(g1g2), g1g2x) = x,

socx(g1g2) = cx(g1) g1cx(g2)

which shows that cx is a cocycle.

2. COHOMOLOGICAL INTERPRETATIONS 95

We now must check that the cohomology class of cx is independent of the choiceof x ∈ X. So choose any other y ∈ X. Then there is a unique a ∈ A such that

y = α(a, x).

We have

α(cy(g), gy) = y

α(cy(g), α(ga, gx)) = α(a, x)

α(cy(g) ga, gx) = α(a, x)

α(a−1cy(g) ga, gx) = x.

Thuscx(g) = a−1cy(g) ga

which shows that cx and cy differ by a coboundary. This shows that we have awell-defined map

T (G,A)→ H1(G,A).We now define the inverse map. Given any cocycle ϕ : G → A, we will define

a corresponding G-action torsor. In order to get the notation straight we define anew set Pϕ to be identical to A as a set; we let

p : A→ Pϕ

be the natural map. We let b ∈ A act on p(a) ∈ Pϕ by

bp(a) = p(ba).

This clearly makes P into a principal homogeneous space for A. We put a twistedG-action on P by

gp(a) = p(gaϕ(g)−1).We leave it to the reader to check that this really is a G-action torsor; it is not

difficult. To show that the isomorphism class of Pϕ depends only on the cohomologyclass of ϕ, one shows that if ψ(g) = bϕ(g) g(b−1), then the map

Pψ → Pϕ

sending pϕ(a) to pϕ(ab) is an isomorphism of G-action torsors. We also leave it tothe reader to check that the maps are inverses.

2.2. Descent. We now apply the formalism of the previous section to theparticular case of interest. We again let V be an algebraic variety defined over Kand let W be an L/K form for V . Recall that IsomL(W,V ) is a Gal(L/K)-actiontorsor for AutL(V ).

This fact can be summarized by saying that we have defined a map

E(L/K, V ) ι−→ T (Gal(L/K),AutL(V ))

given byι(W ) = IsomL(W,V ).

We claim that ι is actually injective. To show this, suppose that we have anL/K form W for V and that ι(W ) is isomorphic to AutL(V ) as an AutL(V ) torsor.Let ϕ be the image in ι(W ) of 1 ∈ AutL(V ). A priori ϕ is a map W → V definedover L. In fact, since the Gal(L/K)-actions are compatible and Gal(L/K) actstrivially on 1, it also acts trivially on ϕ. This implies that ϕ is defined over K, andthus that W is the trivial L/K form. Of course, since neither the domain nor the

96 11. LECTURE 11

range of ι need be a group this does not suffice to prove injectivity; however, thegeneral case is quite similar.

The map ι is sometimes but not always surjective. In any case, we have thefollowing fact.

Proposition 2.2. There is a canonical injection

E(L/K, V ) → H1(Gal(L/K),AutL(V ))

and it is an isomorphism if and only if the natural injection

E(L/K, V ) → T (L/K,AutL(V ))

is an isomorphism.

Proof. This follows immediately from Theorem 2.1 and the above discussion.

If the injection of Proposition 2.2 is actually an isomorphism we will say that wehave descent for V . A fundamental theorem due to Weil is that if V is a projectivevariety then we do have descent.

Theorem 2.3 (Weil). Let V be a projective variety. Then there is a canonicalisomorphism of sets

E(L/K, V )∼=−→ H1(Gal(L/K),AutL(V )).

Proof. We sketch the main ideas. For the complete proof in the case of curves,see [Si-AEC, Chapter 10, Section 2, Theorem 2.2]. The general case is considerablyharder.

Given any cocycle c : Gal(L/K) → AutL(V ), we must define a twist V c of Vsuch that V c becomes isomorphic to V over L and such that V c gives the correctcohomology class.

To do this, consider V ×SpecK SpecL. We define a Gal(L/K) action on thisby first acting on SpecL via g in the usual way, and then acting on the “new”V ×SpecK SpecL by c(g). V c is then the quotient of V ×SpecK SpecL by thisGal(L/K)-action. The hard part, of course, is to show that this quotient can begiven the structure of an algebraic variety. In the case of curves one can use functionfields, but the general case is much more difficult.

There is a rather peculiar situation that arises when one has descent, in thatE(L/K, V ) depends only on AutL(V ) and not at all on the actual structures in-volved. This can be exploited in some circumstances. For example, the octoniansand G2 have the same automorphism groups, so given any twist for one structurewe ought to be able to obtain a corresponding twist for the other. Another exam-ple is central simple algebras and Brauer-Severi varieties; see [Se-LF, Chapter 10,Section 6].

As a final example, suppose that AutL(V ) is cyclic of order 2. (For example, Vcould be a general hyperelliptic curve.) A group of order 2 must have trivial Galoisaction, so E(L/K, V ) is independent of the variety V ! In particular, it is simply

H1(Gal(L/K),±1) = Hom(Gal(L/K),±1).

CHAPTER 12

Lecture 12

1. Torsors for Algebraic Groups

1.1. Algebraic Geometric Torsors. Fix a field K. If V is any variety overK we will write V for V ×SpecK SpecKs, where Ks is the separable closure of K.Let Γ be an algebraic group over K, with multiplication morphism µ:

Γ×SpecK Γµ−→ Γ.

We wish to combine our notion of Γ-torsor from Lecture 5, Section 2.3 with theideas from Lecture 11. We define an algebraic geometric torsor (AG torsor forshort) for Γ to be an algebraic variety X defined over K, together with a morphism

Γ×SpecK Xα−→ X,

which becomes isomorphic to the standard torsor (Γ, µ) over Ks; that is, there isan isomorphism

u : Γ→ X

fitting into a commutative diagram

Γ×SpecKs Γµ//

1×u

Γ

Γ×SpecKs Xα // X

We will say that X is trivial as an AG torsor for Γ is there is such a u which isdefinable over K. In fact, this happens precisely when X has a point defined overK; choosing this point to be the identity and using α to define multiplication easilyestablishes this. Note that in any case u is defined by finite data, so it must actuallybe defined over some finite extension L of K. Thus any AG torsor becomes trivialover some finite extension of K.

For any extension L/K let us define E(L/K, (Γ, µ)) to be the pointed set ofAG torsors for Γ which become trivial over L. Note that there is a natural mapping

E(L/K, (Γ, µ))→ E(L/K,Γ),

where E(L/K,Γ) is simply the set of L/K forms for Γ thought of as an algebraicvariety without its group structure. This map is easily seen to be surjective, but itneed not be injective; we will see an example of this later in the lecture.

Recall that we constructed an injection

E(L/K,Γ) → H1(Gal(L/K),AutL(Γ)),


97

98 12. LECTURE 12

where AutL(Γ) is the group of automorphisms of Γ as an algebraic variety over L.Composing with the forgetful map above defines a map

E(L/K, (Γ, µ))→ H1(Gal(L/K),AutL(Γ)).

This map is not terribly useful, however, as it is not even an injection. We wish todefine a similar map for E(L/K, (Γ, µ)) which is somewhat more germane to oursituation.

Proposition 1.1. Let (Γ, µ) be an algebraic group defined over a field K. Thenfor any L/K there is a natural injection of pointed sets

E(L/K, (Γ, µ)) i−→ H1(Gal(L/K),Γ(L)),

where Γ(L) is the group of L-valued points of Γ.We omit the proof. It is clear from this proof that our various maps fit into a

commutative diagram

E(L/K, (Γ, µ)) //

i

E(L/K,Γ)

ι

H1(Gal(L/K),Γ(L)) // H1(Gal(L/K),AutL(Γ))

where the bottom map is induced from the natural Gal(L/K)-module injection

Γ(L)→ AutL(Γ)

where elements of ΓL are viewed as translations.We will say that Γ/K satisfies the descent propety for torsors if the map of

Proposition 1.1 is an isomorphism.

1.2. Torsors for Abelian Varieties. Let K be a field and let A be an abelianvariety defined over K. We have the following fundamental theorem.

Theorem 1.2. An abelian variety A over a field K has the descent propertyfor torsors.

Proof. The proof of this is quite difficult. For the case of elliptic curves, see[Si-AEC, Chapter 10, Theorem 3.6].

It follows from this and Proposition 1.1 that for any finite L/K we have anidentification

E(L/K, (A,µ)) ∼= H1(Gal(L/K), A(L)).Since A(L) is abelian, H1(GK , A(L)) is an abelian group. The induced groupstructure on E(L/K, (A,µ)) is in fact just the Baer sum; see Lecture 5, Section2.2. However, it is fairly difficult to see that the necessary quotient operations arereally defined.

It is much easier to describe the involution induced on E(L/K, (A,µ)) by in-version in H1(GK , A(L)). This is done as follows: let ι : A → A be the inversionmorphism on A and let (X,α) be an algebraic geometric torsor for A. Then theinverse of X, say X∼, is simply X considered as a torsor with A-action α (ι× 1).Note, however, that X∼ and X give exactly the same element of E(L/K,A), sinceι is defined over K. Thus this gives an example of a case in which the forgetfulmap

E(L/K, (A,µ))→ E(L/K,A)

1. TORSORS FOR ALGEBRAIC GROUPS 99

is not injective. More generally elements of AutK(Γ) which are not translationscan be used to manufacture such examples.

1.3. The Shafarevich-Tate Group. Let F be a number field. We will nowrelate torsors to the Shafarevich-Tate group X(A/F ) defined in Lecture 9, Section2.1. More generally, we let S be any finite set of places of F and we define thepointed set

XS(A/F )to be the set of isomorphism classes of AG torsors for A which are trivial whenviewed over the local fields Fv for v /∈ S. Thus a non-trivial element in XS(A/F )is a torsor for A which has an Fv-rational point for every v /∈ S but has no F -rationalpoint.

Using Theorem 1.2 we have the following immediate cohomological interpreta-tion of XS(A/F ).

Proposition 1.3. There is an exact sequence

0→XS(A/F )→ H1(GF , A(F )) res−→∏v/∈S

H1(Gv, A(Fv)).

In particular, X∅(A/F ) can be identified with the Shafarevich-Tate group X(A/F ).

Proof. All that is necessary is, for every L/F , to identify the map

E(L/F, (A,µ))→ E(Lw/Fv, (A×SpecF SpecFv))

with the restriction map in cohomology, but this follows easily from the defintionsof the isomorphisms of Theorem 1.2.

Proposition 1.3 gives a geometric interpretation of the Shafarevich-Tate group;however, it is still extremely difficult (both in theory and in practice) to computeit. As an example of the results of one such computation we quote the followingtheorem which follows from careful consideration of the work of Kolyvagin andRubin.

Theorem 1.4. Let E be the elliptic curve over Q defined by the equation

x3 + y3 + 60z3 = 0,

with origin (−1, 1, 0). Then

X(E/Q) ∼= Z/3Z× Z/3Z.The non-trivial elements of X(E/Q) are represented by

Σ2 : 3x3 + 4y3 + 5z3 = 0;

Σ3 : 12x3 + y3 + 5z3 = 0;

Σ4 : 15x3 + 4y3 + z3 = 0;

Σ5 : 3x3 + 20y3 + z3 = 0;

together with the inverses Σ∼i .The proof of this theorem is quite difficult.The group X(A/F ) can be interpreted as an “obstruction” to the Hasse prin-

ciple. Recall that the Hasse-Minkowski theorem (see [Se-CA, Chapter 4]) statesthat a quadric curve has a point in Q if and only if it has a point in each Qp and inR. Unfortunately, this results fails for curves of higher genus; this is reflected in the

100 12. LECTURE 12

existence of non-trivial elements of X(A/F ). The curve Σ2 is a famous exampleof Selmer, which he used to demonstrate the explicit failure of the Hasse principlefor cubic curves.

CHAPTER 13

Lecture 13

1. The Picard Group of a Curve

1.1. The Picard Group. Let X be a smooth, projective, geometrically irre-ducible curve defined over a field K. For simplicity assume that K is perfect. Thegeometric irreducibility condition is equivalent to the function field K(X) being lin-early disjoint from K over K. In particular, we can identify G = Gal(K(X)/K(X))with GK .

We recall the definition of the Picard group of X, Pic(X). See [Ha, Chapter2, Section 6] for details. The K points of Pic(X) are easiest to describe: theyare in natural bijective correspondence with isomorphism classes of line bundles onX = X ×SpecK Spec K.

There is another interpretation of Pic(X)(K) which is somewhat simpler. Re-call that a divisor on X is a formal sum of points of X(K); that is, it is an elementof the free abelian group Z[X(K)]. Let K(X)∗ be the function field of X over K.There is a natural map

K(X)∗ div−→ Z[X(K)]

given by

f 7→∑

x∈X(K)

ordx(f)x,

where ordx(f) means the order of the zero or pole of f at x. Then

Pic(X)(K) ∼= Z[X(K)]/div(K(X)∗).

It can be shown that div(K(X)∗) lies in the kernel of the natural summationmap

Z[X(K)]→ Z,

so it descends to a map

Pic(X)(K)deg−→ Z

called the degree map. This agrees with the usual notion of degree for line bundles.We define

Pic0(X)(K)

to be the kernel of the degree map.It can be shown that Pic0(X) can be given the structure of an abelian variety

over K of dimension equal to the genus of X. Often Pic0(X) is called the Jacobianof X. See [Mi-JV].


101

102 13. LECTURE 13

If a rational function on X has no zeros or pole then it is a constant. Thusthere is an exact sequence

0→ K∗ → K(X)∗ div−→ Z[X(K)]→ Pic(X)(K)→ 0.

We rewrite this as a short exact sequence

0→ K(X)∗/K∗ → Z[X(K)]→ Pic(X)(K)→ 0;

it is actually an exact sequence of GK-modules (using our identification of G withGK), as is easy to see.

Proposition 1.1. There is an exact sequence

Z[X(K)]GK → Pic(X)(K)→ ker(BrK → H2(G, K(X)∗)

).

(Here Pic(X)(F ) is just the F -valued “points” Z × Pic0(X)(F ) of Pic(X). Notethat Z[X(K)]GK need not equal Z[X(K)].)

Proof. Taking the GK = G cohomology of

0→ K(X)∗/K∗ → Z[X(K)]→ Pic(X)(K)→ 0

yields an exact sequence

Z[X(K)]GK → Pic(X)(K)→ H1(G, K(X)∗/K∗).

Thus it will be enough to identify H1(G, K(X)∗/K∗) with the kernel of

H2(GK , K∗)→ H2(G, K(X)∗).

To do this we take GK cohomology of

0→ K∗ → K(X)∗ → K(X)∗/K∗ → 0,

yielding

H1(G, K(X)∗)→ H1(G, K(X)∗/K∗)→ H2(GK , K∗)→ H2(G, K(X)∗).

By Hilbert’s theorem 90 the first term vanishes, which gives the desired identifica-tion.

We now restrict to the case where K = F is a number field and relate the aboveresult to the Shafarevich-Tate group.

Proposition 1.2. Along with our running hypotheses, suppose further that Xhas an Fv rational point for all place v of F . Then the natural map

BrF → BrF (X)

is injective.

Proof. By global class field theory we have that BrF injects into ⊕v BrFv .The commutative diagram

BrF // _

BrF (X)

⊕v BrFv //

⊕v BrFv(X)

now shows that it will suffice to prove that each map BrFv → BrFv(X) is injective.

1. THE PICARD GROUP OF A CURVE 103

For this we will need to use some of the theory of Brauer groups of schemesdeveloped by Asumaya and Grothendieck. We have a diagram

SpecFv(X) //

SpecXv

xxppppppppppp

SpecFv

where Xv = X ×SpecF SpecFv and SpecFv(X) is the generic point of Xv. Thistranslates to a diagram

Br(Fv(X)) Br(Xv)oo

Br(Fv)

OO 88qqqqqqqqqq

of Brauer groups. Since Xv is smooth, a theorem of Grothendieck (see [Mi-EC,Chapter 4]) implies that the map

Br(Xv)→ Br(Fv(X))

is injective. Thus it will suffice to show that

Br(Fv)→ Br(Xv)

is injective. For this, we use the fact that Xv has an Fv-rational point. This impliesthat there is a section to the natural map

Xv → SpecFv,

and injectivity of Br(Fv)→ Br(Xv) follows immediately.

Corollary 1.3. Let X be a smooth, projective, geometrically irreducible curveover a number field F . Suppose further that X has an Fv-rational point for all placesv of X. Then any F -rational divisor class is represented by an F -rational divisor;that is, the natural map

Z[X(F )]GF → Pic(X)(F )is surjective.

Proof. By Proposition 1.1 it will suffice to prove that the map

BrF → H2(Gal(F (X)/F (X)), F (X)∗)

is injective. By Hilbert’s theorem 90 and the inflation-restriction sequence, thissecond group injects into BrF (X), so that the map BrF → BrF (X) factors throughthe map we are interested in. Proposition 1.2 thus completes the proof.

1.2. Applications to Shafarevich-Tate Groups. Let E be an elliptic curvedefined over a number field F . Let (X,α) be an AG torsor for E and choose a pointP0 ∈ X(F ). Then the map

Z[X(F )]→ E(F )defined by sending a point P ∈ X(F ) to the unique Q ∈ E(F ) for which

α(Q,P0) = P

descends to an isomorphism of varieties over K

Pic0(X) ∼= E.

104 13. LECTURE 13

(See [Si-AEC, Chapter 10, Theorem 3.8].)More generally, let us define Picm(X) to be the inverse image of m ∈ Z under

the degree map Pic(X)→ Z. Each of these become isomorphic to Pic0(X) over F .In fact, there is a natural action of Pic0(X) on Picm(X), by addition, which is easilyseen to be a GF -equivariant principal homogeneous action. Thus each Picm(X) isan AG torsor for E. In fact, the natural map

X → Pic1(X)

sending a point of X to its divisor class is an isomorphism of AG torsors for E. Wewill write Xm for Picm(X).

Now further suppose that X is everywhere locally trivial. Then each Xm is aswell, simply by “multiplying” the local point on X by m. Thus each Xm yields anelement ξm ∈ X(E/F ). We claim that ξm is just mξ1. To prove this one mustidentify the Baer sum of Xm and Xm′ with Xm+m′ . Since for once we actually havea candidate variety for Xm ⊕Xm′ it is possible to show that Xm ⊕Xm′

∼= Xm+m′

by showing that Xm+m′ satisfies the appropriate universal property; we omit thedetails. In any event, this fact immediately implies our claim.

We know that X(E/F ) is a torsion group, so there is some n > 0 such thatnξ1 = 0. Thus ξn = 0, so Xn

∼= E as E-torsors. In particular, Xn has an F -rationalpoint. Thus X has an F -rational divisor class of degree n. By Corollary 1.3 X hasan F -rational divisor. Choose one such and call it D.

From here we will begin to omit some more details. Suppose that n ≥ 3. (Then = 2 case is quite interesting but doesn’t work for our present arguments.) LetOX(D) be the invertible OX -module of rational functions with poles bounded byD. Then by Riemann-Roch H0(X,OX(D)) has dimension n over F , since X hasgenus 1. Thus we get a map

X → Pn−1F ,

where the Pn−1F is the space of linear functionals on H0(X,OX(D)), given by send-

ing x ∈ X to the “evaluation at x” functional. (The range is projective space overF since D is defined over F .) In fact, one can show that this is actually a closedimmersion with projectively normal image. In particular, we can realize X as acurve in Pn−1

F of genus 1 and degree n.Now let us consider all elements of order n in all Shafarevich-Tate groups of all

elliptic curves over F . Our above construction tells us that every such cohomologyclass is represented by a smooth, projectively normal curve of genus 1 and degree nin Pn−1

F . There happens to exist a quasi-projective variety Cn, called the Chow va-riety, defined over Q, whose F -valued points classify all projectively normal curvesin Pn−1

F of degree n and genus 1. In fact, there are “Chow coordinates” givingan explicit embedding of Cn in some large projective space. Thus each element ofX(E/F ) for some E/F yields a point in Cn and thus a point in this large projec-tive space. If we could somehow bound the heights of such points in terms of thecoefficients of E (as from the example of Lecture 12, Theorem 1.4 may at least seema reasonable hope), the fact that there are only a finite number of F -rational pointsof projective space with bounded height would show that X(E/F ) was finite forall elliptic curves E defined over F .

2. DIRECT LIMITS OF SELMER GROUPS 105

2. Direct Limits of Selmer Groups

2.1. Passage to the Limit. Let B be an abelian variety over a field F andlet N be a positive integer. (The reason for the sudden shift in notation will becomeapparent in the next section.) We consider the N -torsion B[N ] as a GF -modulewith its abelian variety finite/singular structure induced by B. (See Lecture 9,Section 2.2.) Thus we have defined (finite) Selmer groups

H1f (GF , B[N ]) = SN (B/F ).

Now, choose another positive integer M . Then the inclusion B[N ] → B[NM ]induces a map

H1f (GF , B[N ])→ H1

f (GF , B[NM ]).(To check that this map exists one just needs to check that the local maps

H1(Gv, B[N ])→ H1(Gv, B[NM ])

send finite parts to finite parts for all v; this follows easily from the definition of theabelian variety structure.) Thus we can form the direct limit (over the multiplicativesystem of positive integers)

H1f (GF , Btors)

def= lim−→N

H1f (GF , B[N ]).

In other notation, this is

S∞(B/F ) def= lim−→N

SN (B/F ).

Recall that there is an exact sequence

0→ B(F )/NB(F )→ SN (B/F )→X(B/F )[N ]→ 0.

Writing B(F )/NB(F ) as B(F )⊗ 1NZ/Z we have a natural commutative diagram

0 // B(F )⊗ 1NZ/Z

//

SN (B/F ) //

X(B/F )[N ] //

0

0 // B(F )⊗ 1NMZ/Z

// SNM (B/F ) //X(B/F )[NM ] // 0

Taking the direct limit yields the following proposition.Proposition 2.1. There is an exact sequence

0→ B(F )⊗Q/Z→ S∞(B/F )→X(B/F )→ 0.

This construction yields a covariant function in B/F .

2.2. The Shafarevich-Tate Conjecture. We now state the Shafarevich-Tate conjecture, which we have eluded to many times before.

Conjecture 2.2 (Shafarevich-Tate). X(B/F ) is finite for any abelian varietyB defined over a number field F .

Recall that we already know that X(B/F ) is torsion (since it is a subgroup ofthe torsion group H1(GF , B(F ))). Further, we know that X(B/F )[N ] is finite forall N , since SN (B/F ) is. We can break the Shafarevich-Tate conjecture up intotwo somewhat distinct parts.

Conjecture 2.3 (Shafarevich-Tate).

106 13. LECTURE 13

(1) The p-primary component X(B/F )p of X(B/F ) has trivial p-divisiblepart. (We already know that it has finite corank since X(B/F )[p] isfinite.)

(2) X(B/F )[p] = 0 for almost all p.Let E be an elliptic curve over Q. There are two main approaches to the

Shafarevich-Tate conjecture. The first, which one might call the classical Fermatdescent method, is to fix a prime p and to try to compute X(E/Q)p. This some-times works for small p, but can yield neither (1) nor (2) of Conjecture 2.3. Thesecond method, due to Kolyvagin and Rubin, instead is quite good at showing thatX(E/Q)p is trivial for almost all p simultaneously.

We now describe a contravariant limit of the Selmer groups, which works wellunder the assumption that X(B/F ) is finite. We use the injection

Q/Z→ R/Z

to get a mapB(F )⊗Q/Z→ B(F )⊗ R/Z.

Next we push out our exact sequence

0→ B(F )⊗Q/Z→ S∞(B/F )→X(B/F )→ 0;

that is, we define a group S†(B/F ) to be the unique group making

0 // B(F )⊗Q/Z //

S∞(B/F ) //

X(B/F ) // 0

0 // B(F )⊗ R/Z // S†(B/F ) //X(B/F ) // 0

a commutative diagram with exact rows. Since B(F ) is finitely generated, B(F )⊗R/Z is a compact topological group. Since X(B/F ) is assumed to be finite,S†(B/F ) has a natural structure of compact topological group. Thus its dual

Σ(B/F ) = Hom(S†(B/F ),R/Z)

is a finitely generated abelian group, and Σ is contravariant in B/F .

3. Conditions on Galois Representations

3.1. Conditions. In this section we will specify the sorts of Galois represen-tations which we will be considering. Fix a prime p > 2 and let A be a local, finite,faithfully flat Zp-algebra. (Thus A is free of finite rank over Zp.) For example, Acould be the ring of integers of a finite extension of Qp. We will write m for themaximal ideal of A and k for the residue field A/m.

Let F be a number field with our usual conventions and let Σ be a finite set ofplaces of F . We will always assume that Σ contains p and all archimedean places.Let FΣ be the maximal extension of F in the fixed algebraic closure F which isunramified outside of Σ and set GF,Σ = Gal(FΣ/F ).

Let H be a free A-module of rank 2 with an A-linear action of GF,Σ; thusH is an A[GF ]-module which is unramified outside of Σ. We will write ρ for therepresentation

ρ : GF,Σ → AutA(H) ∼= GL2(A)and ρ for the residual representation

ρ : GF,Σ → Autk(H ⊗A k) ∼= GL2(k).

3. CONDITIONS ON GALOIS REPRESENTATIONS 107

We are usually only interested in ρ up to conjugation in GL2(A), so that the choiceof basis is not important.

We wish to specify the determinant of ρ. Recall that the p-cyclotomic characterχ is the map

χ : Gal(F (ζp∞)/F )→ Z∗p

giving the action of the Galois group on the group of p-power roots of unity. Weconsider χ as a map to A∗ by composing with the structure map Z∗p → A∗. Werequire that the determinant

det ρ : GF,Σ → A∗

of ρ equal χ.This requirement has some interesting consequences. First, let λ be a place of

F not in Σ. Then we can choose a Frobenius element Frλ for λ in GF,Σ. We willcall

Tλ = tr ρ(Frλ) ∈ Athe λth Hecke operator for reasons that will become clear later. −Tλ is the coefficentof x in the characteristic polynomial

x2 − Tλx+ det ρ(Frλ)

of Frλ. By the p-cyclotomic determinant condition

det ρ(Frλ) = χ(Frλ) = NF/Q(λ),

since Frλ is just NF/Q(λ) powers on residue fields and thus on roots of unity. Thuswe have an Eichler-Shimura type relation

Fr2λ−Tλ Frλ +NF/Q(λ) = 0.

Next, assume that F has at least one real embedding. Then let c be any choiceof complex conjugation for this embedding; equivalently c is the non-trivial elementof a decomposition group of a real valuation. Since c has order 2, ρ(c) has order2 in GL2(k). However, χ(c) = −1 since complex conjugation acts as inversion onroots of unity, so det ρ(c) = −1. It now follows from an easy computation that ρ(c)is not a scalar matrix, so ρ is called an odd representation.

3.2. An Example : Tate Modules of Elliptic Curves. Let E be an ellipticcurve over F . Let Σ consist of p, the archimedean places and those places at whichE/F has bad reduction. Take A = Zp and set

H = lim←−ν

E[pν ],

the maps of the inverse system being

E[pν+1]p−→ E[pν ].

Then by [Si-AEC, Chapter 3, Proposition 7.1] H is free of rank 2 over Zp. H hasa natural GF action which by [Si-AEC, Chapter 7, Proposition 4.1] is unramifiedoutside of Σ. Further the Weil pairing shows that the determinant

GF,Σ → ∧2AH

is just the p-cyclotomic character χ. (See [Si-AEC, Chapter 3, Proposition 8.3].)So H satisfies all of our conditions. The Hecke operators Tλ for λ /∈ Σ are computedin [Si-AEC, Chapter 5, Proposition 2.3]; they are given by

Tλ = 1 + NF/Q(λ)− |E(kλ)| ∈ Z ⊆ Zp,

108 13. LECTURE 13

where kλ is the residue field of Fλ. (Note that this is independent of the choice ofp.)

3.3. Principal Polarizations. We continue to let A and H be as before. Inthis section we consider pairings

(·, ·) : H ⊗Zp H → Zp(1).

We say that (·, ·) is GF,Σ-equivariant if

(gm, gn) = g(m,n)

for all m,n ∈ H and g ∈ GF,Σ. We say that (·, ·) is A-Hermetian if

(am, n) = (m,an)

for all m,n ∈ H and a ∈ A. Finally, we say that (·, ·) is skew-symmetric if

(m,n) = −(n,m)

for all m,n ∈ H.Lemma 3.1. Let

(·, ·)ψ : H ⊗Zp H → Zp(1)

be a GF,Σ-equivariant, A-Hermetian, skew-symmetric self-pairing on H. Let

ψ : H → HomZp(H,Zp(1))

be the induced map, given by sending m ∈ H to the map sending n ∈ H to (m,n)ψ.Then if ψ is an isomorphism of Zp-modules, it is also an isomorphism of A[GF,Σ]-modules.

Proof. Note that Zp acts on the Zp(1) (or equivalently on theH) in HomZp(H,Zp(1)),while A acts only on the H and GF,Σ acts on both factors in the usual adjoint way.The lemma now follows immediately from some easy computations.

Lemma 3.2. An A-Hermetian, skew-symmetric pairing

(·, ·) : H ⊗Zp H → Zp(1)

is automatically GF,Σ-equivariant.

Proof. A pairing as above is by definition the same as a Zp-homomorphism

∧2AH → Zp(1),

where ∧2AH is the second exterior power of H. Furthermore, it follows immediately

from the definitions that GF,Σ-equivariance of the pairing is the same as GF,Σ-equivariance of the map on the exterior power. But here GF,Σ-equivariance isautomatic: by the p-cyclotomic determinant condition, GF,Σ acts on ∧2

AH via thecyclotomic character, which is exactly how GF,Σ acts on Zp(1). Zp-linearity nowcompletes the proof.

Lemma 3.3 (Schur’s Lemma). Let Π be a profinite group and ρ : Π→ GLn(R)a continuous homomorphism, where R is a complete local ring with residue field k.Suppose that the residual representation ρ is absolutely irreducible. Then the onlymatrices which commute with the image of ρ are the scalar matrices.


Proposition 3.4. Let

(·, ·)ψ : H ⊗Zp H → Zp(1)

be a GF,Σ-equivariant, A-hermetian, skew-symmetric pairing on H. Let

ψ0 : ∧2AH → Zp(1)

be the induced map on the exterior square. Then (·, ·)ψ is perfect if and only ifHomZp(∧2

AH,Zp(1)) is free of rank 1, generated by ψ0.

Proof. First suppose that (·, ·)ψ is perfect. Let

ψ : H∼=−→ HomZp(H,Zp(1))

be the map induced by (·, ·)ψ and let

f : H → HomZp(H,Zp(1))

be any other GF,Σ-equivariant homomorphism induced from some map

f0 : ∧2AH → Zp(1).

Thenψ−1 f : H → H

is a GF,Σ-equivariant endomorphism of H. In other words, the matrix of ψ−1 f commutes with the image of GF,Σ in AutA(H). Thus, by Lemma 3.3, ψ−1 f is a scalar. So f = aψ for some a ∈ A, so f0 = aψ0. Thus ψ0 is an A-generator of HomZp(∧2

AH,Zp(1)). (The above argument in fact shows that everyGF,Σ-equivariant, A-Hermetian self-pairing from H to Zp(1) is skew-symmetric.)

It remains to show that HomZp(∧2AH,Zp(1)) is free of rank 1 over A. This, how-

ever, is clear from the fact that ∧2AH is free of rank 1 overA, so that HomZp(∧2

AH,Zp(1))has the same rank over Zp as A does.

For the converse, we suppose that ψ0 is an A-generator of the free rank 1 A-module HomZp(∧2

AH,Zp(1)). The GF,Σ-action plays no role here, so we choose anA-basis x, y of H and a Zp-generator ζ of Zp(1). Now

∧2AH = (x ∧ y)A

andZp(1) = ζZp.

ψ0 : (x ∧ y)A→ ζZp can thus be simply written as

(x ∧ y)A 7→ ζ tr(a),

wheretr : A→ Zp

is a generator of the free rank 1 A-module Hom(A,Zp). (tr is called a Gorensteintrace.)

We now recover (·, ·)ψ by (x, x)ψ = 0, (y, y)ψ = 0 and (x, y)ψ = ζ tr(1). Themap

ψ : xA⊕ yA→ HomZp(yA, ζZp)⊕HomZp(xA, ζZp)can now be computed as

(xa, yb) 7→ (a · tr, b · tr),and this is an isomorphism since tr is an A-basis for HomZp(A,Zp). This completesthe proof.

110 13. LECTURE 13

Definition 7. AGF,Σ-equivariant, A-Hermetian, skew-symmetric perfect pair-ing

H ⊗Zp H → Zp(1)will be called a principal polarization or simply a polarization.

In fact, it follows from Proposition 3.4 that a principal polarization exists ifand only if A is Gorenstein. See [Ti, Section 1].

3.4. Moment Representations. We let A and H be as before. For anyν > 0, Symν

AH is a free A-module of rank ν+1; however, in order to get the theoryto work properly we need to twist the symmetric powers. Precisely, if ν = 2m+ 1is odd we define

W ν = SymνAH ⊗Zp Zp(−m)

and if ν = 2m+ 2 is even we define

W ν = SymνAH ⊗Zp Zp(−m).

(Note that W 1 and W 2 are still just Sym1AH and Sym2

AH respectively.) So W ν isfree of rank ν + 1 over A.

We will write

Wν = (W ν)∗ = Hom(W ν , µp∞) = Hom(SymνAH,µp∞(m))

for the Cartier dual of W ν . We will also write ρν for the representation

GF,Σ → AutA/J(Wν)

induced by H and ρJν for the restriction of ρν to Wν [J ].If H has a principal polarization, it is not difficult to show that

W2∼= End0

A(H)⊗Zp Qp/Zp,

where End0A(H) is the module of A-linear endomorphisms of H of trace 0. For

relations of this fact to the deformation theory of Galois representations see [Maz,Chapters 4 and 5].

3.5. Properties of Representations. In this section we summarize someproperties of representations which will be useful to us. Let A be a ring as before.Let W ∗ be a free A-module of finite rank n with an A-linear action of G = GF,Σ.Let W = Hom(W ∗, µp∞) be its Cartier dual. W is made into an A-module by theaction of A on W ∗. For any ideal J of A of finite index we define

W [J ] = ∩α∈JW [α],

where W [α] is the kernel of multiplication by α on W . Note that W [J ] andW ∗/JW ∗ are Cartier dual.

Lemma 3.5. Let α ∈ A be a non-zero divisor. Then α divides some power of lin A and αA has finite index in A.

Proof. Since A is finite over Zl and α is a non-zero divisor, α necessarilysatisfies some monic linear equation of the form

αn + an−1αn−1 + · · ·+ a1α+ a0 = 0

with ai ∈ Zl and a0 6= 0. Thus α divides a0, which proves the first statement. Sincea0A ⊆ αA and a0A visibly has finite index in A (as A is free of finite rank over Zl),the second statement is now clear as well.


Lemma 3.6. Let W be an A-module and let t be a non-zero element of Wannihilated by some power of l. Then there exists α ∈ A such that αt 6= 0 andαt ∈W [m].

Proof. Since t is l-power torsion, there is a largest power ln of l such thatt0 = lnt 6= 0; thus t0 ∈ W [l]. Note that W [l] is a module over the artinian ringA/lA. Let α1, . . . , αm be generators of m in A; they also generate the maximal idealof A/lA and are therefore nilpotent in this ring. It follows that some power of α1

annihilates t0; let n1 be the smallest such power, and set t1 = αn1−11 t0. Continuing

in this way, we obtain a non-zero element tm = αt where α = lnαn1−11 · · ·αnm−1

m .This tm is killed by every generator of m, so it is clearly in W [m].

Lemma 3.7. Let W be a Zl-torsion A-module. If W [m] = 0, then W = 0.

Proof. Suppose that W 6= 0 and let t be a non-zero element of W . ByLemma 3.6 there is some α ∈ A such that αt is non-zero and annihilated by m.Thus W [m] 6= 0, which yields the desired contradiction.

CHAPTER 14

Lecture 14

1. Structures on Galois Representations

1.1. Finite/Singular Structures. Let p be a prime number. Fix a globalfield F and a finite set Σ of places of F containing p and all archimedean places. LetA be as in Lecture 13, Section 3.1 and let W ∗ be a free A-module of rank n with anA-linear action of GF,Σ, where GF,Σ is the Galois group of the maximal extensionof F unramified outside of Σ. We can realize this action as a representation

ρ∗ : GF,Σ → AutA(W ∗) ∼= GLn(A),

the last isomorphism given by any choice of any A-basis for W ∗.Since W is free of finite rank over Zp, its Cartier dual W = Hom(W ∗, µp∞) is

a discrete, torsion A-module with an A-linear GF,Σ-action. We will write

ρ : GF,Σ → AutA(W )

for the representation of GF,Σ on W .Let m be the maximal ideal of A and let k be the residue field A/m. We make

the assumption that the residual representation

ρ∗ : GF,Σ → Autk(W ∗ ⊗A k)

is absolutely irreducible; this just means that the representation

ρ∗ ⊗k k : GF,Σ → Autk(W ∗ ⊗A k)

is irreducible. This is in fact equivalent to ρ : GF,Σ → Autk(W [m]) being absolutelyirreducible, as is easily seen.

Suppose that we are given a finite/singular structure on W . That is, for allplaces v of F we have chosen subgroups

H1f (Gv,W ) ⊆ H1(Gv,W )

which agree with the standard choices for almost all v. This in turn induces aCartier dual finite/singular structure on W ∗, so that H1

f (Gv,W ) and H1f (Gv,W ∗)

are orthogonal complements under Tate local duality for all v.Let J be an ideal of A such that A/J is finite, and thus artinian. We will write

W [J ] for the “kernel” of multiplication by J on W ; that is,

W [J ] = ∩α∈JW [α],

where W [α] is the kernel of multiplication by α on W . One can use the givenfinite/singular structures on W to induce structures on W [J ] as follows: one simply


113

114 14. LECTURE 14

defines H1f (Gv,W [J ]) to be the unique subgroup of H1(Gv,W ) making the diagram

H1f (Gv,W [J ]) //

_

H1f (Gv,W )

_

H1(Gv,W [J ])j// H1(Gv,W )

cartesian. That is, one sets

H1f (Gv,W [J ]) = h ∈ H1(Gv,W [J ]) | j(h) ∈ H1

f (Gv,W ).

Of course, we must show that this definition really does give a finite/singularstructure on W [J ]. This follows from the commutative diagram with exact columns

0

0

H1(gv,W [J ])

// H1(gv,W )

H1(Gv,W [J ])

// H1(Gv,W )

Hom(Iv,W [J ]) // Hom(Iv,W )

Specifically, an easy diagram chase shows that the middle square is cartesian; thusif W is unramified and has the standard finite/singular structure at v (both ofwhich are true for almost all v by assumption), then this identifies our choice ofH1f (Gv,W [J ]) with the standard choice H1(gv,W [J ]).

In addition, since W ∗/JW ∗ is Cartier dual to W [J ], we get an induced fi-nite/singular structure on W ∗/JW ∗. We will always assume that all W [J ] andW ∗/JW ∗ have finite/singular structures induced from some fixed structure on Win this way.

We will writeρJ : GF,Σ → AutA/J(W [J ])

andρ∗J : GF,Σ → AutA/J(W ∗/JW ∗)

for the representations induced by ρ and ρ∗.The Selmer groups of W and W [J ] are related in a particularly simple way.Lemma 1.1. For any ideal J of A of finite index we have

H1(GF ,W [J ]) = H1(GF ,W )[J ]

andH1f (GF ,W [J ]) = H1

f (GF ,W )[J ].

Proof. Let 0 6= α ∈ J and consider the exact sequences

0→W [α]→W → αW → 0

and0→ αW →W →W/αW → 0.

1. STRUCTURES ON GALOIS REPRESENTATIONS 115

The rows and columns in the commutative diagram

(W/αW )GF

(αW )GF // H1(F,W [α]) // H1(F,W ) //

α

''NNNNNNNNNNNH1(F, αW )

H1(F,W )

are exact. One cheks that the absolute irreducibility ofW [m] implies that (αW )GF =(W/αW )GF = 0. Thus by the above diagram H1(F,W [α]) injects into H1(F,W )and identifies with H1(F,W )[α]. This gives the first equality in the case that J isprincipal, and since A is noetherian and WGF = 0, Lemma 1.2 below finishes theproof.

For the second equality, we just use the commutative diagram with exactcolumns

0

0

H1f (GF ,W [J ])

// H1f (GF ,W )[J ]

H1(GF ,W [J ])

∼= // H1(GF ,W )[J ]

⊕vH1s (Gv,W [J ]) // ⊕vH1

s (Gv,W )[J ]

A diagram chase and the definition of the induced finite singular structure on W [J ]show that the top square is cartesian, and thus that the first horizontal arrow is anisomorphism.

Lemma 1.2. Let M and N be submodules of a G-module W . Suppose thatH1(G,M) and H1(G,N) inject into H1(G,W ). If also WG = 0, then H1(G,M ∩N) injects into H1(G,W ) and is equal to the intersection of H1(G,M) and H1(G,N).

Proof. Let Z, ZM , ZN , B, BM , BN be the cocycles and coboundaries forW , M and N respectively. ZM and ZN are naturally submodules of Z, and theinjectivity assumption means exactly that BM = ZM ∩B and BN = ZN ∩B. (Thatis, every coboundary gw−w which has image in M or N has a representative withw in M or N respectively.) Thus

H1(G,M)∩H1(G,N) = ZM/(ZM ∩B)∩ZN/(ZN ∩B) = ZM ∩ZN/(ZM ∩ZN ∩B).

Clearly ZM ∩ ZN is just the cocycles for M ∩ N , so it remains to show thatZM ∩ZN ∩B are the coboundaries. It is clear that it contains the coboundaries, sosuppose that we have w ∈ W such that gw − w ∈ M ∩N for every g. Then, sinceBM = ZM ∩B and BN = ZN ∩B, we can find m ∈M and n ∈ N such that

gw − w = gm−m = gn− n

116 14. LECTURE 14

for all g. But theng(m− n) = m− n,

so m − n ∈ WG. By assumption, then, m − n = 0, so m = n ∈ M ∩ N . Thiscompletes the proof.

1.2. The ∆-Hypothesis. We keep the notation and assumptions of the pre-vious section. Since W [J ] is finite (since W has finite corank and J has finite indexin A), ρJ factors through some finite extension FJ of F ; that is, ρJ induces aninjection

ρJ : Gal(FJ/F ) → AutA/J(W [J ]).

We will write ∆J for Gal(FJ/F ).The natural exact sequence

1→ GFJ → GF → ∆J → 1

induces an inflation/restriction sequence

0→ H1(∆J ,W [J ])→ H1(GF ,W [J ])→ Hom(GabFJ ,W [J ])∆J .

In fact, the last term can be though of as ∆J -equivariant homomorphisms if GabFJ

is given the correct ∆J -action. Specifically, if δ ∈ ∆J then choose any lifting δof δ to GF . For any x ∈ Gab

FJ, we define δx to be δxδ−1; since Gab

FJis abelian

this does not depend on the choice of lifting of δ. It follows immediately from therelevant definitions that with this action on Gab

FJand the usual action on W [J ],

Hom(GabFJ,W [J ])∆J is just the ∆J -equivariant homomorphisms from Gab

FJto W [J ].

Now suppose that H1(∆J ,W [J ]) = 0. Then we have an injection

H1(GF ,W [J ]) → Hom∆J(Gab

FJ ,W [J ]),

which we write as h 7→ ϕh. Given this, to show that an h ∈ H1(GF ,W [J ]) is 0it is in turn enough to show that ϕh is 0; to do this it is enough to show that ϕhvanishes on a set of Frobenius elements of density 1. Thus the assumption thatH1(∆J ,W [J ]) = 0 is a very useful one; by the ∆-hypothesis we will mean theassumption that H1(∆J ,W [J ]) = 0 for every ideal J of A of finite index.

In fact, this is not a particularly restrictive assumption. To give an example, weconsider a slightly more general situation: let M be a free R-module of finite rankn for some finite local ring R of residue characteristic p and let ∆ be a subgroupof GLn(R). (To relate this to the above situation just take R = A/J , M = W [J ]and ∆ = ∆J ; ∆J is a subgroup of AutA/J(W [J ]) = GLn(R).) We will give onecondition under which H1(∆,M) = 0.

Lemma 1.3 (Sah). Suppose that there is a non-trivial subgroup C of ∆, of orderprime to p, which consists entirely of scalar matrices. Then Hr(∆,M) = 0 for allr.

Proof. We have a Hochschild-Serre spectral sequence

Hp(∆/C,Hq(C,M))⇒ Hp+q(∆,M);

this makes sense since C is in the center of ∆ and thus normal. This shows that itwill suffice to show that Hq(C,M) = 0 for all q. If q > 0 this follows from Lecture5, Lemma 2.4.

2. DEPTH AND KOLYVAGIN-FLACH SYSTEMS 117

For q = 0 we must show that MC = 0. By hypothesis C ⊆ R∗. So supposethat there is a non-zero m ∈M such that cm = m for all c ∈ C. Then

(c− 1)m = 0,

so c− 1 is not a unit. Thus c− 1 lies in the maximal ideal of R, so C is a subgroupof the multiplicative group 1 + m. But this group has order a power of p, while Chas order prime to p, which is a contradiction. So MC = 0.

2. Depth and Kolyvagin-Flach Systems

2.1. Depth. We continue to let A and W be as before. Fix a non-unit idealJ of A of finite index. The induced finite/singular structure on W ∗/JW ∗ yields anexact sequence

0→ H1f (GF ,W ∗/JW ∗)→ H1(GF ,W ∗/JW ∗)→ H1

s (GF ,W ∗/JW ∗)→ 0.

We will now begin to use the notation of Lecture 8, Section 1.2. Let C besome set of elements of the Kolyvagin group H1

s (GF ,W ∗/JW ∗). Fix a place v ofF and let C ′ be the A/J-submodule of H1

s (Gv,W ∗/JW ∗) generated by resv,s(c)for c ∈ C. If I is an ideal of A containing J , we will say that C is of depth I at vif the quotient of H1

s (Gv,W ∗/JW ∗) by C ′ is annihilated by I.Let L be a (not necessarily finite) set of places of F disjoint from Σ. If I is

some ideal of A containing J , a Kolyvagin-Flach system of classes of depth I forW ∗/JW ∗ for L is an assignment of some subset C(v) of H1

s (GF ,W ∗/JW ∗) for eachv ∈ L such that

• Supp c = v for all c ∈ C(v);• C(v) is of depth I at v.

2.2. The Main Theorem. We will need two more hypotheses in order toget the main theorem. First, we assume that p 6= 2; this seems to be a necessarycondition for this exposition and not just the result of a lack of care. Second, weassume that there is an involution τ in the image of

ρ : GF,Σ → Autk(W [m])

which is not a scalar.In particular, this implies that such a non-scalar involution exists for the rep-

resentationρJ : GF,Σ → Autk(W [J ])

for any ideal J of finite index in A. To see this, first note that

AutA/J(W [J ])→ Autk(W [m])

is surjective with kernel a p-group. Thus we can lift our given τ to an element ofAutA/J(W [J ]) in the image of GF,Σ. It will be of order 2 times a power of p, sotaking an appropriate power of this yields the desired non-scalar involution. (It isnon-scalar since it still reduces to τ modulo m, as p is odd.)

Fix such a τ for our fixed ideal J . Let Lτ be the set of places of F overwhich there is a place whose associated Frobenius in FJ is τ ; equivalently, Lτ isthe set of places of F with Frobenius in FJ the conjugacy class of τ . Note that theTchebatorev density theorem ([La-ANT, Chapter 8, Theorem 10]) implies that Lτhas positive density; in particular, it is infinite.

118 14. LECTURE 14

Theorem 2.1. Let A, W and J be as above. Suppose that the residual rep-resentation ρ is absolutely irreducible; that the ∆-hypothesis holds; and that thereexists a non-scalar involution τ in the image of ρ. Further suppose that there existsa Kolyvagin-Flach system of classes of depth I for W ∗/JW ∗ for Lτ , for some idealI containing J . Then

H1f (GF ,W [I]) = H1

f (GF ,W [J ]).

Note that the statement of this theorem is very much in the spirit of Mazur’scomments in Lecture 8, Section 2.

CHAPTER 15

Lecture 15

1. The Main Theorem

1.1. Statement of the Main Theorem. We briefly recall our running hy-potheses. Fix a prime p 6= 2. Let F be a number field with absolute Galois groupGF and let GF,Σ be the maximal quotient of GF unramified outside of a finite setΣ of places of F (containing p and the archimedean places). Let A be a local, finite,faithfully flat Zp-algebra with maximal ideal m and residue field k. Let W ∗ be afree A-module of finite rank (at least 2) with a continuous A-linear action of GF,Σ.Let W = Hom(W ∗, µp∞) be its Cartier dual. We let

ρ : GF,Σ → AutA(W )

be the representation of GF,Σ on W , and for any ideal J of A of finite index we let

ρJ : GF,Σ → AutA/J(W [J ])

be the restriction of ρ to the submodule

W [J ] = ∩α∈JW [α].

We make three hypotheses on this data. First, we assume that ρ is absolutelyirreducible. This implies that WG = 0, by Corollary 3.7 of Lecture 13. (Here weneed the fact that W has rank at least 2.) Second, for any ideal J , let FJ be thesplitting field of ρJ ; it is a finite extension of F . Set ∆J = Gal(FJ/F ). We assumethat

H1(∆J ,W [J ]) = 0for all J . Lastly, we suppose that there is a non-scalar involution τ in the image of

ρ : GF,Σ → Autk(W [m]);

we can lift this τ to AutA/J(W [J ]) for any ideal J ⊆ m of A of finite index, sincep is odd. We fix such a τ .

Theorem 1.1 (Kolyvagin). With the above notation and hypotheses, supposein addition that there is a Kolyvagin-Flach system of depth I for W ∗/JW ∗ for Lτ .Then

H1f (GF ,W )[I] = H1

f (GF ,W )[J ].

If one does not assume that H1(∆J ,W [J ]) = 0, then one finds that the productof I and any annihilator of H1(∆J ,W [J ]) annihilates H1

f (GF ,W )[J ]. This will beclear from the proof below.

We will prove this theorem in two stages. Recall that for any place v of F thereis a natural restriction map

resv : H1(GF ,W [J ])→ H1(Gv,W [J ]).


119

120 15. LECTURE 15

This map sends H1f (GF ,W [J ]) to H1

f (Gv,W [J ]), so we also obtain a natural map

ress,v : H1s (GF ,W [J ])→ H1

s (Gv,W [J ]).

Lemma 1.2. With the hypotheses of Theorem 1.1, if x ∈ H1f (GF ,W [J ]) and

α ∈ I, thenresv(αx) = 0

for all v ∈ Lτ .Lemma 1.3. Continue to suppose the hypotheses of Theorem 1.1. Let X ∈

H1f (GF ,W [J ]) be such that resv(X) = 0 for all v ∈ Lτ . Then X = 0.

It is clear that Lemma 1.2 and Lemma 1.3 suffice to prove the theorem. We notealso that the proof of Lemma 1.3 will not require the existence of the Kolyvagin-Flach system.

As a corollary, we obtain the following:Corollary 1.4. In addition to the hypotheses of Theorem 1.1, suppose that

mI ⊇ J . ThenH1f (GF ,W )[I] = H1

f (GF ,W ).

Proof. Everything in H1f (GF ,W ) is killed by some power of m, so we prove

the corollary for each H1f (GF ,W )[mm] by induction on m. The result is clear for

m = 1, since J ⊆ m. So suppose that we know the result for m < n. Takex ∈ H1

f (GF ,W )[mn]. Then mn−1(mx) = 0, so by the induction hypothesis we haveI(mx) = 0. So, since Im ⊇ J , we have Jx = 0. Now, by Theorem 1.1, we haveIx = 0. This complete the induction.

1.2. Proof of Lemma 1.2. Fix v ∈ Lτ . SinceW [J ] andW ∗/JW ∗ are Cartierdual, Tate local duality (Lecture 6, Theorem 2.2) yields a perfect cup productpairing

H1f (Gv,W [J ])⊗H1

s (Gv,W ∗/JW ∗)→ Q/Z.

Since v ∈ Lτ , we have a set C(v) of Kolyvagin-Flach classes such that ress,v′(c) =0 for c ∈ C(v) and v 6= v′, and if C ′ is the A-submodule of H1

s (Gv,W ∗/JW ∗)generated by C(v), then

H1s (Gv,W ∗/JW ∗)/C ′

is annihilated by I.Next recall that under our global pairing (see Lecture 8, Section 2)

H1f (GF ,W [J ])⊗⊕vH1

s (Gv,W ∗/JW ∗)→ Q/Z,

H1f (GF ,W [J ]) and H1

s (GF ,W ∗/JW ∗) are orthogonal. Since ress,v′(c) = 0 forc ∈ C(v) and v′ 6= v, the definition of the pairing and the above orgthogonalityimply that

resv(x) ∪ ress,v(c) = 0

for all c ∈ C(v). Thus, resv(x) is contained in the orthogonal complement of C ′ inH1f (Gv,W [J ]) under Tate local duality. Since Tate local duality is a perfect pairing

and I annihilatesH1s (Gv,W ∗/JW ∗)/C ′,

it follows immediately that I annihilates resv(x) ∈ H1f (Gv,W [J ]), as desired.

1. THE MAIN THEOREM 121

1.3. Proof of Lemma 1.3. Consider the exact sequence

1→ GFJ → GF → ∆J → 1.

This yields an inflation-restriction sequence

0→ H1(∆J ,W [J ])→ H1(GF ,W [J ])→ Hom∆J(Gab

FJ ,W [J ]).

(See Lecture 14, Section 1.2 for a discussion of the ∆J action on GabFJ

.) By hypoth-esis, H1(∆J ,W [J ]) = 0, so it will be enough to show that the image

ϕ : GabFJ →W [J ]

of X in Hom∆J(Gab

FJ,W [J ]) is trivial.

Let L be the fixed field of kerϕ, and set Γ = Gal(L/FJ).

GabFJ

ϕ//

AAAAAAAA

W [J ]

Γ.

ϕ==

There is an exact sequence

1→ Γ→ Gal(L/F )→ ∆J → 1.

Since Γ injects into W [J ], it is a finite abelian p-group.Let τ be our fixed non-scalar involution in ∆J . Since Γ has odd order we

can lift τ to a non-scalar involution τ ∈ Gal(L/F ). Let g be any element of Γand consider τ g ∈ Gal(L/F ). By the Tchebatorev density theorem ([La-ANT,Chapter 8, Theorem 10]) there exists an unramified place w of L such that

FrL/F (w) = τ g.

Set w = w|FJ and v = w|F .

L

Γ

_

Gal(L/F )

_

w

FJ

∆J

w

F v

By standard properties of Frobenius, we have

FrFJ/F (w) = FrL/F (w)|FJ = τ g|FJ = τ

(so in particular v ∈ Lτ ), and

FrL/FJ (w) = (FrL/F (w))deg(w/v) = (τ g)2.

Here by deg(w/v) we mean the degree of the local field extension FJ,w/Fv; it is 2since τ has order 2 and w/v is unramified.

We claim that since resv(x) = 0, we have

ϕ(FrL/FJ (w)) = 0.

122 15. LECTURE 15

To see this, begin with the commutative diagram

X ∈ H1(GF ,W [J ])resv //

H1(Gv,W [J ])

ϕ ∈ Hom(GFJ ,W [J ]) // Hom(GFJ ,w,W [J ])

Since ϕ factors through Γ = Gal(L/FJ), ϕ|GFJ ,w factors through Gal(Lw/FJ,w),which is generated by FrL/FJ (w) since w/w is unramified. But resv(X) = 0, soϕ|GFJ ,w = 0; thus ϕ(FrL/FJ (w)) = 0 as claimed.

Combining all of this we have

ϕ(τ gτg) = 0.

But τ2 = 1, so τ gτg = τ gτ−1g. By the definition of the ∆J -action on Γ, this isnone other than τg · g. Thus

ϕ(τg) + ϕ(g) = 0,

so by ∆J -equivarianceτϕ(g) = −ϕ(g) ∈W [J ].

This is true for all g ∈ Γ, so if we let H be the A-submodule of W [J ] generatedby ϕ(Γ), we have

H ⊆W [J ]−.Here W [J ]− is the minus eigenspace for τ acting on W [J ]:

W [J ]− = z ∈W [J ] | τz = −z.

Note that W [J ]− 6= W [J ] since τ is non-scalar. Since ϕ is ∆J -equivariant and ∆J

acts A-linearly, H is ∆J -stable. But if we consider H[m] ⊆ W [m]−, we must haveH[m] = 0, since W [m] is irreducible as a GF,Σ-representation space and W [m]− 6=W [m]. But if H[m] = 0, then H = 0, by Corollary 3.7 of Lecture 13. Thus ϕ = 0as desired.

2. The Main Theorem for Rank 1 Modules

The end of the proof of Lemma 1.3, and thus the proof of Theorem 1.1, reliedvery heavily on the fact that WG = 0. Thus need not be true in the case that Whas rank 1, even if ρ is absolutely irreducible. In fact, the version of the theoremwhich is useful for rank 1 A-modules is somewhat simpler than the main theorem;essentially, one is able to take τ = 1. We state the result for all W even though itis only useful in practice for W of rank 1.

We keep all of the definitions of the first paragraph of Section 15.1, exceptthat we now allow W to have rank 1. We need not assume that ρ is absolutelyirreducible or that there is a non-scalar involution τ . The only extra hypothesis weneed is the ∆-hypothesis

H1(∆J ,W [J ]) = 0.We let L = L1 be the set of unramified places of F with trivial Frobenius in FJ .

Theorem 2.1. With the above notation and hypotheses, suppose in additionthat there is a Kolyvagin-Flach system of depth I for W ∗/JW ∗ for L. Then

H1f (GF ,W )[I] = H1

f (GF ,W )[J ].

2. THE MAIN THEOREM FOR RANK 1 MODULES 123

Proof. The proof is actually somewhat simpler than the proof of Theorem 1.1.Lemma 1.2 goes through with no changes; Lemma 1.3 in fact simplifies greatly, asone finds that

FrL/FJ (w) = g,

and thus ϕ(g) = 0 immediately by Lemma 1.2.

This theorem is only useful in the case where W has rank 1; in the higher rankcases there are no Kolyvagin-Flach systems for trivial τ .

CHAPTER 16

Lecture 16

1. Other Versions of the Main Theorem

1.1. Characters. We briefly review the theory of eigenspaces of group mod-ules; since it will be sufficient for our applications we restrict attention to thesimplest case. So let p be a fixed prime number and let G be an abelian group oforder dividing p− 1. We will write G for the group of characters of G. Since G isof order dividing p− 1, the image of any character of G lies in µp−1. Thus, in thisset-up, we can identify G with Hom(G,Z∗p).

Let ε be a character of G and define the ε-idempotent of Zp[G] by

eε =1|G|

∑g∈G

ε−1(g)g ∈ Zp[G].

One checks easily that

eεeε′ =

eε ε = ε′

0 ε 6= ε′.

Thus the eε form a system of idempotents of Zp[G], yielding a decomposition

Zp[G] = ⊕ε∈GZp[G]eε.

Since each Zp[G]eε is non-zero (it contains eε), it follows from the fact that G hasthe same order as G that each Zp[G]eε is free of rank 1 over Zp. One also notesthat for all g ∈ G,

geε = ε(g)eε,

so that G acts on Zp[G]eε via ε.Now let M be a G-module which is a finite p-group. Then M can be viewed as

a Zp[G]-module, since each element of M is killed by some power of p. The abovedecomposition of Zp[G] yields a decomposition

M = ⊕ε∈GMeε.

In fact, one shows easily from the fact that geε = ε(g)eε that Meε can also berealized as

Meε = m ∈M | gm = ε(g)m for all g ∈ G.We will also write Mε for Meε, and call it the ε-eigenspace of M .

In the case that G has order 2, we will write M+ and M− for the eigenspaceof the trivial character and the eigenspace of the non-trivial character respectively.


125

126 16. LECTURE 16

1.2. Galois Groups Acting on Cohomology. We now return to the nota-tion of the previous lecture, although we do not yet assume any hypotheses on W .Let us suppose that we are also given a subfield F0 of F of index 2, so that F/F0

is a quadratic extension. SetG = Gal(F/F0).

We assume that the GF -action on W is the restriction of a continuous, A-linearaction of GF0 on W .

Let J be an ideal of A of finite index. We claim that H1(GF ,W [J ]) has thestructure of a G-module. In fact, it has the structure of GF0-modules, where GF0

acts on GF by inverse conjugation and on W [J ] by the given action. Further, theaction of GF on H1(GF ,W [J ]) in this way is trivial ([Se-LF, Chapter 7, Section 5,Proposition 3]), so that the GF0-action factors through G, as claimed. In the sameway we can consider H1(GF ,W ∗/JW ∗) as a G-module. Given this, the discussionof the previous section yields eigenspace decompositions

H1(GF ,W [J ]) = H1(GF ,W [J ])+ ⊕H1(GF ,W [J ])−

andH1(GF ,W ∗/JW ∗) = H1(GF ,W ∗/JW ∗)⊕H1(GF ,W ∗/JW ∗)−.

Unfortunately, H1f (GF ,W [J ]) is not necessarily a G-module. In order to give

conditions under which it is, we first must consider the G-action induced on thelocal cohomology groups. So let v be a place of F . Then the action of τ ∈ G yieldsa map

H1(Gτ−1v,W [J ]) = H1(τ−1Gvτ,W [J ])→ H1(Gv,W [J ]).

Thus, if v|F0 is inert or ramified in F/F0, G does act on H1(Gv,W [J ]). However,if v|F0 splits over F into v and v′, then G acts on the direct sum

H1(Gv,W [J ])⊕H1(Gv′ ,W [J ])

by interchanging the factors. In any event, there is a natural G-action on

⊕vH1(Gv,W [J ])

and the natural map

res : H1(GF ,W [J ])→∏v

H1(Gv,W [J ])

is G-equivariant, essentially by definition of the action on the local factors.We make the assumption for the remainder of this section that the G-action

respects the finite/singular structure, in the sense that

⊕vH1f (Gv,W [J ])

is G-stable. The next lemma shows that this is a reasonable condition; the proof isstraightforward and we omit it.

Lemma 1.1. Let M be a GF0-module. Let v0 be a place of F0 which is unramifiedin F and let v be a place of F above v0. Then H1(gv,MIv ) is G-stable. (In the casethat v0 is inert the meaning of this is clear; if v0 splits into v and v′, this meansthat the G-action interchanges H1(Gv,MIv ) and H1(Gv′ ,MIv′ ).)

With this assumption on our finite/singular structures,

⊕vH1s (Gv,W [J ])

1. OTHER VERSIONS OF THE MAIN THEOREM 127

is also a G-module, and G-equivariance of

ress : H1(GF ,W [J ])→ ⊕vH1s (Gv,W [J ])

implies that H1f (GF ,W [J ]) is a G-module as well. In the same way one can im-

pose conditions on the finite/singular structure on W ∗/JW ∗ in order to insurethat H1

f (GF ,W ∗/JW ∗) is a G-module; we make this assumption as well. We willshow in a moment that this assumption is compatible with the usual Cartier dualfinite/singular requirements.

Let v be a place of F so that v|F0 is either inert or ramified in Fv0 . Then Gacts on H1(Gv,W [J ]), and we claim that the Tate local pairing

H1(Gv,W [J ])⊗H1(Gv,W ∗/JW ∗)→ Qp/Zp

is G-invariant. (Equivalently, it is G-equivariant when Qp/Zp is given the trivial G-action.) To do this it suffices to check equivariance of cup product, Cartier dualityand the invariant map. The first follows from functoriality and the second froman easy direct computation. For the last one must go back to the definition ofthe invariant map (see [Se-LF, Chapter 13, Section 3]) and see that each step isG-equivariant; we omit the details.

It follows immediately from this invariance and the fact that p is odd thatH1(Gv,W [J ])ε and H1(Gv,W ∗/JW ∗)−ε pair to 0, where ε is some choice of sign.Thus Tate local duality restricts to perfect pairings

H1(Gv,W [J ])+ ⊗H1(Gv,W ∗/JW ∗)+ → Qp/Zp

andH1(Gv,W [J ])− ⊗H1(Gv,W ∗/JW ∗)− → Qp/Zp.

These in turn restrict to perfect pairings

H1f (Gv,W [J ])+ ⊗H1

s (Gv,W ∗/JW ∗)+ → Qp/Zp

andH1f (Gv,W [J ])− ⊗H1

s (Gv,W ∗/JW ∗)− → Qp/Zp.

Among other things these show that, at least when v|F0 is inert or ramified,the various requirements on finite/singular structures are compatible. We also notethat in the same way one can show that the Tate local pairing is G-equivariant forv|F0 which split completely, in the sense that the pairing(H1(Gv,W [J ])⊗H1(Gv,W ∗/JW ∗)

)⊕(H1(Gv′ ,W [J ])⊗H1(Gv′ ,W ∗/JW ∗)

)→ Qp/Zp ⊕Qp/Zp

is G-equivariant, where the non-trivial element of G acts on Qp/Zp⊕Qp/Zp by inter-changing the factors. This again shows that the requirements on the finite/singularstructures at these places are compatible.

Given the above pairings, it is now possible to define Kolyvagin-Flach systemsfor these eigenspaces. We let Σ be a finite set of places of F and L a (not necessarilyfinite) set of places of F disjoint from Σ. We also let I be some ideal of A containingJ and ε some choice of sign.

Definition 8. A Kolyvagin-Flach system of classes of depth I for (W ∗/JW ∗)ε

for L is an assignment of some subset C(v) of H1s (GF ,W ∗/JW ∗)ε for each v ∈

L such that Supp c = v for all c ∈ C(v) and, if C ′ is the A-submodule ofH1s (Gv,W ∗/JW ∗)ε generated by C(v), then

H1s (Gv,W ∗/JW ∗)ε/C ′

128 16. LECTURE 16

is annihilated by I.

1.3. The Main Theorem for ± Eigenspaces. We continue with the usualnotation. We also continue to assume that ρ is absolutely unramified and thatH1(∆J ,W [J ]) = 0 for all ideals J of finite index in A. We substitute a newcondition for the existence of a non-scalar involution.

Specificially, let ∆J be the Galois group of FJ over F0. There is an exactsequence

1→ ∆J → ∆J → G → 1

and a field diagramFJ

∆J

_

∆J

_

F

G

F0

We make the hypothesis that there is an involution τ ∈ ∆J which projects to thenon-trivial element in G and acts on W [m] (and thus on W [J ]) as a non-scalar. Wedefine Lτ to be the set of places v of F such that there is a place w of FJ lyingabove v such that

FrFJ/F0(w) = τ ∈ ∆J .

Note that any place v ∈ Lτ is inert in F/F0 since τ projects to the non-trivialelement of G = Gal(F/F0).

Theorem 1.2 (Kolyvagin). Let ε be some choice of sign. With the abovenotation and hypotheses, suppose in addition that there is a Kolyvagin-Flach systemof depth I for (W ∗/JW ∗)ε for Lτ . Then

H1f (GF ,W )[I]ε = H1

f (GF ,W )[J ]ε.

Note that the hypothesis of Theorem 1.2 are neither clearly weaker nor strongerthan those of Theorem 1.1 of Lecture 15, but the conclusion is weaker. Nevertheless,this version is the appropriate one to apply in many situations.

1.4. Proof of Theorem 1.2. The proof of Theorem 1.2 is quite similar to theproof of Theorem 1.1 of Lecture 15. We will again prove it through two lemmas.

Lemma 1.3. With the hypotheses of Theorem 1.2, if x ∈ H1f (GF ,W [J ])ε and

α ∈ I, thenresv(αx) = 0

for all v ∈ Lτ .The proof of Lemma 1.3 is almost identical to that of Lemma 1.2 of Lecture

15; we leave the details to the reader.Lemma 1.4. Continue to suppose the hypotheses of Theorem 1.2. Let X ∈

H1f (GF ,W [J ])ε be such that

resv(X) = 0

for all v ∈ Lτ . Then X = 0.


Proof. This proof is the same as that of Lemma 1.3 of Lecture 15 at manypoints. We again begin with the exact sequence

1→ GFJ → GF → ∆J → 1,

which, by the inflation-restriction sequence and the ∆-hypothesis, yields an injec-tion

H1(GF ,W [J ]) → Hom∆J(Gab

FJ ,W [J ]).In fact, this map is G-equivariant (it is just restriction), so we get an injection

H1(GF ,W [J ])ε → Hom∆J(Gab

FJ ,W [J ])ε.

Thus it will again be enough to show that the image

ϕ : GabFJ →W [J ]

of X in Hom∆J(Gab

FJ,W [J ])ε is trivial.

We again let L/F be the splitting field of ϕ and set Γ = Gal(L/FJ). Thus ϕ isnow thought of as an injection

ϕ : Γ →W [J ].

So Γ is a p-group, and as before it follows that we can lift τ to an involutionτ ′ ∈ Gal(L/F0). Now, choose some g ∈ Γ and let w be some place of L such that

FrL/F0(w) = τ ′g ∈ Gal(L/F0).

(w exists by the Tchebatorev density theorem, as usual.) We let w, v and v0 bethe restriction of w to FJ , F and F0 respectively.

L

Γ

w

FJ

∆J

_

∆J

_

w

F

G

v

F0 v0

We now have

FrFJ/F0(w) = FrL/F0(w)|FJ = τ ′g|FJ = τ ∈ ∆J ,

so v ∈ Lτ . Also,

FrL/FJ (w) = (FrL/F0(w))deg(w/v0) = (τ ′g)2,

since τ = FrFJ/F0(w) has order 2. Furthermore, exactly as in the proof of Lemma 1.3of Lecture 15, we have

ϕ(FrL/FJ (w)) = 0,so that

ϕ(τ ′gτ ′g) = 0.Thus,

ϕ(τg) + ϕ(g) = 0.

130 16. LECTURE 16

Now, ϕ is ∆J -equivariant, but not necessarily ∆J -equivariant. However, it does liein

Hom∆J(Gab

FJ ,W [J ])ε.From this it follows easily, since τ projects non-trivially to G, that

ϕ(τg) = ετϕ(g).

Thus ϕ(g) lies in the subspace of W [J ] on which τ acts by −ε and the A-moduleH generated by ϕ(Γ) lies in W [J ]−ε. (Here by W [J ]−ε we mean the −ε eigenspacefor τ .) H is also ∆J -stable, since ϕ is ∆J -equivariant, so from here one proceedsas before, using the fact that τ acts as a non-scalar on W [m] to conclude thatW [m]−ε 6= W [m].

1.5. A Generalization of the Main Theorem. In this section we give onelast version of the main theorem. We keep the usual notation and continue toassume that ρ is absolutely irreducible and that H1(∆J ,W [J ]) = 0 for all ideals Jof finite index in A. We no longer assume that p 6= 2, and instead of assuming thatwe have a non-scalar involution τ , we simply assume that there is some τ ∈ ∆J

such thatW [m]/(τ − 1)W [m] 6= 0.

Theorem 1.5 (Rubin). With the above notation and hypotheses, suppose inaddition that there is a Kolyvagin-Flach system of depth I for W ∗/JW ∗ for Lτ .Then

H1f (GF ,W )[I] = H1

f (GF ,W )[J ].

Proof. The proof is quite similar to that of Theorem 1.1 of Lecture 15.Lemma 1.2 of Lecture 15 goes through with no changes. For Lecture 15, Lemma 1.3,we proceed as follows. As usual, it suffices to show that the homomorphismϕ : GFJ → W [J ] is trivial. However, now we choose also a representative co-cycle c : GF → W [J ] for the cohomology class X. Let L be some finite extensionof F through which c factors; then ϕ necessarily factors through Gal(L/FJ) = Γ.That is, we now have maps

c : Gal(L/F )→W [J ]

andϕ : Γ→W [J ].

Note that ϕ need no longer be injective on Γ.Choose any g ∈ Γ and some lifting τ of τ to Gal(L/F ). By the Tchebatorev

density theorem we can find some place w of L such that

FrL/F (w) = τ g.

Let w and v be the restriction of w to FJ and F respectively. As usual we have

FrFJ/F (w) = τ,

so that v ∈ Lτ .Now, c|Gal(Lw/Fv) is a coboundary, since resv(X) = 0. Thus

c(FrL/F (w)) ∈ (FrL/F (w)− 1)W [J ];

that is,c(τ g) ∈ (τ g − 1)W [J ] = (τ − 1)W [J ],


since τ acts on W [J ] as τ and g acts trivially on W [J ]. Since c is a cocycle,

c(τ g) = c(τ) + τ c(g) = c(τ) + τc(g) ∈ (τ − 1)W [J ].

Furthermore, this is true for all g ∈ Γ. Taking g = 1 we find that

c(τ) ∈ (τ − 1)W [J ].

Thusτc(g) ∈ (τ − 1)W [J ].

Since also(τ − 1)c(g) ∈ (τ − 1)W [J ],

we conclude thatc(g) ∈ (τ − 1)W [J ].

Thus the image of c, and therefore the image of ϕ, lies in (τ − 1)W [J ]. LettingH be the A-submodule of (τ − 1)W [J ] generated by ϕ(Γ), the proof continues inthe usual way, using the fact that

(τ − 1)W [m] 6= W [m]

to show that H = 0.

CHAPTER 17

Lecture 17

1. Modular Curves

In this section we summarize the main properties of modular curves whichwill be of interest to us. We will refer primarily to [DI], which provides moreextensive references and some proofs. Other standard references are (in roughorder of difficulty) [Se-CA, Chapter 7], [Si-2, Chapter 1], [La-MF] and [Shi].

1.1. Definitions. Let H be the upper half plane in C; that is,

H = z ∈ C | Im z > 0.Recall that there is a natural left action of SL2(Z) on H, given by(

a bc d

)z =

az + b

cz + d.

(In fact, −1 ∈ SL2(Z) acts trivially, so that this action can be viewed as an actionof PSL2(Z).)

Fix an integer N and let Γ0(N) be the subgroup of SL2(Z)

Γ0(N) =(

a bc d

)| c ≡ 0 (mod N)

.

Then one can show that the quotient

Γ0(N)\Hhas the structure of a Riemann surface. (See [DI, Section 7].) In fact, this Riemannsurface can be given the structure of a smooth, affine algebraic curve defined overQ, which we denote by Y0(N). (See [DI, Section 8].) Thus

Y0(N)(C) ∼= Γ0(N)\H.The projective closure of Y0(N) is denoted by X0(N); the complement of Y0(N)

in X0(N) are called the cusps. X0(N) is a smooth, projective curve defined overQ.

The importance of X0(N) and Y0(N) to us comes from the fact that they area (partial) solution to the problem of classifying modular pairs (E/S,CN ), whereS is some Q-scheme, E is an elliptic curve over S (that is, E is an abelian schemeover S of dimension 1) and CN is a finite, flat subgroup scheme of E, all of whosegeometric fibers are cyclic groups of order N . (If S is the spectrum of a field K,then this is just a pair of an elliptic curve defined over K and a cyclic subgroup oforder N of E(K). For simplicity we shall always refer to CN as above as “cyclicsubgroups of order N”.) More precisely, we define a functor FN from Q-schemesto sets by sending a Q-scheme S to the set of isomorphism classes of modular pairs


133

134 17. LECTURE 17

(E/S,CN ). (Two modular pairs (E/S,CN ) and (E′/S,C ′N ) are isomorphic if thereis an isomorphism (over S) of E and E′ taking CN to C ′N .) This functor is notactually representable, but Y0(N) is what is called the representable hull. Thismeans that if FY0(N) is the functor from Q-schemes to sets

S 7→ HomSpecQ(S, Y0(N)) = Y0(N)(S),

then there is a natural transformation of functors

j : FN → FY0(N)

with the following universal property: If Z is any other Q-scheme such that thereis a natural transformation

i : FN → FZ ,

where FZ is the functor sending Q-schemes to HomSpecQ(S,Z), then i factorsuniquely through j:

FNi //

j##GGGGGGGG FZ

FY0(N)

;;xx

xx

Note that any natural transformation FY0(N) → FZ must come from some map ofschemes

Y0(N)→ Z;

specifically, the image of the identity map in FY0(N)(Y0(N)) yields an element ofFZ(Y0(N)) = HomSpecQ(Y0(N), Z), and one easily checks that this map Y0(N)→ Zinduces the given natural transformation FY0(N) → FZ .

Now, j yields a map jS from pairs (E/S,CN ) to Y0(N)(S). In general this mapneed be neither injective nor surjective. However, if K is an extension of Q (notnecessarily finite), then the map jK from (E/K,CN ) to Y0(N)(K) is surjective. IfK is algebraically closed, then it is a bijection.

1.2. Modular Curves over C. In order to better illustrate the modularinterpretation of Y0(N), let us return to the case of modular curves over the complexnumbers. We assume in this section that the reader is familiar with the theory ofelliptic curves over C as in [Si-AEC, Chapter 6].

Recall that an elliptic curve over C is isomorphic to the quotient of C by alattice Λ ⊆ C. From this point of view, Y0(N)(C) classifies isomorphism classes ofpairs of lattices (Λ ⊆ Λ′) where Λ′/Λ is cyclic of order N . Specifically, to such apair we associate the pair

(C/Λ,Λ′/Λ)

of an elliptic curve and a subgroup of order N , and two pairs of lattices are isomor-phic if the give rise to isomorphic pairs. By [Si-AEC, Chapter 6, Corollary 5.1.1]this occurs if and only if the pairs are homothetic.

Let us now consider pairs (Λ ⊆ Λ′) more explicitly. It is not difficult to seethat one can choose a Z-basis ω1, ω2 ∈ C of Λ such that

Im(ω1/ω2) > 0

1. MODULAR CURVES 135

and ω1,ω2N is a Z-basis of Λ′. Since lattice pairs are defined only up to homothety,

we obtain a map τ from pairs (ω1, ω2) ∈ C × C with Im(ω1/ω2) > 0 to the upperhalf plane, given by

(ω1, ω2) 7→ ω1

ω2.

To complete our analysis, we must determine what other pairs (ω1, ω2) give riseto the same lattice pair

(Zω1 ⊕ Zω2,Zω1 ⊕ Zω2

N).

(The map τ already takes care of homothety.) If (ω′1, ω′2) is some other pair (with

Im(ω′1/ω′2) > 0 as always), it is a basis for Λ if and only if there is some matrix(

a bc d

)∈ SL2(Z)

such that (a bc d

)(ω1

ω2

)=(ω′1ω′2

).

(The matrix must be in SL2(Z) rather than GL2(Z) in order to preserve the upperhalf-plane condition.) Furthermore, it is easy to see that it gives rise to the sameΛ′ if and only if

c ≡ 0 (mod N);that is, if (

a bc d

)∈ Γ0(N).

Thus the set of isomorphism classes of lattice pairs is in bijection with the quotientof the upper half-plane by the action of Γ0(N); that is, they are in bijection with

Γ0(N)\H ∼= Y0(N)(C).

Since isomorphism classes of lattice pairs correspond to isomorphism classes of mod-ular pairs (E/C, CN ), this shows that the points of Y0(N)(C) really do correspondto such pairs, in a canonical way.

One can also give an explicit description of the cusps

X0(N)(C)− Y0(N)(C).

To do this, one considers the extended upper half-plane

H∗ = H∐P

1(Q).

One defines an action of Γ0(N) on P1(Q) in the natural way: thinking of elements

of P1(Q) as homogeneous pairs(ab

), the action is just by multiplication on the

left; thinking of P1(Q) as Q∐∞ (as we usually will), one has(a bc d

)τ =

aτ + b

cτ + d,

with the convention thata∞+ b

c∞+ d=a

c.

With this action, one has

X0(N)(C) = Γ0(N)\H∗,so that the cusps correspond to the Γ0(N)-orbits on P1(Q).

136 17. LECTURE 17

In the case that N is prime, one easily checks that there are only two cusps:

0 =ab∈ Q | b 6≡ 0 (mod N)

∞ =

ab∈ Q | b ≡ 0 (mod N)

∐∞.

Any time that we need a base point for X0(N) (whether N is prime or not) we willuse the cusp containing the class of ∞.

1.3. Functions on Modular Curves. We consider briefly here the theoryof functions and differential forms on modular forms; this a moderately importantarea in modern number theory1, and we give here a complete treatment2.

Note that for any integer N , the matrix(

1 10 1

)lies in Γ0(N). This matrix

acts on H byτ 7→ τ + 1.

Thus any function or differential form on X0(N), when considered as a Γ0(N)-invariant function or differential form on H∗, can be expanded in terms of theparameter

q = e2πiτ .

We can therefore expand functions f : X0(N)(C)→ C as Laurent series

f(τ) =∑n

anqn

(these are modular forms of weight 0 and level N) and differential forms fdτ as

f(τ)dτ =∑n

anqn dq

q

(these are modular forms of weight 2 and level N).

2. Operators on Modular Curves

2.1. Degeneracy operators. We are going to construct operators on modu-lar curves by defining the operators on the underlying problems which the modularcurves “classify”. It will be useful to refer to the following formal argument. LetF1 and F2 be two functors from Q-schemes to sets which have Q-schemes Y1 andY2 as representable hulls; thus, if FY1 and FY2 are the functors represented by Y1

and Y2 respectively, we have natural transformations

j1 : F1 → FY1

andj2 : F2 → FY2 .

Suppose that we define a natural transformation

i : F1 → F2.

Composition with j2 yields a natural transformation

j2 i : F1 → FY2 ,

1Sarcasm.2Extreme sarcasm.

2. OPERATORS ON MODULAR CURVES 137

so since Y1 is a representable hull for F1 we obtain a commutative diagram ofnatural transformations

F1j2i //

j1

BBBBBBBB FY2

FY1

i′==

i′ in turn comes from a mapY1 → Y2.

Thus any natural transformation of functors yields a canonical map of their rep-resentable hulls. If in addition the Yi are smooth with projective closures Xi, oneobtains a map

X1 → X2.

For any integer N , we let FN be the functor from Q-schemes to sets associat-ing to a Q-scheme S isomorphism classes of pairs (E,CN ) as in Section 1.1. ThusFN has Y0(N) as a representable hull. Also, for simplicity we will only give con-structions over fields, or sometimes even only over algebraically closed fields, withreferences to [DI] for the general construction.

Now, let N be an integer dividing another integer M . Let d be an integerdividing the quotient M/N ; d = 1 is permitted. We will define a map

Bd : X0(M)→ X0(N)

called a degeneracy operator. We will do this by defining a natural transformationof the functors FM and FN . So let K be a field containing Q and let (E/K,CM )be a pair of an elliptic curve defined over K and a cyclic subgroup of order M ofE(K). CM has a unique subgroup Cd of order d. Furthermore, the cyclic subgroupCM/Cd of order M/d on the elliptic curve E/Cd has a unique subgroup CN oforder N , since N divides M/d. (See [Si-AEC, Chapter 3, Section 4] for basics onquotients of elliptic curves.) We let Bd (as a natural transformation of functors) bethe correspondence

(E,CM ) 7→ (E/Cd, CN ).As above, Bd induces a map, which we shall also call Bd, from

X0(M)→ X0(N).

(See [DI, Sections 6.3 and 7.3] for the general case.)In the case K = C, one can ask what effect Bd has on q-expansions. If qM is

the parameter on X0(M) and qN is the parameter on X0(N), it is not difficult toshow (from the analysis in Section 1.2) that under Bd, qN pulls back to qdM .

2.2. Atkin-Lehner Involutions. We now define maps from modular curvesto themselves. So fix an integer N , and let N = ab be a factorization of N with aand b relatively prime. We will define a map

wa : X0(N)→ X0(N)

called an Atkin-Lehner operator, again by defining a natural transformation fromFN to FN . (See [DI, Section 4 and Remark 8.4.1] for the general case.)

So let K be a field containing Q and let (E/K,CN ) be a modular pair. Thegroup CN has a unique decomposition (since a and b are relatively prime)

CN = Ca + Cb

138 17. LECTURE 17

as a product of a cyclic group of order a and a cyclic group of order b. Also recallthat the full a-torsion E[a] of E is free of rank 2 over Z/aZ. We define wa to bethe natural transformation

(E,CN ) 7→ (E/Ca, E[a] + Cb/Ca);

since E[a]/Ca is cyclic of order N and Ca ∩ Cb = 1 this makes sense.We further claim that

w2a = 1.

To see this, note that

w2a(E,CN ) = wa(E/Ca, E[a] + Cb/Ca) = (E/E[a], (E/Ca)[a] + Cb/E[a]).

One checks easily that multiplication by a

a : E/E[a]→ E

yields the desired isomorphism with (E,CN ).In the case K = C, we can also interpret wa in terms of an action on the upper

half-plane. Going through the analysis in Section 1.2, one finds that there is acommutative diagram

H //

H

Y0(N)(C)wn // Y0(N)(C)

where the vertical maps are the quotient maps and the top horizontal map is

τ 7→ − 1aτ.

Note that it is completely unclear what the operation on q-expansions is; to deter-mine it, one must find the q-expansion

e−2πi/aτ ,

which is rather mysterious.

2.3. Jacobians of Curves. We briefly recall some mapping properties ofjacobians of curves. (See Lecture 13, Section 1.1 for the basics on jacobians.) So letX and Y be smooth, proper, irreducible curves over some field K with jacobiansJX and JY respectively. Suppose that we have a non-constant map

ϕ : X → Y ;

thus ϕ has finite degree. It induces both a covariant map

ϕ∗ : JX → JY

and a contravariant mapϕ∗ : JY → JX .

See [Si-AEC, Chapter 2, Section 3] for the corresponding maps on divisors. Thecomposition

ϕ∗ϕ∗ : JY → JY

is simply multiplication by the degree of ϕ.We will write J0(N) for the jacobian of the modular curve X0(N). We fix the

embeddingX0(N) → J0(N)

2. OPERATORS ON MODULAR CURVES 139

so that the cusp∞ of X0(N) maps to the origin of J0(N). Thus an arbitrary pointx ∈ X0(N) maps to the divisor class of

(x)− (∞).

2.4. Hecke Correspondences. Fix an integer N and a prime number l notdividing N . We will define a correspondence Tl on X0(N) called the lth Heckecorrespondence or the lth Hecke operator. For our purposes, a correpondence willsimply be a many-valued

X0(N) //___ X0(N).

Alternately, and somewhat more accurately, a self-correspondence on X0(N) is asubscheme Tl of the product X0(N)×SpecQ X0(N). See [DI, Section 3.2] for moredetails on correspondences, and [DI, Section 3 and Section 8.3] for details on Heckeoperators.

For our first definition, we work on the level of the functor FN over an al-gebraically closed field K. Let (E,CN ) be a modular pair over K. Recall thatE[l] ∼= Z/lZ × Z/lZ. We consider the subgroups of E[l] of order l + 1; there arel+ 1 of them and they are naturally classified by P1(Fl). (See [Se-CA, Chapter 7,Section 5] for all combinatorial analysis in this section.) We define Tl by sending(E,CN ) to the l + 1 values

(E/Cl, Cl + CN/Cl),

where Cl runs over the subgroups of E[l] of order l.In the case K = C we have X0(N)(C) = Γ0(N)\H and we can be a bit more

explicit. If τ ∈ Γ0(N)\H, Tl(τ) is the set of values

τ

l,τ + 1l

, · · · , τ + l − 1l

, lτ.

The advantage of this expression is that the number N does not appear; the Heckecorrespondence Tl is thus in some sense the same on all modular curves.

We can also use the above analysis to determine the effect of Tl on q-expansions.We consider the case of a differential form

fdq

q=∑

anqn dq

q.

The Hecke correspondence is clearly linear, so it will suffice to consider the case of

qndq

q= e2πinτ dq

q.

Under Tl, this goes to

1l

l−1∑j=0

e2πin(τ+j)/l dq

q+ le2πinlτ dq

q=

1l

l−1∑j=0

e2πinj/l · e2πinτ/l dq

q+ le2πinlτ dq

q.

Butl−1∑j=0

e2πinj/l =

0 l - n

l l | n,

so the above expression reduces to

qn/ldq

q+ lqnl

dq

q,

140 17. LECTURE 17

where we interpret qn/l as 0 if l does not divide n. In general, then,

Tl

(∑anq

n dq

q

)=∑

(anl + lan/l)qndq

q,

where we set an/l = 0 if l does not divide n.One can also define Tl in terms of the operator

B1 : X0(Nl)→ X0(N)

and the Atkin-Lehner involution wl on X0(Nl). We write π for B1 for this section.Consider the (many-valued) composition

X0(N) π−1

−→ X0(Nl) wl−→ X0(Nl) π−→ X0(N).

The first map sends a modular pair (E,CN ) to the set of all pairs (E,CN + Cl),where Cl is a cyclic subgroup of E of order l. wl then sends (E,CN + Cl) to(E/Cl, E[l] + CN/Cl), which is mapped by π to (E/Cl, Cl + CN/Cl). Thus thiscomposition is precisely the Hecke correspondence Tl.

This interpretation allows us to interpret Tl as an endomorphism of the Jaco-bian J0(N). Specifically, the commutative diagram

X0(Nl)wl //

π

X0(Nl)

π

X0(N)Tl //___ X0(N)

is reinterpreted as a commutative diagram

J0(Nl)w∗l // J0(Nl)

π∗

J0(N)Tl∗ //

π∗

OO

J0(N)

(That is, we take π∗wl∗π∗ as the definition of Tl∗.) We could also define T ∗l by thismethod, but note that since wl is an involution we have wl∗ = w∗l , so

T ∗l = π∗w∗l π∗ = Tl∗.

We will often just write Tl for the map on J0(N) induced by the Hecke correspon-dence.

There is also a double coset decomposition definition of Hecke operators, whichis quite useful for generalizations. Ask the local Gross student or see [DI, Section3.1].

CHAPTER 18

Lecture 18

1. Representations on Torsion Points

1.1. The Tree of a Modular Pair. For this section fix a modular pair(E,CN ) defined over the complex numbers C. (The precise choice of CN will onlycome into play later.) We associate to this pair a graph (actually a building)Γ(E,N) in the following way. The vertices of Γ(E,N) are in bijection with thecyclic subgroups of E(C) of order relatively prime to N . Two vertices C1 and C2

are joined by an edge if one is contained in the other with index some prime l. Wewill consider this edge to have weight l.

For any prime l not dividing N , we also define a subgraph Γl(E) consisting ofthose vertices of Γ(E,N) corresponding to cyclic subgroups of l-power order (andall of the edges in Γ(E,N) connecting them). Γl(E) is in fact a tree, as one easilysees.

For any l not dividing N , we define a Hecke correspondence Tl on the verticesvert(Γ(E,N)) by sending a vertex v to all vertices which are joined to v by an edgeof weight l. This induces a homomorphism on divisor groups

Tl : Z[Γ(E,N)]→ Z[Γ(E,N)];

here by Z[Γ(E,N)] we mean the free abelian group on the vertices of Γ(E,N).Most Hecke correspondences do not restrict to Γl(E); the only one that does is

Tl.

1.2. Galois Actions on Γ(E,N). Suppose now that the modular pair (E,CN )is defined over a subfield K of C. We let E(N)

tors be the subgroup of the torsion pointsof E of order relatively prime to N . Define K(E,N) to be the splitting field ofthe natural GK-action on E

(N)tors ; equivalently, K(E,N) is generated over K by the

coordinates of all of the prime to N torsion points of E(C). If K is a number field,then K(E,N) is certainly an infinite extension of K.

Note that Aut(E(N)tors) also acts on the graph Γ(E,N): if ϕ ∈ Aut(E(N)

tors) andv ∈ vert(Γ(E,N)) corresponds to the cyclic group C ⊆ E(N)

tors , then ϕv is the vertexcorresponding to ϕ(C). One easily checks that this action preserves edges, so thatit really is an automorphism of the graph Γ(E,N).


Figure 1. The Beginning of the Graph Γ for l = 2

141

142 18. LECTURE 18

Now, any choice of isomorphism

E(N)tors∼=∏l-N

Ql/Zl ×Ql/Zl

yields an isomorphism

Aut(E(N)tors) ∼=

∏l-N

GL2(Zl).

Note that the subgroup of scalar matrices is independent of the choice of “basis”ofE

(N)tors (it is just those elements of the automorphism group which act on each E[M ]

as a scalar), and furthermore that it acts trivially on Γ(E,N), since any cyclicsubgroup is taken to itself by scalars. Thus the map∏

l-N

GL2(Zl) ∼= Aut(E(N)tors)→ Aut(Γ(E,N))

factors through∏

PGL2(Zl):∏l-N GL2(Zl) //

((PPPPPPPPPPPPAut(Γ(E,N))

∏l-N PGL2(Zl)

66nnnnnnnnnnnnn

Of course, Gal(K(E,N)/K) acts in a natural way on E(N)tors , yielding a repre-

sentationρE,N : Gal(K(E,N)/K)→ Aut(E(N)

tors) ∼=∏l-N

GL2(Zl).

The action of Aut(E(N)tors) on Γ(E,N) yields a second representation

pρE,N : Gal(K(E,N)/K)→ Aut(Γ(E,N)) ∼=∏l-N

PGL2(Zl);

of course, pρE,N is just the composition of ρE,N with the natural quotient map∏l-N

GL2(Zl)→∏l-N

PGL2(Zl).

Let us now connect all of this. Let Γ(E,N) be the graph associated to amodular pair (E,CN ) defined over a subfield K of C. Then we can define a naturalmap

vert(Γ(E,N))→ X0(N)(C)

by sending a vertex corresponding to a cyclic group C of order prime to N to themodular pair

(E/C,CN + C/C).

In fact, the modular pair is defined over K(E,N), since C is, so the image of themap is contained in X0(N)(K(E,N)). Furthermore, this mapping is compatiblewith the Hecke correspondences on Γ(E,N) and X0(N)(C), as is clear from thedefinitions. Composing with the natural map

X0(N)(K(E,N))→ J0(N)(K(E,N)),

1. REPRESENTATIONS ON TORSION POINTS 143

(this map is an embedding if X0(N) has genus greater than zero, and is trivial ifX0(N) has genus 0) we obtain a Hecke equivariant map

vert(Γ(E,N))→ J0(N)(K(E,N)).

1.3. The Image of ρE,1. We now consider the case N = 1 above. Thus wehave an elliptic curve E, defined over some field K, and we are considering therespresentation of GK on Etors, which we will just write as ρE . The image of thisrepresentation turns out to depend very much on whether or not E has complexmultiplication. Recall that E is said to have complex multiplication (or CM forshort) if EndC(E) is strictly larger than Z. It follows that EndC(E) is an orderin an imaginary quadratic field. (See [Si-AEC, Chapter 6, Theorem 5.5]. For anice introduction to the theory of complex multiplication, see [Si-2, Chapter 2].Unfortunately, it does not cover complex multiplication in the generality in whichwe will need it. We will comment more on that later.) In the non-CM case, wehave the following theorem of Serre.

Theorem 1.1 (Serre). Let K be a number field and let E be an elliptic curveover K without complex multiplication. Then the representation

ρE : GK →∏

GL2(Zl) ∼= GL2(Z)

has open image. In particular, the image has finite index, and the composition

GK → GL2(Z)→ GL2(Zl)

is surjective for almost all primes l.

Proof. See [Se-EC] or [La-EF, Chapter 17].

In the CM case, ρE is very far from surjective; to see this, note that the imageof ρE must commute with all elements of EndK(E). Thus

ρE(GK) ⊆ AutEndK(E)(Etors) ⊆ Aut(Etors).

If EndK(E) is larger than Z, then this subgroup is much smaller than the wholeof Aut(Etors). Furthemore, although EndK(E) need not be larger than Z eventhough E has complex multiplication, there exists a finite extension L/K such thatEndL(E) is larger than Z (see [Si-2, Chapter 2, Theorem 2.2]), which is enough toforce ρE(GL) to be small.

One can actually characterize ρE(GK) fairly precisely in the CM case. Let F bethe quadratic imaginary field by which E has complex multiplication, with Galoisgroup g = Gal(F/Q) = 1, τ. Consider the group

(OF ⊗Z Zl)∗/Z∗l ;

here OF is the ring of integers in F , and

Z∗l → (OF ⊗ Zl)∗

is induced by the natural inclusion

Z → OF .

We define Dl to be the semi-direct product

Dl = gn (OF ⊗ Zl)∗/Z∗l ,

where τ ∈ g acts on (OF ⊗ Zl)∗/Z∗l by inversion.

144 18. LECTURE 18

Dl can actually be realized as a subgroup of PGL2(Zl). This is done by firstchoosing a Z-basis of OF , which is a free rank 2 Z-module. The action of (OF ⊗Zl)∗on OF ⊗ Zl now yields an embedding

(OF ⊗ Zl)∗ → GL2(Zl)

which in turn yields(OF ⊗ Zl)∗/Z∗l → PGL2(Zl).

Next, one observes that this group is a Cartan subgroup of PGL2(Zl), meaning thatit becomes a maximal torus over the algebraic closure. (Recall that a torus is agroup isomorphic to a product of multiplicative groups.) In this situation, it isa general fact that the group has index 2 in its normalizer, and that non-trivialelements of the normalizer act by inversion. Thus we have realized

Dl → PGL2(Zl)

as the normalizer of(OF ⊗ Zl)∗/Z∗l → PGL2(Zl).

The relevance of Dl to us rests on the fact that ρE(GK) is conjugate to asubgroup of Dl.

2. Heegner Points

2.1. Ring class fields. For an excellent introduction to orders and ring classfields and proofs of the results in this section, see [Cox, Chapter 2, Sections 7 and9].

Let F be an imaginary quadratic field. A subring R of the ring of integers OFis called an order if it is finitely generated and if it generates all of F over Q. Itfollows easily that R is a lattice in C. One can also show that all orders are of theform

OF,n = Z+ nOFfor some integer n.

The ideal theory of OF,n is quite similar to that of OF ; in particular, if a isan ideal of OF relatively prime to n, then a ∩ OF,n is an invertible ideal in OF,n.Furthermore, the ideal class group Pic(OF,n) of OF,n (this is not defined exactly asone might expect; see [Cox, p. 136]) is an extension of the ideal class group of OFby the abelian group

(OF /nOF )∗/(Z/nZ)∗.Orders have important connections to the class field theory of F . Specifically,

for any positive integer n we define the ring class field F (n) of conductor n to bethe unique abelian extension of F , unramified away n, such that a prime p of OFsplits completely in F (n) if and only if p is principal with a generator α such thatα is congruent to a rational integer modulo n. (That F (n) exists follows from theusual statements of class field theory; see [Cox, p. 145 and pp. 179–180].)

Note that F (1) is just the Hilbert class field of F . Furthermore, it follows fromthe definition that Gal(F (n)/F ) is isomorphic to the ideal class group of OF,n; fromthis and the computation of the ideal class group of OF,n, we see that

Gal(F (n)/F (1)) ∼= (OF /nOF )∗/(Z/nZ)∗.

F (n) is also Galois over Q; here one finds that

Gal(F (n)/Q) ∼= go Pic(OF,n)

2. HEEGNER POINTS 145

where τ ∈ g operates on Pic(OF,n) in the natural way, which works out to just beby inversion. In fact, F (n) can also be characterized as the maximal generalizeddihedral extension of Q of conductor n; see [Cox, Chapter 2, Section 9, Part D].

2.2. Heegner Points. We will now need some of the theory of complex mul-tiplication over non-integrally closed fields. Most of what we need is contained in[Cox, Chapter 3, Section 11, esp. Theorem 11.1]; alternately, [La-EF] and [Shi]have extensive treatments of the theory of complex multiplication.

Let N be a positive integer and let F be a quadratic imaginary field in whichevery prime dividing N splits completely. (The existence of such an F follows easilyfrom the quadratic reciprocity law.) For simplicity, we assume that F is not Q(i)or Q(

√−3); the existence of non-trivial units complicates things somewhat. We

also must fix an embedding F → C. Our hypothesis on F implies that there is anideal N of OF such that OF /N is cyclic of order N . Fix such an ideal N . We alsohave that N−1/OF is cyclic of order N .

Now, let n be an integer relatively prime to N . We consider the order OF,nand the ideal Nn = N ∩OF,n. Nn is invertible in OF,n (see the previous section),and we still have that N−1

n /OF,n is cyclic of order N .We associate to each pair (OF,n,N−1

n ) the modular pair

(C/OF,n,N−1n /OF,n)

defined over C; note that OF,n is a lattice, so C/OF,n really is an elliptic curve, andthat N−1

n /OF,n is a cyclic subgroup of order N . This modular pair then defines aHeegner point

x(n) ∈ X0(N)(C).For our applications, we will need the following theorem, which is essentially

[Cox, Theorem 11.1].Theorem 2.1. Let n be relatively prime to N . Let E be the elliptic curve over

C defined by C/OF,n and let C be the cyclic subgroup of order N of E defined byN−1n /OF,n. Then the modular pair (E,C) is defined over the ring class field F (n).

In particular,x(n) ∈ X0(N)(F (n)).

CHAPTER 19

Lecture 19

1. Hecke Operators on Heegner Points

1.1. Galois Actions on Cyclic Subgroups. Let us review briefly our nota-tion and hypotheses. N is a fixed positive integer and F is a quadratic imaginaryextension of Q in which each prime dividing N splits completely. We further requirethat F is not Q(i) or Q(

√−3). We fix an embedding F → C and an ideal N of F

such that OF /N is cyclic of order N . For each n relatively prime to N we let OF,nbe the order Z+ nOF of conductor n in OF , and we let F (n) be the ring class fieldof F of conductor n. Thus

Gal(F (n)/F ) ∼= Pic(OF,n)

andGal(F (n)/F (1)) ∼= (OF /nOF )∗/(Z/nZ)∗.

We now further assume that X0(N) has positive genus and we fix an embeddingX0(N) → J0(N). x(n) is the point of X0(N) corresponding to the modular pair(E(n), C

(n)N ), where E(n) = C/OF,n, Nn = N ∩ OF,n and C

(n)N = N−1

n /OF,n. x(n)

actually lies in X0(N)(F (n)), and we will also regard it as an element of J0(N)(F (n))via our fixed embedding.

Fix a prime l not dividing N . We claim that for any n relatively prime to lN ,

Gal(F (nl)/F (n)) = Gal(F (l)/F (1)).

This follows from the fact that F (n) ∩ F (l) = F (1), which is true since the twofields have relatively prime conductor (in the sense of class field theory; see [Cox,Exercise 9.20]). We denote this Galois group by g(l).

F (nl)

g(l)

xxxxxxxx

FFFFFFFF

F (n)

GGGGGGGG F (l)

g(l)xxxxxxxx

F (1)

Fix such an n. As we have said, the elliptic curve E(n) has a model over F (n); infact, it is defined over an index 2 subfield of F (n) (see [Shi, Theorem 5.7]) and thusthe model over E(n) is only well-defined up to twisting by a quadratic character. Inany event, fixing such a model yields a Galois representation

GF (n) → Aut(E(n)[l]).


147

148 19. LECTURE 19

The image of GF (n) must commute with the endomorphisms of E(n) defined overF (n), which is all of OF,n (see [Si-2, Theorem 2.2.6]). Thus the image of the Galoisrepresentation actually lies in

AutOF,n/lOF,n(E(n)[l]).

E(n)[l] is free of rank 1 over OF,n/lOF,n ∼= OF /lOF , so we have a Galois represen-tation

GF (n) → AutOF,n/lOF,n ∼= (OF,n/lOF,n)∗ ∼= (OF /lOF )∗.

We are actually only interested in the action of GF (n) on the cyclic subgroupsof E(n) of order l, so we need only consider the induced map

GF (n) → (OF /lOF )∗/F∗lsince scalars act trivially on the set of cyclic subgroups. This map is surjective andfactors through Gal(F (nl)/F (n)) ([Shi, Theorem 5.7]), yielding a map

g(l) → (OF /lOF )∗/F∗lwhich is now an isomorphism, since each group has the same order.

Let us now assume that l is inert in F/Q. Then OF /lOF ∼= Fl2 . Now, E(n)[l]is an Fl2 vector space of dimension 1, and thus an Fl vector space of dimension 2.One thus sees immediately that F∗l2/F

∗l acts simply transitively on the collection

PFl(E(n)[l]) of one dimensional Fl subspaces of E(n)[l], which are precisely the cyclicsubgroups of E(n) of order l. Combining all of this, we have seen that g(l) actssimply transitively on the collection of cyclic subgroups of E(n) of order l.

In order to restate our above analysis in a more useful way we first must definea trace map

Tr(nl,n) : J0(N)(F (nl))→ J0(N)(F (n)).

This is nothing more than the natural map

x 7→∑σ∈g(l)

σx.

Proposition 1.1. With the above notation and hypothesis (in particular, l isinert in F/Q), we have

Tlx(n) = Tr(nl,n) x

(nl),

where Tl is the usual Hecke operator on J0(N).

Proof. By definition,

Tlx(n) =

∑Cl⊆E(n)[l]

(E(n)/Cl, C(n)N + Cl/Cl),

where the sum is over all cyclic subgroups of E(n) of order l. On the other hand,

Tr(nl,n) x(nl) =

∑σ∈g(l)

σx(nl)

=∑σ∈g(l)

σ(E(nl), C(nl)N )

=∑σ∈g(l)

σ(C/OF,nl,N−1nl /OF,nl)

1. HECKE OPERATORS ON HEEGNER POINTS 149

Note that

(C/OF,nl,N−1nl /OF,nl) = ((C/OF,n)/Cl, (N−1

n /OF,n)/Cl)

for some unique cyclic group of order l Cl ∈ E(n), since OF,n/OF,nl is cyclic of orderl. Furthermore, by definition

σ((C/OF,n)/Cl, (N−1n /OF,n)/Cl) = (C/OF,n)/σCl, (N−1

n /OF,n)/σCl);

combined with the two expressions above and the transitivity of the action of g(l)

on the cyclic subgroups of E(n) of order l, this proves the proposition.

Although we will not use it, we sketch the corresponding result in the case thatl splits in F/Q. Let λ and λ be the primes of F lying over l. In this case,

g(l) = F∗l × F∗l /F∗l ,

the last F∗l being the diagonal copy of F∗l . So g(l) is cyclic of order l − 1, andit acts on the PFl(E(n)[l]) of cyclic subgroups of order l in E(n). Of course, g(l)

can not act on this projective space principally homogeneously. In fact, it fixesthe two distinguished subgroups E(n)[λ] and E(n)[λ′], and then acts principallyhomogeneously on PFl(E(n)[l])− E(n)[λ], E(n)[λ′]. In this case, then, the formulaof Proposition 1.1 must be modified slightly. Specifically, if one denotes by σλ andσλ the elements of Gal(F (n)/F ) produced by λ ∩ OF,n and λ ∩ OF,n under theisomorphism

Pic(OF,n) ∼= Gal(F (n)/F ),then one has the formula

Tr(nl,n) x(nl) = (Tl − σλ − σλ)x(n)

= (Tl − σλ − σ−1λ )x(n).

With a little care, then, one can derive the general formula

Tr(nl,n)

(−σλx(nl) + σ2

λx(n))

=(1− Tlσλ + lσ2

λ

)x(n)

of Mazur and Rubin, which works independently of the behaivor of l in F/Q.

1.2. Hecke Algebras. We will now relate all of our data to our earlier nota-tion. For simplicity we now require that N is squarefree. Let T ⊂ EndQ(J0(N)) bethe subalgebra generated by 1, Tl for l prime not dividing N , and Uq for q dividingN . T is a commutative, finite, free Z-algebra. (See [DI, Sections 3.3-3.4].)

Let m be a maximal ideal of T such that T/m has characteristic p 6= 2. Let Bbe the abelian variety J0(N), and let

W =⋃r

B(Q)[mr]

be the m-divisible subgroup of B(Q). W is of course also p-divisible. Both T andGQ act on W , and the actions commute since T is defined over Q.

W is actually a module over

Tm = lim←− T/mr.

We will write A for this ring; it satisfies all of our usual hypotheses. Let H bethe Cartier dual of W ; it is a free A module of finite rank. We have a Galoisrepresentation

ρ : GQ → AutA(H)

150 19. LECTURE 19

and a residual representation

ρ : GQ → Autk(H ⊗A k),

where k = T/m = Tm/mTm.We make two hypothesis on the maximal ideal m. First, we assume that H is

free of rank 2 over Tm. Second, we assume that

ρ : GQ → GL2(k)

is surjective. We call any maximal ideal with these properties good. We will seelater that these hypotheses imply all of the hypotheses of Kolyvagin’s theorem.

We also assume that there is an η ∈ T such that ηT ∈ m, η : B → B is anisogeny and

Z ∩ ηT = psZ

for some s ≥ 1. We will say that a prime l (inert in F/Q and not dividing N) isgood for η if

Tl ∈ ηAand

l + 1 ∈ ηA.Note that in particular this implies that ps divides l + 1, and thus that ps dividesthe order of g(l).

1.3. Computations. We continue with the above notation, and we supposethat we have a good maximal ideal m, an η as above, and a prime l which is goodfor η. We define

P (l) =∑g∈g(l)

gx(l) ⊗ g ∈ B(F (l))⊗Z Z[g(l)].

Let P (l)η be the “reduction of P (l) modulo η”; that is, P (l)

η is the image of P (l) in

T/η ⊗T B(F (l))⊗Z Z[g(l)].

Note thatT/η ⊗T B(F (l))

is killed by ps since T/η is, so

T/η ⊗T B(F (l))⊗Z Z[g(l)] = T/η ⊗T B(F (l))⊗Z/psZ Z/psZ[g(l)].

Next, let I(l) be the augmentation ideal of Z/psZ[g(l)], so that there is an exactsequence

0→ I(l) → Z/psZ[g(l)]→ Z/psZ→ 0,

the last map being given by evaluating each g ∈ g(l) at 1. Each of these groupsis free over Z/psZ, so the exact sequence splits over Z/psZ. In particular, thesequence

0 −−−−→ T/η ⊗T B(F (l))⊗Z/psZ I(l) −−−−→ T/η ⊗T B(F (l))⊗Z/psZ Z/psZ[g(l)]

−−−−→ T/η ⊗T B(F (l))⊗Z/psZ Z/psZ −−−−→ 0

is still exact.We claim that

P (l)η ∈ T/η ⊗T B(F (l))⊗Z/psZ I(l).

1. HECKE OPERATORS ON HEEGNER POINTS 151

To prove this, it suffices to show that P (l)η is in the kernel of the summation map

T/η ⊗T B(F (l))⊗Z/psZ Z/psZ[g(l)]→ T/η ⊗T B(F (l))⊗Z/psZ Z/psZ.

Equivalently, we must show that the image of P (l) under

B(F (l))⊗Z Z[g(l)]→ B(F (l))⊗Z Z = B(F (l))

is contained in ηB(F (l)). This image is∑g∈g(l)

gx(l) ⊗ 1 =∑g∈g(l)

gx(l)

= Tr(l,1) x(l)

= Tlx(l)

by Proposition 1.1. But by hypothesis,

Tl ∈ ηT,so this establishes the claim.

Next, let P(l)η be the image of P (l)

η in

T/η ⊗T B(F (l))⊗Z/psZ I(l)/(I(l))2.

Note that the mapg(l) → I(l)

sending g to g − 1 induces an isomorphism

g(l) ⊗Z Z/psZ→ I(l)/(I(l))2.

Thus we can (and do) regard P(l)η as an element of

T/η ⊗T B(F (l))⊗Z/psZ(g(l) ⊗Z Z/psZ

).

Note also that g(l) ⊗ Z/psZ is cyclic of order ps, so

T/η ⊗T B(F (l))⊗Z/psZ(g(l) ⊗Z Z/psZ

)∼= T/η ⊗T B(F (l)),

at least as abelian groups.We claim that P(l)

η is fixed under the action of g(l) on the B(F (l)) coordinate.To see this, we compute for σ ∈ g(l)

σP (l) − P (l) =∑g∈g(l)

σgx(l) ⊗ g −∑g∈g(l)

gx(l) ⊗ g

=∑g∈g(l)

gx(l) ⊗ gσ−1 −∑g∈g(l)

gx(l) ⊗ g

=∑g∈g(l)

gx(l) ⊗ g(σ−1 − 1)

CHAPTER 20

Lecture 20

1. The Euler System for X0(11)

1.1. The Set-Up. For simplicity we now restrict our investigations to thecase N = 11; it is in many ways typical of the general case, and it will help to cutdown on the notation. Recall that X0(11) is a curve of genus 1 with good reductionaway from 11, and we make it into an elliptic curve by using the cusp at infinity asthe base point. This yields a canonical isomorphism

X0(11) ∼= J0(11),

and we will also use B to denote this elliptic curve.X0(11) can actually be realized by the Weierstrass equation

y2 + y = x3 − x2 − 10x− 20,

and the group B(Q) of rational points is cyclic of order 5. The standard everywhereregular differential on X0(11) has a Fourier expansion

ω(q) = η(q)2η(11q)2

= q∞∏n=1

(1− qn)2(1− q11n)2

=∞∑n=1

anqn

= q − 2q2 − q3 + 2q4 + q5 + 2q6 − 2q7 − 2q9 − 2q10 + q11 + · · · .

The image T of the Hecke algebra in EndQ(J0(11)) = Z is just Z, and the Heckeoperator Tl (for l 6= 11) on J0(11) can be identified with its eigenvalue al. Forl = 11, U11 is identified with a11 = 1. The Atkin-Lehner involution w11 is thenegative of this, and thus is the involution −1.

For every p, we obtain a Galois representation

ρp : GQ → Aut(B[p]) ∼= GL2(Fp).

ρp is surjective for all p 6= 2, 5, 11. For p = 5 this lack of surjectivity is forced by thefact that B has a rational point of order 5; the image of ρ5 in GL2(F5) is simplythe subgroup of diagonal matrices with 1 in the upper left coordinate.

We let F be any quadratic imaginary field Q(√−d) in which 11 is inert; by

quadratic reciprocity, this condition says precisely that

d ≡ −1, 2,−3,−4,−5 (mod 11).


153

154 20. LECTURE 20

(Note that this automatically eliminates the fields Q(√−1) and Q(

√−3).) We also

fix a factorization 11 = N · N in OF .Now fix a prime p 6= 2, 3, 5, 11. Our maximal ideal m is just (p), and it is easily

seen to be good. Our element η is a power ps of p, and a prime l, inert in F/Q, isgood for η if and only if

l + 1 ≡ al ≡ 0 (mod ps).

Fix such an l.For later use, we define the basic Heegner point P ∈ B(F ) by

P = TrF 1/F x1.

1.2. The Construction. We now briefly review our construction of Lecture19 in this simplified notation. We begin with a point

P (l) =∑g∈g(l)

gx(l) ⊗ g ∈ B(F (l))⊗Z Z[g(l)].

We now “reduce P (l) modulo ps”; that is, we take P (l)η to be image of P (l) in

B(F (l))⊗Z Z/psZ[g(l)].

As discussed in Lecture 19, Section 1.3, P (l)η actually lies in the submodule

B(F (l))⊗Z I(l).

We define P(l)η to be the image of P (l)

η in

B(F (l))⊗Z I(l)/(I(l))2 ∼= B(F (l))⊗Z (g(l) ⊗Z Z/psZ).

We also fix an isomorphism

g(l) ⊗Z Z/psZ ∼= Z/psZ;

this allows us to consider P(l)η as an element of

B(F (l))⊗Z Z/psZ = B(F (l))/psB(F (l)).

Letting g(l) act on B(F (l))/psB(F (l)) in the natural way, we have as before

P(l)η ∈

(B(F (l))/psB(F (l))

)g(l)

.

We now relate P(l)η to Galois cohomology and in particular to our abelian variety

finite/singular structure on B[ps]. We begin with the exact sequence

0→ B[ps]→ Bps−→ B → 0.

Taking the long exact sequence for GF (1) and GF (l) cohomology yields a commuta-tive diagram with exact rows

0 // B(F (1))/psB(F (1)) //

H1(GF (1) , B[ps]) //

H1(GF (1) , B)[ps] //

0

0 // B(F (l))/psB(F (l)) // H1(GF (l) , B[ps]) // H1(GF (l) , B)[ps] // 0

1. THE EULER SYSTEM FOR X0(11) 155

Note that the two horizontal sequences are simply our finite/singular sequences forthe Galois module B[ps] over F (1) and F (l) respectively. In particular, we have anidentification

H1(GF (1) , B)[ps] = H1s (GF (1) , B[ps]).

In fact, the vertical maps all have image in the g(l)-invariant subgroups, yieldinga diagram

0 // B(F (1))/psB(F (1)) //

H1(GF (1) , B[ps]) //

ϕ

H1(GF (1) , B)[ps] //

ψ

0

0 // (B(F (l))/psB(F (l)))g(l) // H1(GF (l) , B[ps])g(l) // H1(GF (l) , B)[ps]g(l)

Consider the map

ϕ : H1(GF (1) , B[ps])→ H1(GF (l) , B[ps])g(l).

By the inflation-restriction sequence, this has kernel H1(g(l), B(F (l))[ps]) and cok-ernel H2(g(l), B(F (l))[ps]). However, we must have B(F (l))[ps] = 0, since if B hadrational ps-torsion in a dihedral extension of Q, the image of the representationρp would factor through this dihedral group and an abelian subgroup of GL2(Fp),and thus be solvable. Since GL2(Fp) is not solvable for p 6= 2, 3, our assumptionsimply that this is not possible. In particular, this implies that ϕ is an isomor-phism. The same is not true of ψ, but at the very least we can identify its kernelas H1(g(l), B(F (l)))[ps].

Let us now push our element P(l)η of (B(F (l))/psB(F (l)))g(l)

around the dia-gram. It yields an element of of H1(GF (l) , B[ps])g(l)

, and thus via ϕ−1 an elementof H1(GF (1) , B[ps]). Pushing this forward we obtain an element

c(l)η ∈ H1(GF (1) , B)[ps]

which maps to 0 under ψ, and thus also an element

d(l)η ∈ H1(g(l), B(F (l)))[ps].

We are now in a position to define our long awaited Kolyvagin-Flach system.We define

κ(l)η ∈ H1

s (GF , B[ps])

to be the image of c(l)η under the corestriction map

cor : H1s (GF (1) , B[ps])→ H1

s (GF , B[ps]).

We will now summarize the main remaining results; proofs will be given (hope-fully) in later lectures.

Theorem 1.1. Let ε be the eigenvalue −a11 of W11. Then κ(l)η is in the −ε

eigenspace of H1s (GF , B[ps]).

Recall that P is the basic Heegner point and lies in B(F ). We let Fl be thecompletion of F at l; since l is inert this makes sense. We also restrict now to thecase η = p.

Theorem 1.2. If P ∈ pB(Fl), then Suppκ(l)η is empty. If P /∈ pB(Fl), then

Suppκ(l)η = l and the depth of κ(l)

η is 1.

156 20. LECTURE 20

Corollary 1.3. Let F = Q(√−d) where d ≡ −1, 2,−3,−4,−5 (mod 11). Let

p be a prime distinct from 2, 3, 5, 11. Then if P is not divisible by p in J0(11)(F ),we have

X(J0(11)/Q) = 0.Corollary 1.4. If there exists a field F as above for which P is of infinite

order, then the p-primary component of X(J0(11)/Q) vanishes for almost all p.

CHAPTER 21

Lecture 21

1. Local Behavior of Cohomology Classes

1.1. X0(11) over Finite Fields. We continue with the analysis of the classes

c(l)η ∈ H1(GF (1) , B)[p]

d(l)η ∈ H1(g(l), B(F (l)))[p]

κ(l)η ∈ H1

s (GF , B[p])

for the modular curve B = X0(11) = J0(11). We begin by considering the actionof Frl on B[p]. (Here by Frl we mean the Frobenius of l in the extension of Fgenerated by the coordinates of B[p]. This makes sense since B has good reductionat l and l 6= p.) Frl has characteristic polynomial x2−alx+l on the two-dimensionalFp-vector space B[p]; since l is good for p, we have

x2 − alx+ l ≡ x2 − 1 (mod p).

In particular, we see immediately that Frl is a non-scalar involution on B[p]. (It isnon-scalar since it has distinct eigenvalues.)

Next consider the points of B over the residue fields k0 and k of Ql and Flrespectively; of course, k0 = Fl and k = Fl2 since l is inert in F/Q. It is a standardfact that

|B(k0)| = 1 + l − aland

|B(k)| = (1 + l)2 − a2l .

In particular, since l is good for p we see that B(k0) has a point of order p, and allof B[p] is contained in B(k). Note that B(k0) does not contain all of B[p], sinceFrl projects to the non-trivial element of Gal(k/k0) and acts as a non-scalar, andthus in particular is non-trivial.

If we consider the eigenspaces of B(k) under the action of Frl (or equivalentlyunder the action of the non-trivial element of Gal(F/Q) = Gal(k/k0)), we obtain adecomposition

B(k)p = B(k)+p ⊕B(k)−p

= B(k0)p ⊕B(k)−p

where B(k)p is the p-primary component of B(k); we do not actually obtain such adecomposition of all of B(k) because of problems with 2. In any event, we do findthat

B(k)[p] = B(k0)[p]⊕B(k)[p]−


157

158 21. LECTURE 21

and our above analysis shows that both direct summands are cyclic of order p.This implies that B(k0)p and B(k)−p are themselves cyclic, and combining all ofthis yields the following lemma.

Lemma 1.1. B(k)+p = B(k0)p is cyclic of order

pordp(1+l−al)

and B(k)−p is cyclic of order

pordp(1+l+al).

1.2. The Kolyvagin-Flach Theorem. Recall that our eventual goal is toprove Lecture 20, Theorem 1.2 and to derive Lecture 20, Corollary 1.3 from it. Forthe moment let us discuss this second issue. The difficulty is that Theorem 1.2only yields Kolyvagin-Flach classes for primes l such that the basic Heegner pointP is not a pth-power in B(Fl). Using the notation from earlier in the course, ourKolyvagin-Flach theorems to this point have required that we have such classes forall l ∈ Lτ , where Lτ is the set of places of F for which there exists a place w of FJover v such that FrFJ/F (w) is the fixed non-scalar involution τ ∈ Gal(FJ/Q). Inour case we can identify Lτ with the set of l which are good for p. (This essentiallycomes from the fact that for any l 6= p, the action of Frl on B[p] has characteristicpolynomial x2− alx+ l; which ones yield non-scalar involutions is then clear.) We,however, only have Kolyvagin-Flach classes for the subset L′τ of Lτ for which P isnot a pth-power locally.

Let us define this set L′τ purely field theoretically. Let F ′ be a (probably non-Galois) extension of F generated by some pth root of P . Then L′τ is precisely theset of place of F which are good for p and which have no degree 1 places of F ′ overthem. (The existence of such a place means exactly that P is a pth-power over thecompletion of F .)

We will sketch the proof of the Kolyvagin-Flach theorem in this new situation.Abstractly, we add a new field F ′/F and we define F ′J = F ′FJ and L′ = F ′L.(Again we are using our earlier notation.) We make the hypothesis that FJ 6= F ′J .

L′

AAAAAAAA

F ′J

AAAAAAAA L

Γ

F ′

BBBBBBBB FJ

∆J

F

F0

We claim that in this situation it is enough to have a Kolyvagin-Flach systemfor L′τ (as defined above). Specifically, we want to show that if ϕ(Frw) = 0 forsome w dividing each v ∈ L′τ , then ϕ = 0. To see this, we first lift τ to an element

1. LOCAL BEHAVIOR OF COHOMOLOGY CLASSES 159

τ ′ ∈ Gal(L′/F0). We now consider ϕ((g′τ ′)2) only for those g′ ∈ Gal(L′/FJ) −Gal(L′/F ′J). The exact same proof works for these g′ to show that

ϕ(Gal(L′/FJ)−Gal(L′/F ′J)) = 0.

But this implies that ϕ vanishes on all of Gal(L′/FJ), since we already know thatit vanishes on the group minus a proper subgroup (since we assumed F ′J 6= FJ) andϕ is a homomorphism. This completes the proof of the Kolyvagin-Flach theoremin this restricted situation.

1.3. Finiteness of the Kolyvagin-Flach Classes. The first step in provingTheorem 1.2 is the following lemma. For the remainder of the lecture we assumethat F (1) = F , for purely notational reasons; the general proofs are the same exceptthat we can avoid a few corestrictions.

Lemma 1.2. If v 6= l, then resv(κ(l)η ) ∈ H1

s (Gv, B[p]) is trivial.

Proof. Recall that we had a class d(l)η ∈ H1(g(l), B(F (l))) which mapped to

κ(l)η = c

(l)η ∈ H1

s (GF , B[p]). The commutative diagram

H1(g(l), B(F (l))) //

H1(Dw, B(F (l)))

H1(GF , B) // H1(Gv, B)

where w is a place of F (l) lying over v and Dw is the decomposition group of win g(l), shows that it will suffice to show that resw(d(l)

η ) = 0. We will do this byshowing that all of H1(Dw, B(F (l))) vanishes.

More generally, let Φ/Φ0 be an unramified, Galois extension of local fields. (Inthe case above we take Φ = F

(l)w and Φ0 = Fv.) Let B be an abelian variety over

Φ0 with good reduction over OΦ0 . We claim that

H1(Gal(Φ/Φ0), B(Φ)) = 0.

This is essentially a consequence of Lang’s theorem; we begin by noting that byproperness

B(Φ) = B(OΦ) = lim←− B(OΦ/mn)

where m is the maximal ideal of OΦ. Thus we must show that

H1(Gal(Φ/Φ0), B(OΦ/mn)) = 0

for all n. For n = 1 this is precisely Lang’s theorem; for n > 1, one uses theGreenberg functor to express B(OΦ/m

n) as the k-valued points of an n-dimensionalsmooth, connected algebraic group. Lang’s theorem again completes the proof.

The above argument works for v 6= 11, l. For v = 11, B has bad reduction,so one must use the Neron model of B and a more refined analysis in order toapply Lang’s theorem. The key is that the Neron model is just the product of themultiplicative group and a cyclic group of order 5; since 5 is prime to 11, this groupdoes not enter in to the calculations, and Lang’s theorem applies directly to Gm;we omit the details.

160 21. LECTURE 21

1.4. Singular Depth of the Kolyvagin-Flach Classes. We next mustcompute the singular depth of κ(l)

η at v = l. Recall the defining diagram

0

H1(g(l), B(F (l)))[p]

0 // B(F )/pB(F ) //

H1(GF , B[p]) //

ϕ

H1(GF , B)[p] //

ψ

0

0 // (B(F (l))/pB(F (l)))g(l) // H1(GF (l) , B[p])g(l) // H1(GF (l) , B)[p]g(l)

(Recall that we are assuming F (1) = F .) We have

d(l)η ∈ H1(g(l), B(F (l)))[p]

andκ(l)η = c(l)η ∈ H1(GF , B)[p] = H1

s (GF , B[p]).

Let d(l)η be the image of d(l)

η under the map

H1(g(l), B(F (l)))→ H1(g(l), B(F (l)l )).

(Note that F (l)/F is totally ramified at F , so g(l) is the decomposition group of l.)The residue field of F0,l is k0 = Fl, while those of Fl and F

(l)l are k = Fl2 .

Consider the mapB(F (l)

l )→ B(k).

The kernel is pro-l, so, since d(l)η maps to an element of order p in the cohomology,

we find that d(l)η = 0 if and only if ˜

d(l)η = 0, where ˜

d(l)η is the image of d(l)

η under

H1(g(l), B(F (l)l ))→ H1(g(l), B(k)).

Since g(l) acts trivially on k, we have that

H1(g(l), B(k)) = Hom(g(l), B(k)).

Now, however, our earlier notational simplifications come back to haunt us. Reallywe must consider

Hom(g(l), B(k))⊗ I(l)/(I(l))2 = Hom(g(l), B(k)⊗ I(l)/(I(l))2)

and ask what˜d(l)η : g(l) → B(k)⊗ I(l)/(I(l))2

is.Lemma 1.3. For all g ∈ g(l),

˜d(l)η (g) = ±

((l + 1) Frl−al

p

)P ⊗ (1− g).

CHAPTER 22

Lecture 22

1. Completion of the Proofs

1.1. Bounding of the Shafarevich-Tate Group. We continue with thenotation of the previous lectures. We assume the following lemmas for this section;we will prove Lemma 1.1 later in the lecture. (Recall that the cohomology class˜d

(l)η lies in H1(g(l), B(k)), where k is the residue field of F (1)

l (and of F (l)l and F ).

Lemma 1.1. For all g ∈ g(l),

˜d(l)η (g) = ±

((l + 1) Frl−al

p

)P ⊗ (1− g).

Lemma 1.2.

κ(l)η ∈ H1

s (GF , B[p])+.

We now prove the main theorem of the course. Recall that P ∈ B(F ) is thebasic Heegner point.

Theorem 1.3. If P ∈ pB(Fl), then Suppκ(l)η = ∅. If P /∈ pB(Fl), then

Suppκ(l)η = l, and κ(l)

η has singular depth 1 at l.

Proof. Recall from Lecture 21 that we had reduced this to determining thebehavior of any representative crossed homomorphism of

˜d(l)η ∈ H1(g(l), B(k)).

In fact, since g(l) acts trivially on B(k), the homomorphism

˜d(l)η : g(l) → B(k)

is well-defined, and we must determine when it vanishes.This can be done easily from Lemma 1.1 and Lemma 1.1 of Lecture 21. Specif-

ically, Frl acts on P by −1, so

˜d(l)η (g) = ±

(l + 1 + al

p

)P ⊗ 1− g.

But since P ∈ B(k)−p and B(k)−p is cyclic of order pordp(l+1+al), ˜d

(l)η vanishes if and

only if P is divisible by p in B(k)−p . Our analysis at the end of Lecture 21 shows

that this happens if and only if P is divisible by p in B(F (1)l ), which completes the

proof.


161

162 22. LECTURE 22

Corollary 1.4. If P is of infinite order in B(F ), then

X(J0(11)/Q)p = 0

for all but finitely many p.

Proof. If P is of infinite order in B(F ), then P lies in pB(F ) for only finitelymany p. For any p outside of this set and distinct from 2, 5 and 11, Theorem 1.3 andour Kolyagin-Flach theorems apply to yield the vanishing of X(J0(11)/Q)p.

In fact, P is of infinite order and is a generator of B(F ), as can be shownby various (non-elementary) methods. In particular, this means that Corollary 1.4applies for all p 6= 2, 5, 11. A more refined analysis can show that X(J0(11)/Q)p = 0for these p as well, so that

X(J0(11)/Q) = 0.

1.2. Explicit Determination of Cocycles 1. In preparation for the proofof Lemma 3 we consider the following situation: let H be a subgroup of a groupG with quotient G/H = Γ. Let M be a G-module such that MH = 0. Then theinflation-restriction sequence yields an isomorphism

H1(G,M)∼=−→ H1(H,M)Γ.

We wish to give an explicit inverse map, on the level of cocycles.So let c : H →M be a crossed homomorphism representing a class inH1(H,M).

Suppose further that the cohomology class of c is invariant under the action of G,which can be given explicitly by

cg(h) = g−1c(ghg−1).

Thus for each g ∈ G, there is an element b(g) ∈M such that

(cg − c)(h) = (h− 1)b(g)

for all h ∈ H. In fact, b(g) is unique since MH = 0. We now define a cocycle

C : G→M

by C(g) = b(g−1). It is not difficult to check that C(g) is a crossed homomorphismand its class in H1(G,M) is independent of the choice of crossed homomorphismc. Furthermore, C restricts to c on H, so that C is the desired extension of c.

1.3. Explicit Determination of Cocycles 2. We abstract the situation ofLemma 1.1 somewhat before giving the proof. Let p be a fixed prime number andlet B be an abelian variety defined over a number field K. Let L/K be a finiteextension such that B(L) has no non-trivial p-torsion. Set G = GK , H = GL andΓ = G/H. Suppose that we are given Q ∈ B(L) such that the image of Q lies in(B(L)/pB(L))Γ. We wish to determine a crossed homomorphism

κQ : Γ→ B(L)

1. COMPLETION OF THE PROOFS 163

to which Q maps in the following commutative diagram.

0

H1(Γ, B(L))

H1(G,B[p]) //

H1(G,B)

0 // (B(L)/pB(L))Γ // H1(H,B[p])Γ // H1(H,B)

Now, since Q is Γ-invariant in B(L)/pB(L), for every γ ∈ Γ there exists R(γ) ∈B(L) such that

γQ−Q = pR(γ).

Furthermore, R(γ) is unique, since B(L) has no p-torsion. We claim that thecrossed homomorphism

γ 7→ −R(γ)

represents the class κQ in H1(Γ, B(L)) corresponding to Q.To see this, recall first that the Kummer map

B(L)/pB(L)→ H1(H,B[p])

sends Q to the cocycle

h 7→ (h− 1)S ∈ B[p]

where S ∈ B(Q) is such that pS = Q. The analysis of the previous section tellsus how to pull this cocycle back to H1(G,B[p]); in our new notation, we find thecocycle

g 7→ (g − 1)S −R(g).

This maps next to H1(G,B). Here, however, (g− 1)S is trivial, so we are left withthe cocycle

g 7→ −R(g).

This restricts to

γ 7→ −R(γ)

in H1(Γ, B(L)), as claimed.

164 22. LECTURE 22

1.4. Proof of Lemma 1.1. Let us now apply the above analysis to the specificsituation of Lemma 1.1. We have a diagram

0

H1(g(l), B(F (l))) //

H1(g(l), B(k))

H1(F (1), B[p]) //

H1(F (1), B)

0 (B(F (l))/pB(F (l)))g(l) // H1(F (l), B[p])g(l)

(which one should really view as tensored with I(l)/(I(l))2) and we wish to determinethe cocycle of

H1(g(l), B(k))

corresponding to

P (l)η =

∑γ∈g(l)

γx(l) ⊗ γ ∈ (B(F (l))/pB(F (l)))g(l)⊗ I(l)/(I(l))2.

Note that since ∑γ∈g(l)

γx(l) ⊗ 1 ∈ (B(F (l))/pB(F (l)))g(l)⊗ (I(l))2,

we can rewrite P (l)η as

P (l)η =

∑γ∈g(l)

γx(l) ⊗ (γ − 1).

Let us begin by rewriting P(l)η . We fix a generator γ0 of g(l). We write any

γ ∈ g(l) as γlog γ0 , where 0 ≤ log γ < l + 1 = |g(l)|. Then an easy calculation shows

that

γ − 1 = (γ0 − 1)log γ−1∑j=0

(γj0 − 1)

in the group ring Z[g(l)]. Furthermore, since γ0−1 ∈ I(l), one easily computes fromthis that

γ − 1 ≡ log γ(γ0 − 1) (mod (I(l))2).

Thus we can rewrite P (l)η as

P (l)η =

∑γ∈g(l)

log γ · γx(l) ⊗ (γ0 − 1).

Now, note that in Z[g(l)], we have the identity

(γ0 − 1)∑γ∈g(l)

log γ · γ = l + 1−∑γ∈g(l)

γ.

1. COMPLETION OF THE PROOFS 165

We now compute

(γ0 − 1)P (l)η =

(γ0 − 1)∑γ∈g(l)

log γ · γ

x(l) ⊗ (γ0 − 1)

=

1 + l −∑γ∈g(l)

γ

x(l) ⊗ (γ0 − 1).

Our expression for γ − 1 in terms of γ0 − 1 shows that this extends to

(γ − 1)P (l)η =

1 + l −∑γ∈g(l)

γ

x(l) ⊗ (γ − 1)

for all γ ∈ g(l).We can use this expression to determine R(γ), using the notation of the previous

section. Using the fact that l is good for p and that∑γ∈g(l)

γx(l) = alx(1),

we find that

R(γ) =(

1 + l

px(l) − al

px(1)

)⊗ (γ − 1).

The analysis of the previous section shows that the cocycle

d(l)η ∈ H1(g(l), B(F (l)))

is given byd(l)η (γ) = −R(γ).

To complete the proof of Lemma 1.1 we must consider this formula over theresidue field k. Here we have the reductions x(l) and x(1) in B(k), since k is theresidue field of both F

(l)l and F

(1)l . The following lemma, which comes from a

careful analysis of complex multiplication over finite fields, completes the proof ofLemma 1.1.

Lemma 1.5.

x(l) = Fr x(1)

where Fr is the generator of Gal(k/Fl).

Bibliography

[AW] Micahel Atiyah and C.T.C. Wall, Cohomology of groups in Algebraic number fields, Ian

Cassels and A. Frohlich (ed.), pp. 94–115.[BFH] D. Bump, S. Friedberg and J. Hoffstein, Nonvanishing theorems for L-functions of modular

forms and their derivatives, Inventiones mathematicae, vol. 102 (1990), pp. 543–618.[Car] Roger Carter, Finite groups of Lie type, John Wiley and Sons, New York, 1985.

[Cas] Ian Cassels, Global fields in Algebraic number fields, Ian Cassels and A. Frohlich (ed.), pp.

42–84.[CF] Ian Cassels and A. Frohlich (ed.), Algebraic number fields, Academic Press, London, 1967.

[Co] Brian Conrad, personal correspondence, March 4, 1998.[Co2] Brian Conrad, personal correspondence, March 5, 1998.[CS] Gary Cornell and Joseph Silverman (ed.), Arithmetic geometry, Springer-Verlag, New York,

1986.[CSS] Gary Cornell, Joseph Silverman and Glenn Stevens (ed.), Modular forms and Fermat’s last

theorem, Springer-Verlag, New York, 1997.[Cox] David Cox, Primes of the form x2 + ny2, John Wiley and Sons, New York, 1989.[DI] Fred Diamond and John Im, Modular forms and modular curves in Seminar on Fermat’s

last theorem, V. Kumar Murty (ed.), pp. 39–113.[FD] Benson Farb and R. Keith Dennis, Noncommutative algebra, Springer-Verlag, New York,

1993.[Fr] A. Frohlich, Local fields in Algebraic number fields, Ian Cassels and A. Frohlich (ed.), pp.

1–41.[GZ] Benedict Gross and Donald Zagier, Heegner points and derivatives of L-series, Inventiones

mathematicae, vol. 84 (1986), pp. 225–320.

[Ha] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977.[Ko] Victor Kolyvagin, Finiteness of E(Q) and X(E/Q) for a class of Weil curves, Math. of

the USSR Izvestiya, vol. 32 (1989), pp. 523–542.

[La-Al] Serge Lang, Algebra, Addison-Wesley.[La-ANT] Serge Lang, Algebraic number theory, Springer-Verlag, New York, 1986.

[La-EF] Serge Lang, Elliptic functions, Springer-Verlag, New York, 1987.[La-MF] Serge Lang, Introduction to modular forms, Springer-Verlag, Heidelberg, Germany, 1995.

[Ma] Hideyuki Matsumura, Commutative ring theory, Cambridge University Press, Cambridge,

1986.[Maz] Barry Mazur, An introduction to the deformation theory of Galois representations in Mod-

ular forms and Fermat’s last theorem, Gary Cornell, Joseph Silverman and Glenn Stevens(ed.), pp. 243-311.

[Mi-AV] J.S. Milne, Abelian Varieties in Arithmetic Geometry, Gary Cornell and Joseph Silver-

man (ed.), pp. 103–150.[Mi-ADT] J.S. Milne, Arithmetic duality theorems, Academic Press, Boston, 1986.

[Mi-EC] J.S. Milne, Etale cohomology, Princeton University Press, Princeton, New Jersey, 1980.

[Mi-JV] J.S. Milne, Jacobian varieties in Arithmetic Geometry, Gary Cornell and Joseph Silver-

man (ed.), pp. 167–212.[Mu] David Mumford, Abelian Varieties, Oxford University Press, Oxford, 1970.

[MM] M.R. Murty and V.K. Murty, Mean values of derivatives of modular L-series, Annals ofMathematics, vol. 133 (1991), pp. 447–475.

[Mur] V. Kumar Murty (ed), Seminar on Fermat’s last theorem, American Mathematical Society

Press, Providence, 1995.

167

168 BIBLIOGRAPHY

[PR] Vladimir Platonov and Andrei Rapinchuk, Algebraic groups and number theory, AcademicPress, Boston, 1994.

[Pi] Richard Pierce, Associative algebras, Springer-Verlag, New York, 1982.[Roh] David Rohrlich.[Ro] Michael Rosen, Abelian varieties over C in Arithmetic Geometry, Gary Cornell and Joseph

Silverman (ed.), pp. 79–101.[Ru-TS] Karl Rubin, Tate-Shafarevich groups and L-functions of elliptic curves with complex

multiplication, Inventiones mathematicae, vol. 89 (1987), pp. 527–560.[Ru-ES] Karl Rubin, Euler systems and Iwasawa theory.[Se-CA] Jean-Pierre Serre, A course in arithmetic, Springer-Verlag, New York, 1973.

[Se-GC] Jean-Pierre Serre, Galois cohomology, Springer-Verlag, New York, 1997.[Se-LF] Jean-Pierre Serre, Local fields, Springer-Verlag, New York, 1979.

[Se-EC] Jean-Pierre Serre, Proprietes galoisiennes des points d’ordre fini des courbes elliptiques,Inventiones mathematicae, vol. 15 (1972), pp. 259–331.

[ST] Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Annals of Mathe-

matics, vol. 68 (1968), pp. 492–517.[Sh] Stephen Shatz, Group schemes, formal groups and p-divisible groups in Arithmetic geom-

etry, Gary Cornell and Joseph Silverman (ed.), pp. 29–78.

[Shi] Goro Shimura, Introduction to the arithmetic theory of automorphic functions, PrincetonUniversity Press, Princeton, New Jersey, 1971.

[Si-AEC] Joseph Silverman, The arithmetic of elliptic curves, Springer-Verlag, New York, 1986.[Si-H] Joseph Silverman, The theory of height functions in Arithmetic geometry, Gary Cornell

and Joseph Silverman (ed.), pp. 151–166.

[Si-2] Joseph Silverman, Advanced topics in the arithmetic of elliptic curves, Springer-Verlag,New York, 1993.

[Ta] John Tate, Global class field theory in Algebraic number fields, J.W.S Cassels and A.Frohlich (ed.), pp. 162–203.

[Ti] Jacques Tilouine, Hecke algebras and the Gorenstein property in Modular forms and Fer-

mat’s last theorem, Gary Cornell, Joseph Silverman and Glenn Stevens (ed.), pp. 327–342.[Wei] Charles Weibel, An introduction to homological algebra, Cambridge University Press, Cam-

bridge, 1994.[Wes-Ide] Tom Weston, The idelic approach to number theory.

[Wes-IR] Tom Weston, The inflation-restriction sequence: an introduction to spectral sequences.

Euler Systems and Arithmetic Geometry Barry Mazur and Tom...

Documents

Transcript of Euler Systems and Arithmetic Geometry Barry Mazur and Tom...