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Invent mathDOI 10.1007/s00222-013-0448-1

The Iwasawa Main Conjectures for GL2

Christopher Skinner Eric Urban

Received: 22 October 2009 / Accepted: 15 March 2012

Springer-Verlag Berlin Heidelberg 2013

Abstract We prove the one-, two-, and three-variable Iwasawa-GreenbergMain Conjectures for a large class of modular forms that are ordinary withrespect to an odd prime p. The method of proof involves an analysis of anEisenstein ideal for ordinary Hida families for GU(2, 2).

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.1 The Iwasawa-Greenberg Main Conjecture . . . . . . . . . . .1.2 Applications to elliptic curves . . . . . . . . . . . . . . . . . .1.3 The nature of the proof . . . . . . . . . . . . . . . . . . . . .1.4 An outline of the proof . . . . . . . . . . . . . . . . . . . . .

2 Basic notations and conventions . . . . . . . . . . . . . . . . . . .2.1 Fields and Galois groups . . . . . . . . . . . . . . . . . . . .2.2 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C. Skinner ()Department of Mathematics, Princeton University, Fine Hall, Washington Road,Princeton, NJ 08544-1000, USAe-mail: [email protected]

E. UrbanInstitut Mathmatiques de Jussieu, 175, rue de Chevaleret, 75013 Paris, France

e-mail: [email protected]

E. UrbanDepartment of Mathematics, Columbia University, 2990 Broadway, New York,NY 10027, USAe-mail: [email protected]
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


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C. Skinner, E. Urban

3 Selmer groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1 Sorites on Selmer groups . . . . . . . . . . . . . . . . . . . .3.2 Iwasawa theory of Selmer groups . . . . . . . . . . . . . . . .3.3 Selmer groups and modular forms . . . . . . . . . . . . . . .

3.4 p-adic L-functions . . . . . . . . . . . . . . . . . . . . . . . .3.5 The Main Conjectures . . . . . . . . . . . . . . . . . . . . . .3.6 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Constructing cocycles . . . . . . . . . . . . . . . . . . . . . . . . .4.1 Some notations and conventions . . . . . . . . . . . . . . . .4.2 The set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.3 The canonical lattice . . . . . . . . . . . . . . . . . . . . . . .4.4 The cocycle construction . . . . . . . . . . . . . . . . . . . .

4.5 Local conditions . . . . . . . . . . . . . . . . . . . . . . . . .5 Shimura varieties for some unitary groups . . . . . . . . . . . . . .5.1 The groups Gn . . . . . . . . . . . . . . . . . . . . . . . . . .5.2 The Shimura varieties over C . . . . . . . . . . . . . . . . . .5.3 Moduli of Abelian schemes with CM . . . . . . . . . . . . .5.4 Compactifications . . . . . . . . . . . . . . . . . . . . . . . .5.5 Automorphic forms . . . . . . . . . . . . . . . . . . . . . . .

6 Hida theory for unitary groups . . . . . . . . . . . . . . . . . . . .6.1 The Igusa tower and p-adic automorphic forms . . . . . . . .

6.2 Ordinary automorphic forms . . . . . . . . . . . . . . . . . .6.3 -adic ordinary automorphic forms . . . . . . . . . . . . . .6.4 Universal ordinary Hecke algebras . . . . . . . . . . . . . . .6.5 The Eisenstein ideal for GU(2, 2) . . . . . . . . . . . . . . .

7 Galois representations . . . . . . . . . . . . . . . . . . . . . . . . .7.1 Galois representations for Gn . . . . . . . . . . . . . . . . . .7.2 Families of Galois representations . . . . . . . . . . . . . . .7.3 Selmer groups and Eisenstein ideals . . . . . . . . . . . . . .

7.4 Putting the pieces together: the proof of Theorem 3.26 . . . .8 More notation and conventions . . . . . . . . . . . . . . . . . . . .8.1 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.2 Groups and measures . . . . . . . . . . . . . . . . . . . . . .8.3 Automorphic forms and modular forms . . . . . . . . . . . .

9 Some cuspidal Eisenstein series . . . . . . . . . . . . . . . . . . . .9.1 Induced representations and Eisenstein series: generalities . .9.2 Induced representations again: good sections . . . . . . . . .9.3 Good Eisenstein series . . . . . . . . . . . . . . . . . . . . . .

9.4 The classical picture I . . . . . . . . . . . . . . . . . . . . . .9.5 Hecke operators and L-functions . . . . . . . . . . . . . . . .9.6 The classical picture II . . . . . . . . . . . . . . . . . . . . . .

10 Hermitian theta functions . . . . . . . . . . . . . . . . . . . . . . .10.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . .


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10.2 Some useful Schwartz functions . . . . . . . . . . . . . . . .10.3 Connections with classical theta functions . . . . . . . . . . .

11 Siegel Eisenstein series and their pull-backs . . . . . . . . . . . . .11.1 Siegel Eisenstein series on Gn: the general set-up . . . . . . .

11.2 Pull-backs of Siegel Eisenstein series . . . . . . . . . . . . .11.3 Fourier-Jacobi expansions: generalities . . . . . . . . . . . .11.4 Some good Siegel sections . . . . . . . . . . . . . . . . . . .11.5 Good Siegel Eisenstein series . . . . . . . . . . . . . . . . . .11.6 ED via pull-back . . . . . . . . . . . . . . . . . . . . . . . . .11.7 The classical picture III . . . . . . . . . . . . . . . . . . . . .11.8 A formula for CD(,x) . . . . . . . . . . . . . . . . . . . . .11.9 Identifying ED and D, (h, ; u) . . . . . . . . . . . . . . .

12 p-adic interpolations . . . . . . . . . . . . . . . . . . . . . . . . . .12.1 p-adic families of Eisenstein data . . . . . . . . . . . . . . .12.2 Key facts, lemmas, and interpolations . . . . . . . . . . . . .12.3 Application I: p-adic L-functions . . . . . . . . . . . . . . .12.4 Application II: p-adic Eisenstein series . . . . . . . . . . . .

13 p-adic properties of Fourier coefficients ofED . . . . . . . . . . .13.1 Automorphic forms on some definite unitary groups . . . . .13.2 Applications to Fourier coefficients . . . . . . . . . . . . . . .13.3 p-adic properties of the cD(,x)s . . . . . . . . . . . . . . .

13.4 Independence of constant terms and non-singular Fouriercoefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . .References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 Introduction

In this paper we prove the Iwasawa-Greenberg Main Conjecture for a largeclass of elliptic curves and modular forms.

1.1 The Iwasawa-Greenberg Main Conjecture

Let p be an odd prime. Let Q C be the algebraic closure ofQ in C. We fixan embedding Q Qp. For simplicity we also fix an isomorphism Qp = Ccompatible with the inclusion ofQ into both. We let Q be the cyclotomicZp-extension ofQ and Q

:=Gal(Q

/Q) its Galois group. The reciprocity

map of class field theory identifies 1 + pZp with Q; we let Q be thetopological generator identified with 1 + p.

Suppose f Sk(N,) is a weight k 2 newform of level N and Neben-typus . The Hecke eigenvalues of f (equivalently, the Fourier coefficientsof f) generate a finite extension Q(f ) of Q in C. Suppose f is ordinary;


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that is, a(p,f) is a p-adic unit, a(p,f) being the pth Fourier coefficientoff. Let L be any finite extension ofQp containing Q(f ) and the roots ofx2 a(p,f)x + (p)pk1. In this setting, Amice-Vlu [1] and Vishik [74](see also [47]) have constructed a p-adic L-function for f. This is a power

series Lf Q,OL := OL[[Q]], OL the ring of integers ofL, with the prop-erty that if : Q,OL Qp is a continuous OL-homomorphism such that() = (1+ p)m with a primitive pt1th root of unity and 0 m k 2an integer, then

Lf() := (Lf) = e()p

t (m+1)m!L(f,1 m, m + 1)(2 i)mG(1 m)

sgn((1)m)f

,

where is the primitive Dirichlet character modulo pt of p-power ordersuch that (1 + p) = 1; is the cyclotomic character modulo p (nor-malized as in Sect. 2); pt

is the conductor ofm; G() denotes the usual

Gauss sum for a Dirichlet character ; f are the canonical periods off; ande() is an interpolation factor that involves m (p), (p), k, m, and theroots of the aforementioned polynomial. The p-adic L-function Lf is one ofthe two main ingredients of the Iwasawa-Greenberg Main Conjecture for f.

The other main ingredient is the characteristic ideal of the p-adic Selmer

group of f over Q. Recall that there exists a continuous p-adic Galoisrepresentation f : Gal(Q/Q) AutL(Vf), Vf a two-dimensional L-space,such that the L-function off is the L-function off (we take geometric con-ventions for all Galois representations). Furthermore, iff is ordinary, then itis known that there exists a unique Gp-stable unramified line V

+f Vf; here

Gp is a decomposition group at p. Let Tf Vf be a Gal(Q/Q)-stable OL-lattice and let T+f := Tf V+f . Let T := Tf(det 1f ) and T+ := T+f (det 1f )be their respective twists by det 1

f

. We define a p-adic Selmer group offover Q to be the subgroup

SelQ,L(f ) ker

H1(Q, T Zp Qp/Zp) H1Q,p, T / T + Zp Qp/Zp

of classes unramified at all finite places not dividing p, where the map isthat induced by restriction and where Q,p is the completion ofQ at theunique prime above p. This is a discrete Q,OL -module. Its Pontrjagin dual

XQ,L(f ) := Homcont(SelQ,L(f ), Qp/Zp) is a finitely-generated Q,OL -module. The characteristic ideal ChQ,L(f ) of SelQ,L(f ) is defined tobe the characteristic ideal in Q,OL of the module XQ,L(f ). The groupSelQ,L(f ) depends on the choice of the lattice Tf, but this dependency is re-flected in the characteristic ideal ChQ,L(f ) only at its valuation at the prime


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containing p. In particular, ChQ,L(f ) is well-defined in Q,OL Zp Qp .Furthermore, if f is residually irreducible then the isomorphism class ofthe Q,OL -module SelQ,L(f ) is independent of the choice of Tf and soChQ

,L(f )

Q,OL is well-defined. For more precise definitions and refer-

ences regarding these Selmer groups and p-adic L-functions see Sects. 3.3and 3.4 below.

Iwasawa-Greenberg Main Conjecture for f

ChQ(f ) = (Lf) in Q,OL Zp Qp,and furthermore, if f is residually irreducible then this equality holds inQ,OL .

Kato [36, Theorem 17.4] has proven that Lf ChQ,L(f ) under certainhypotheses on f and f. The following theorem, establishing the main con-

jecture in many cases, is one of the main results of this paper.

Theorem 1 (Theorem 3.29) Suppose

= 1 andk 2 mod p 1; the reduction f off modulo the maximal ideal of OL is irreducible; there exists a prime q = p such thatq||N and f is ramified atq;

p

N.

Then ChQ,L(f ) = (Lf) in Q,OL Zp Qp. If furthermore there exists an OL-basis ofTf with respect to which the image of f con-

tains SL2(Zp),

then the equality holds in Q,OL ; that is, the Iwasawa-Greenberg Main Con-jecture for f is true.

This theorem is deduced by combining Katos result with the main theorem

of this paper (Theorem 3.26; see also Theorem 3 below) which proves oneof the divisibilities (p-adic L-function divides characteristic ideal) of theIwasawa-Greenberg Main Conjecture for a Hida family of eigenforms and animaginary quadratic field. This main theorem should be thought of as partof a three-variable main conjecture, one variable being the variable in theHida family and the two other variables being cyclotomic and anti-cyclotomiccharacters of the maximal Zp-extension of the imaginary quadratic field. Formore on the general Iwasawa-Greenberg Main Conjectures, the reader shouldconsult Sect. 3 for the special cases of interest for this paper and the papers

of Greenberg more generally, especially [16].The hypotheses of Theorem 1 intervene at various points in the proof,

occasionally only to shorten an argument. Following the statement of ourmain theoremsTheorems 3.26 and 3.29we have attempted to indicate theplaces in the proof where these hypotheses have been used.


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1.2 Applications to elliptic curves

When the f in Theorem 1 is the newform associated with an elliptic curveE/Q the hypotheses of this theorem are frequently satisfied. For example they

are always satisfied if E has semistable reduction and p 11 is a primeof good ordinary reduction for E. In any case, the above theorem impliesthe main conjecture1 for many elliptic curves (see Theorem 3.32). As shownby Greenberg, this has consequences for the Birch-Swinnerton-Dyer formulafor E.

Theorem 2 (Theorem 3.35) LetE be an elliptic curve overQ with conductorNE . Suppose

E has good ordinary reduction at p; E,p is irreducible; there exists a prime q = p such thatq||NE and E,p is ramified atq .(a) IfL(E, 1) = 0 and E,p is surjective then

L(E, 1)

E

1

p

= #X(E/Q)p

|NEc(E).

(b) If L(E, 1) = 0 then the corank of the Selmer group Selp(E/Q) is atleast one.

Here E,p is the representation of Gal(Q/Q) on E[p], X(E/Q)pis the p-primary part of the Tate-Shafarevich group of E/Q, c(E) :=|#E(Q)/E0(Q)|1p is the maximal power of p that divides the Tamagawanumber of E at the prime , and Selp(E/Q) is the p-Selmer group ofE/Q. Again we note that the hypotheses of the theorem are satisfied if E is

semistable and p 11 is a prime of good ordinary reduction.1.3 The nature of the proof

Iwasawas original Main Conjecture (cf. [15]) identified the Kubota-Leopoldtp-adic L-function of an even Dirichlet character as a generator of the char-acteristic ideal of the p-adic Selmer group over Q of ( is the p-adiccyclotomic character; is identified with a Galois character with the sameL-function as the Dirichlet character). This conjecture was first proved by

Mazur and Wiles [49] by analyzing the cuspidal subgroup of quotients ofJacobians of modular curves; we refer the interested reader to the original

1In the case of elliptic curves this conjecture was first stated by Mazur and Swinnerton-Dyer[46].


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paper of Mazur and Wiles for more history regarding this conjecture and itsproof. Then in [76] Wiles proved the Iwasawa Main Conjecture for all to-tally real fields. The proof in [76] involves an extensive generalization of theconstruction in [55], replacing the analysis of cuspidal subgroups of Jaco-

bians in [49] with congruences between p-adic families of Eisenstein seriesand p-adic families of cuspforms. The resulting relation between these con-gruences and the constant term of the Eisenstein family (essentially the p-adic L-function) is combined with the Galois representations associated withfamilies of cuspforms to prove that the p-adic L-function divides the char-acteristic ideal. When combined with the analytic class number formula, thisimplies equality. Subsequent to the work of Mazur and Wiles, another proofof the Main Conjecture for Q was given by Rubin (based on work of Kolyva-

gin and Thaine) using an Euler system. The Euler system argument yields aresult opposite to that obtained via congruences: the characteristic ideal con-tains the p-adic L-function. But again, together with the analytic class num-ber formula this implies equality. Rubin also used Euler systems and the an-alytic class number formula to prove the one- and two-variable main conjec-tures for imaginary quadratic fields [57]; this includes the Iwasawa-GreenbergMain Conjecture for CM forms. Using an Euler system constructed from el-ements in K-groups of modular curves, Kato proved what amounts to half ofthe Iwasawa-Greenberg Main Conjecture for modular forms (Theorem 3.25).

Lacking an analog of the analytic class number formula, Katos result doesnot imply the main conjecture in general.

The main result of this paperTheorem 3 below (also Theorem 3.26)isproved following the strategy used by Wiles in his proof of the main conjec-ture for totally real fields [76]. In essence the result is an inclusion (divisi-bility) in the opposite direction from that in Katos theorem. Combining theresults then yields equality. The strategy of studying congruences betweenEisenstein families and cuspidal families has also been employed by the sec-

ond named author [72] to prove many cases of the Iwasawa-Greenberg MainConjecture for the adjoint of a modular form; there the class number formulais the one that appears in the theory of Galois deformations as in the work ofWiles [77] and its various extensions to totally real fields. There have beenother results proved in the direction of various main conjectures, too many tosurvey here. While most have made use of Euler systems, some, most notably[48] and [34], have exploited congruences between cuspforms and variousspecial modular forms, analogously to the approach employed in this paper.

In this paper we work in the context of automorphic forms on the unitary

group G := GU(2, 2)/Q defined by a Hermitian pairing of signature (2, 2) ona four-dimensional space over an imaginary quadratic fieldK. The connectionwith L-functions for elliptic modular forms comes through constant terms ofEisenstein series. Let P be the maximal Q-parabolic subgroup ofG fixing anisotropic line. Then P has Levi decomposition P = MN with Levi subgroup


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M = GU(1, 1)/Q ResK/Q Gm. The Eisenstein series on G(AQ) inducedfrom cuspforms on M(AQ) have constant terms along P that involve L-seriesof the form LS(BC( ) , s ) where is a cuspidal automorphic represen-tation of GL2(AQ), BC( ) is its base change to GL2(AK), and is an idele

class character ofAK. When interpreted classically, for an appropriate choiceof inducing data this yields holomorphic scalar-valued Hermitian Eisensteinseries E on H := {Z M2(C) : i(Z tZ) > 0} (the Hermitian upper half-space) whose singular Fourier coefficients are simple multiples of products ofnormalized L-values of the form L := LS(f,,k 1)LS(1, k 2)/.Here is a finite idele class character, := |A

Q, f is a weight k eigen-

form of character , LS(f,,k 1) := LS(BC((f )) ,(k 1)/2) with(f) the usual unitary representation associated with f, and is an explicit

algebraic multiple of a product of periods. Also, LS() denote the partial L-function with the Euler factors at the places in S removed. One might hopethat the Fourier coefficients of the Eisenstein series E so-constructed are p-adic integers and that the singular Fourier coefficients are divisible by L. IfLis not a p-adic unit, then one would expect that E is congruent modulo L to acusp form, provided some non-singular Fourier coefficient is a p-adic unit. Ifsuch were the case, then this congruence could be combined with the p-adicGalois representations associated with cuspidal eigenforms on H to constructclasses in a Selmer group related to f and . This last part is a generalizationof the Galois arguments in [55] and [76]. Carrying this argument out for a p-adic family of Eisenstein series, where L is replaced by a product ofp-adicL-functions, leads to the main theorem of this paper. We do this as outlinedin the following.

1.4 An outline of the proof

The proof of the main result of this paper can be divided into two parts. The

first part, comprising Sects. 27, explains how the index of a certain idealthe Eisenstein idealin the cuspidal p-ordinary Hecke algebra for the uni-tary group GU(2, 2) divides the characteristic ideal of a certain three-variableSelmer group and how this implies the Iwasawa-Greenberg Main Conjec-ture if the index is divisible by a certain three-variable p-adic L-function. Thesecond part, comprising Sects. 813, constructs a p-adic family of Eisensteinseries for GU(2, 2) with singular Fourier coefficients divisible by the three-variable p-adic L-function, and this family is then used to relate the index ofthe Eisenstein ideal to the p-adic L-function.

1.4.1 Selmer groups

After introducing in Sect. 2 the notation and conventions necessary to de-scribe many of the objects studied in this paper and the main results about


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them, in Sect. 3 we develop the theory of Selmer groups as used in this paper,particularly the relations between Selmer groups over various fields and ringsand the corresponding relations between characteristic ideals. This includesthe results that allow the deduction of Theorem 1 from the main theorem of

this paper (in combination with Katos theorem). In Sect. 3 we state the mainresults of this paper about Selmer groups and their connections with p-adicL-functions and explain how they follow from the main theorem.

The Selmer groups that appear in the main results of this paper are as-sociated with Hida families of ordinary cuspidal eigenforms of tame levelN and character (a Dirichlet character modulo Np). Let L be a finiteextension of Qp containing the values of . Such a family is a formal q-expansion f=

n=1 a(n)q

n I[[q]], I a local reduced finite integral exten-sion of W,O

L :=OL

[[W

]], with the property that for a continuous OL-

homomorphism : I Qp with (1+W ) = (1+p)2 and 2 an inte-ger, f :=

n=1 (a(n))q

n is a p-ordinary cuspidal eigenform of weight ,level Np, and character 2. Associated with a Hida family fare contin-uous two-dimensional semisimple Galois representations f : Gal(Q/Q) GLFI (Vf), FI being the fraction field of I, and f : Gal(Q/Q) GLF(V ),F being the residue field ofI, such that for all primes Np the trace of theserepresentations on a geometric Frobenius element for is the image of theFourier coefficient a(). Assume that

f is irreducible. (irred)f

Then there is a basis such that f takes values in GLI(Tf) with Tf a free I-module of rank two. The image ofTf under a homomorphism as above is

just a lattice Tf in the usual p-adic Galois representation f associated withthe ordinary eigenform f . If we also assume that

f is Gp-distinguished, (dist)f

meaning that the semisimplification of f|Gp , Gp being a decompositiongroup at p, is a sum of two distinct characters, then there exists an unram-ified Gp-stable rank-one I-summand T

+f Tf.

Let K be an imaginary quadratic field in which the prime p splits. Let Kbe the composite of all Zp-extensions ofK and K its Galois group over K(so K = Z2p). Then to f and any finite set of primes we attach a Selmergroup SelK(f), defined analogously to the Selmer group for an eigenform.This is a discrete module over IK

:=I

[[K

]]. We explain how it is a conse-

quence of Katos work and the relation ofSelK(f) with other Selmer groupsthat the Pontrjagin dual X

K(f) := Homcont(SelK(f), Qp/Zp) is a finitely-generated torsion IK-module. We can therefore define the characteristic idealChK(f) in IK of X

K(f), and having done so we can then state the main

result of this paper:


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Theorem 3 (Theorem 3.26) Let f be an I-adic ordinary eigenform of tamelevel N and trivial character. Assume that L contains Q[Np, i , D1/2K ]. Sup-pose N = N+N with N+ divisible only by primes that split in K and Ndivisible only by primes inert in K. Suppose also

(irred)f and(dist)f hold; N is square-free and has an odd number of prime factors; the reduction f off modulo the maximal ideal ofI is ramified at all |N.Let be a finite set of primes containing all those that divide N DK and some

prime = p that splits in K. Then

ChK(f) Lf,K

.

Here Lf,K IK is a certain three-variable p-adic L-function (described inSect. 3.4.5 but constructed later on). We explain how to deduce Theorem 1from this theorem and Katos theorem using the relations between SelK(f)and the Selmer groups for the various f together with the corresponding re-lations between Lf,K and the usual p-adic L-functions Lf . From this we arealso able to deduce that the inclusion in Theorem 3 is often an equality (seeTheorem 3.31). We also explain other consequences, including Theorem 2.

We follow the discussion of Selmer groups in Sect. 3 with an exposition

in Sect. 4 of an abstract set-up and concomitant construction of subgroupsof group cohomology classes. This construction provides a means for relat-ing orders of certain congruence ideals to orders of characteristic ideals; thesetting abstracts that obtained from the Eisenstein ideals studied later. Thisgeneralizes and formalizes the construction of Selmer classes using Galoisrepresentations and congruences that was alluded to in Sect. 1.3.

1.4.2 Hida theory, the Eisenstein ideal, and Galois representations

In Sect. 5 and Sect. 6 we describe Hida theory for p-adic modular formsfor G = GU(2, 2). In particular we explain the surjectivity of certain -adicSiegel operators. This last point is the key to connecting the divisibility prop-erties of singular Fourier coefficients of the p-adic families of Eisenstein se-ries to congruences with cusp forms. Here = Zp[[T (Zp)]] with T G acertain maximal torus. Our exposition of Hida theory generally follows [32],but whereas Hida restricts attention to cuspidal forms, we require a theoryfor modular forms with non-zero constant terms. The explanation of this aug-

mentation of Hidas results necessitates that we review (or sketch proofs of)some facts about the construction and nature of the arithmetic toroidal andminimal compactifications of the Shimura varieties associated with the uni-tary similitude groups GU(n,n). After defining the spaces of ordinary -adicforms (Hida families) for the groups GU(n,n), we define an ideal in a Hida


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Hecke algebra for G and explain how, given the existence of a suitable Hidafamily of Eisenstein series, its indexthe Eisenstein idealis related to p-adic L-functions.

A Hida family f and a set of primes as in Theorem 3 gives rise to a

tuple D that we term a p-adic Eisenstein datum. We put D := IK[[K ]],K

the Galois group of the anticyclotomic Zp-extension ofK; this has thestructure of a finite -algebra. A Hida family over D (of prescribed tamelevel KD G(Apf)) is then a collection of formal series F = (Fx ) (indexedby certain cusps x) with Fx D[[q]], where runs over a lattice of Her-mitian matrices in M2(K) that depends on x and KD, such that for a certainclass of continuous OL-homomorphisms : D Qp, the specializationof F at the collection of formal series obtained from applying to the

coefficients of the Fx is the collection ofq-expansions at the cusps x of ap-ordinary holomorphic Hermitian modular form on H G(Af). The set ofsuch forms is a finite free D-module and there is a natural Hecke action onthis space. We let hD be the D-algebra generated by the Hecke operatorsacting on the D-cuspforms; this is a finite D-algebra. We define an idealID hDdetermined by Dand consider the quotient hD/ID. By the def-inition ofID, this is a quotient of D via the structure map. The Eisensteinideal ED D is the kernel of this surjection: D/ED hD/ID. The ideal IDis defined with the expectation that there exists a D-eigenform ED such thatID is generated by the image in hD of the annihilator of ED in the abstractHecke algebra and such that the singular coefficients ofED are divisible bythe three-variable p-adic L-function Lf,K IK. Assuming such a form EDexists, from the aforementioned surjectivity of the D-Siegel operators weconclude that ifP IK is a height one prime containing Lf,K but ED is non-zero modulo P, then ordP(ED) ordP(Lf,K).

In Sect. 7 we recall the Galois representations associated with cuspidalrepresentations ofG and explain the existence of Galois representations as-sociated with Hida eigen-families and particulary with components of hD.Using this we show that the ring hD, the ideal ID, and a prime P IK as inthe preceding paragraph give rise to a set-up as formalized in Sect. 4. Assum-ing the existence of the Hida family ED, the main theorem then follows fromthe abstract construction given there and the inequality relating the orders ofED and Lf,K.

The rest of the paper is taken up with proving the existence of the three-variable p-adic L-function Lf,K and the Hida family ED.

1.4.3 Eisenstein series

After introducing more notation in Sect. 8, in Sect. 9 we define the Eisensteinseries that belong to the family ED and describe their Hecke eigenvalues and


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constant terms. More specifically, for certain of the homomorphisms wedefine an Eisenstein datum D and thence an Eisenstein series ED . Thisseries is induced from a cuspform on M(A) associated with f . The singularFourier coefficients turn out to be essentially Fourier coefficients off .

In Sect. 10 and Sect. 11 we recall some auxiliary functionstheta func-tions and Siegel Eisenstein seriesthat show-up in our analysis of the p-adic properties of the ED . In particular, in Sect. 11 we recall a formulaof Garrett and Shimura that essentially expresses a multiple GD of EDas an inner-product of f and the pull-back to h H of a Siegel Eisen-stein series on GU(3, 3). The multiple GD is essentially L

(f , , k 1)L(1f

, k 2)ED , with a finite idele class character ofK. We use

the pull-back formula to express the Fourier coefficients of GD as inner-

products of modular formsthe inner-products of f with the restrictionsof Fourier-Jacobi coefficients of the Siegel Eisenstein series (these Fourier-Jacobi coefficients are essentially products of theta functions and Eisensteinseries on h). Similarly, the L-function L(f, , k 1) is realized as aninner product off with the pull-back to h h of a Siegel Eisenstein serieson GU(2, 2). Much of Sect. 11 is taken up with the definitions of the ramifiedlocal sections used to define the Siegel Eisenstein series and the computationsof the Fourier-Jacobi coefficients of these series.

To help orient the reader amidst the notationally dense calculations in

Sects. 9, 10, and 11, we have included a more detailed summary of theircontents at the start of each of these sections.

The use of the pull-back formula to express the Fourier coefficients of cus-pidal Eisenstein series like GD as inner-products of modular forms was alsoexploited in [72],andtheideagoesbackatleastto[4].Thework[72] containsa similar analysis of the ramified local sections defining the Siegel-Eisensteinseries. In [78], a similar use was made of the pull-back formula to analyze theFourier-Jacobi expansions of cuspidal Eisenstein series on GU(3, 1).

The non-singular Fourier coefficients of GD

are indexed by positive-definite Hermitian matrices GL2(K), and the coefficients for a given define automorphic functions on the unitary group U() of the Hermitianpairing on K2 defined by . Pairing these functions with a suitable auto-morphic form on U()which amounts to taking a linear combination ofFourier coefficients of GD results in a Rankin-Selberg convolution of fwith a theta lift to GL2 of the form on U() (this lift has weight 2). If all thedata has been chosen well (and one is lucky) the resulting formulas are es-sentially products ofL-functions whose arithmetic properties are sufficiently

understood.1.4.4 p-adic interpolations

In Sect. 12 we use the formulas from Sect. 11 to construct the D Hida fam-ilies ED (which essentially specialize to GD ) and the three-variable p-adic


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L-function Lf,K (which we relate to the p-adic L-functions Lf and otherp-adic L-functions such as the anticyclotomic L-functions of the f ). Thatthe singular coefficients ofED are divisible by Lf,K is immediate. The key tothis interpolation is that the Fourier coefficients of Siegel Eisenstein are par-ticularly simple; they p-adically interpolate by inspection. Via the pull-backformulas, this p-adic interpolation carries over to the L-functions and to theFourier coefficients of the induced Eisenstein series.

1.4.5 Co-primality of the p-adic L-functions and Eisenstein series

Finally, in Sect. 13 we show that the Fourier coefficients of the family ED

havethe needed properties; essentially, the non-singular coefficients are prime tothe p-adic L-function. Our proof of this involves some of the formulas fromSect. 11 and appeals to the mod p non-vanishing results of Vatsal [73] andFinis [11]. The appeal to the former in particular is responsible for some ofthe hypotheses in the main theorem as well as Theorems 1 and 2. An appeal toVatsals non-vanishing result is also made in [72]. There the resulting formulafor the Fourier coefficient of the Eisenstein series is essentially the Rankin-

Selberg convolution of an eigenform with a weight one theta function (asopposed to the weight two theta functions that show up in the formulas inthis paper), and so a suitable linear combination of Fourier coefficients is aspecial value of the twist of the L-function of an eigenform by a finite Heckecharacter; Vatsals theorem applies directly to this last L-value. The situationin this paper is less straightforward.

Using our formulas for linear combinations of Fourier coefficients ofGD ,we show that there is a p-adic family g of CM forms such that a suitable

D-combination of non-singular coefficients ofED factors as AD,gBD,g withAD,g I[[+K ]] IK, +K the Galois group of the cyclotomic Zp-extension ofK, and BD,g IK. The factor AD,g interpolates Rankin-Selberg convolutionsof the f with weight two specializations ofg and so is easily observed to benon-zero. That AD,g is co-prime to Lf,K under the hypotheses of the maintheorem then follows from Vatsals result on the vanishing of the anticyclo-tomic -invariant (and the relation ofLf,K with the p-adic anticyclotomicL-functions). The factor BD,g specializes under some to a convolution of

a specialization ofg and a weight one Eisenstein series, and we show by ap-peal to the results of Finis that g can be chosen so that this convolutionandhence BD,gis a unit.

We have included an index to important notation not defined in Sect. 2 orSect. 8


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2 Basic notations and conventions

In this section we collect the notation and conventions for fields, characters,and Galois representations needed to describe the basic framework and main

results of this paper. For the most part these follow conventional practice. Ad-ditional notation will, of course, be introduced later in the paper; particularlysignificant notation or conventions will be given at the start of each section.

Throughout this paper p is a fixed odd prime number.

2.1 Fields and Galois groups

2.1.1 Number fields

We fix algebraic closures Q and Qp of Q and Qp , respectively. For thepurpose of p-adic interpolation we fix embeddings p : Q Qp and : Q C and a compatible isomorphism p : Qp

C (so that = p p).The adeles of a number field F are denoted by AF and the finite adeles are

denoted by AF,f. When F = Q we will often drop it from our notation for theadeles. We let Z := Z Af (so Af =Z Q). Ifv is a place ofQ andx AQ then we write xv for the v-component ofx, and similarly for AF. Fora place v ofQ,

| |v will denote the usual absolute value on Qv (

|

|

=1)

and | |Q the corresponding absolute value on A (|x|Q =v |xv|v). We definean absolute value | |F on AF by |x|F := |NF /Q(x)|Q, where NF /Q(x) is thenorm from AF to A. We reserve | | to denote the usual absolute value on Cand R.

We let K Q be an imaginary quadratic extension ofQ in which p splitsand denote the ring of integers ofK by O. The absolute discriminant, classnumber, and different ofK are denoted by DK, hK, and d, respectively. Welet K :=

DK (so this generates d). The action of the nontrivial automor-phism ofK is often denoted by a bar (thus x

K is sent to

x by this auto-

morphism). For any Z-algebra A this extends to O A and K A throughits action on the first factor.

We let v0 be the place ofK over Q determined by the fixed embeddingQ Qp. We denote its conjugate place by v0. We let p be the prime ideal ofO corresponding to v0 and let p be its conjugate ideal (so in O, (p) = pp).

If v is a finite place of Q then Kv := K Q Qv and Ov := O Zv . Ifw is a finite place ofK then Kw and Ow have their usual meanings. For aprime , D is the absolute discriminant ofK over Q. If splits in K then

we fix a Z-algebra identification ofO with Z Z and hence ofK withQ Q. With respect to these identifications, if (a,b) K then (a,b) =(b,a). We assume that the identification Kp = Qp Qp has been made sothat p = K (pZp Zp). We identify K R with C by x y (x)y.We similarly identify K C with C C by x y ((x)y,(x)y).


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We let Q Q be the unique Zp-extension of Q and let K Qbe the unique Z2p-extension of K. We let K

+ and K be, respectively,the cyclotomic and anti-cyclomotic Zp-extensions (so K+ = KQ andK+

K

=K). In the context of these extensions we will write Qn, Kn,

and Kn to mean the maximal subfields of conductor pn+1.

2.1.2 Galois groups

For any subfield F Q we let GF := Gal(Q/F ). Given a set of finiteplaces ofF, we let GF, := Gal(F/F ), F being the maximal extensionofF unramified at all finite places not in . If is a set of finite places ofa subfield ofF we write GF, for GF, , where is the set of places ofFover those in .

A decomposition group at a place v of F will be denoted GF,v and itsinertia subgroup will be denoted IF,v. If the choice of decomposition groupis important, the choice will be made clear in the text. When F is understoodwe will often drop it from our notation for decomposition and inertia groups.

If F is finite over Q, then for a finite place v of F we will write frobvfor a geometric Frobenius element (well-defined only in GF,v/IF,v). The L-function of a Galois representation ofGF will always be defined with respect

to geometric Frobenius elements.When F = Q or K we fix choices of decomposition groups. When v =

we assume that GF,v is the decomposition group determined by the fixedembedding Q C. We let c GQ, be the unique nontrivial element; thisis a complex conjugation which agrees with the usual complex conjugation onC via the fixed embedding Q C. The restriction ofc to K is the nontrivialautomorphism ofK, so no confusion should result from our also denotingthe action of c on an element of C by bar (i.e., writing x to mean c(x)).

When v is the place determined by the fixed embedding Q Qp we assumeGF,v is the decomposition group determined by this embedding. We chooseGK,v0 = cGK,v0 c1.

2.1.3 Reciprocity maps

For a local or global field F we normalize the reciprocity map recF of classfield theory so that uniformizers get mapped to geometric Frobenius elements.

2.1.4 Hodge-Tate weights

Unless otherwise stated, whenever we discuss Hodge-Tate weights for ap-adic Galois representation ofGK they are for the place v0.


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2.1.5 The groups Q, K, andK

We let Q := Gal(Q/Q) and K := Gal(K/K) and let K K be thesubgroup on which conjugation by c acts as

1. Then K

=+K

K

. Via

the canonical projection K Gal(K/K), K is identified with the tar-get. Via the canonical projection K Q, +K is identified with Q. Wefix topological generators K and let Q be the topological gener-ator identified with +. To simplify matters we will assume these have beenchosen so that recQp (1 + p) = and recKp ((1 + p)1/2, (1 + p)1/2) = .

2.2 Characters

Let F be a number field.2.2.1 Idele class characters

For an idele class character : AF C (so trivial on F) we write f forthe conductor of . IfF = Q then we make no distinction between the idealf and the positive integer that generates it. For any finite set of places S ofF we set

LS

(,s) := vf ,v /S

1 v(v)qsv 1, = v ,where v is a uniformizer at v. IfS is a finite set of places of a subfield ofF we use the same notation for the product over places that do not divide aplace in S. The same convention will be used for Euler products throughoutthis paper.

Suppose F = Q or K and (z) = sgn(z)bza ifF = Q and (z) = za zbifF

=K, a and b being integers. We denote by the (unique) p-adic Galois

character : GF Qp

such that

(frobv) = v(v), vpf .For any finite set of places S {v|p}

LS( , s)

=LS(,s).

IfF = Q then the motivic weight of is 2a and its Hodge-Tate weight isa. IfF =K then the motivic weight of is (a + b) and its Hodge-Tateweight with respect to the place v0 is a. We let c be the composition ofwith conjugation ofGF by c.


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Given an idele class character of AK

, we let c(x) := (x) and := |A

Q, where AQ AK is the canonical inclusion. Then c = NK/Q,

so

c = c = NK/Q ,

provided exists. Note that c = c .

2.2.2 Hecke characters

If is a Hecke character ofF of conductor f , then we associate with anidele class character v of AF in the usual way. In particular, for a finite

placevfv ,

v

(v

)=

(pv

), where

v is a uniformizer at

vand pv is theprime ideal corresponding to v. We will continue to denote v by ; this

should cause no confusion.If is a Dirichlet character of conductor N then we associate with it a

Hecke character, which we also denote , of conductor N such that (()) =() for all N. Then () = () for all primes N.

2.2.3 The cyclotomic character

We denote by the p-adic Galois character associated with the character| |Q. Then |GF is the Galois character associated with | |F. The character gives the action of Galois on p-power roots of unity: ifvp is a place ofFthen (frobv) = NF /Qv1. The Hodge-Tate weight of is 1 and its motivicweight is 2.

2.2.4 The Teichmller character

Let be the composition GQZp (Zp/pZp) Zp , where the second

arrow is reduction modulo p and the third is the Teichmller lift. Via thereciprocity map, induces a character ofA, which we continue to denoteby . In this way, we can view as a character of Zp (via the inclusionQp

AQ).

2.2.5 The character associated withK

Let K : A C be the usual quadratic character associated with K. Wealso write K for the p-adic Galois character associated with this idele classcharacter; this should cause no confusion as which character is meant willalways be clear from the context.


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3 Selmer groups

In this section we define the Selmer groups that are the main focus of this pa-per. The first two subsections below contain Sorites on p-adic Selmer groups;

these should be well-known to experts. In the third subsection we define theSelmer groups studied in this paper. In the fourth and fifth subsections werecall the p-adic L-functions associated with modular forms, the main con-

jectures for the Selmer groups associated with an ordinary cuspidal eigenformand their corollaries, and the results of Kato [36] about these conjectures. Thelast subsection is devoted to the statements of the main results of this paper.

We make no claim of originality for the results in Sects. 3.1 and 3.2 below.Most of the results therein can be found in different guises elsewhere in the

literature, especially in the papers of Greenberg. In particular, we learned theproof of Proposition 3.7 from [16].

3.1 Sorites on Selmer groups

The following develops the theory of Selmer groups as needed to understandthe main results of this paper and their proofs.

3.1.1 Shapiros lemma

Let E/F be a finite Galois extension of fields. By Shapiros lemma, for anydiscrete GE-module M there is a canonical isomorphism

Hi (E,M) = HiF, IndGFGE M (3.1)with IndGFGE M := { : GF M : (gg) = g(g) g GE}. If the GE-action on M is the restriction of a GF-action, then we have an isomorphism

ofGF-modules

IndGFGE MHomZ

ZGal(E/F)

, M

given by

gGal(E/F)ngg

gGal(E/F)

ngg

g1

.

The action of Gal(E/F) on HomZ(Z[Gal(E/F)], M) is ( f)(x) =f(1x). Therefore, ifM is a GF-module then there is a canonical isomor-phism

Hi (E,M) = HiF, HomZZGal(E/F), M.


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3.1.2 Shapiros lemma and restriction

For analyzing Selmer groups, it is useful to know how Shapiros lemma in-teracts with restrictions at finite places. To this end, let v be a finite place of

F and GF,v a decomposition group for v. Let w0|v be a place of E fixedby GF,v . Then GE,w0 := GF,v GE is a decomposition group for w0. Foreach place w|v of E fix gw GF such that gww0 = w and put GF,w :=gwGF,vg

1w and GE,w := gwGE,w0 g1w . The latter are decomposition groups

for v and w, respectively. There is a GF,v-module isomorphism

IndGFGE Mw|v

IndGF,wGE,w

M,

(w)w|v, w(g) := (ggw).(3.2)

The GF,v-action on the right-hand side is given by (g w)(g) =w(g

gwgg1w ). The isomorphism (3.2) induces an isomorphism

Hi

Fv, IndGMGE

M

w|vHi

Fv, IndGF,wGE,w

M

w|v HiGF,w, IndGF,wGE,w M=w|v Hi (Ew,M),with the equality denoting the canonical identification coming from Shapiroslemma and the middle isomorphism given on cocycles by (cw)w|v (cw)w|vwith cw(g) = cw(g1w ggw). This isomorphism fits into a commutative dia-gram

Hi (F, IndGFGE M)

res

H1(E,M)

res

Hi (Fv, IndGMGE

M)

w|v Hi (Ew,M).

(3.3)

Let Iv GF,v, IF,w GF,w, and Iw GE,w be the respective inertiasubgroups. From (3.2) we obtain an isomorphism

IndGFGE MIv w|vIndGF,wGE,w MIF,wand that the natural inclusion

IndGkvGkw

MIw IndGF,wGE,w MIF,w


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is an isomorphism. Here kv and kw are, respectively, the residue fieldsof Fv and Ew. It follows that the bottom isomorphism of (3.3) identi-fies H1(kv, (Ind

GFGE

M)Iv ) with

w|v H

1(kw, MIw ). In particular, a class in

H1(F, IndGFGE

M) is unramified at v if and only if the corresponding class in

H1(E,M) is unramified at all w|v.Suppose M is a GF-module. The isomorphism (3.1) can be rewritten as

IndGFGE MMZ HomZ

ZGal(E/F)

, Z

.

IfM+ M is a GF,v-submodule, then from (3.2) we obtain an isomorphismofGF,v-submodules

M+

Z HomZZGal(E/F), Z

w|v M+w Z HomZZGal(Ew/Fv), Z,where M+w := gwM+g1w ,

and hence the bottom isomorphism of (3.3) identifies the image of H1(Fv ,M+Z HomZ(Z[Gal(E/F)], Z)) in H1(Fv, MZ HomZ(Z[Gal(E/F)], Z))with the image of

w|v H

1(Ew, M+w ) in

w|v H

1(Ew, M). In particular, the

restriction of a class in H1(F, IndGFGE M) to

H1Fv, IndGFGE M= H1Fv, MZ HomZZGal(E/F), Zis in the image ofH1(Fv, M+ Z HomZ(Z[Gal(E/F)], Z)) if and only if therestriction of the corresponding class in H1(E,M) to

w|v H

1(Ew, M) is in

the image of

w|v H1(Ew, M

+w ).

3.1.3 -primitive Selmer groups

Let F

Q. Let T be a free module of finite rank over a profinite Zp-algebra

A and assume that T is equipped with a continuous action ofGF. We assumethat for each place v|p ofF we are given a Gv-stable free A-direct summandTv T. Let be a set of finite places ofF. We denote by SelF (T,(Tv)v|p)the kernel of the restriction map

SelF

T,(Tv)v|p := kerH1F, T A A

v /vp H1

Iv, T A A

v|p H1

Iv, T / T v A A

,where A := Homcont(A, Qp/Zp) is the Pontrjagin dual of A. (We willsimilarly denote by M the Pontrjagin dual of any locally compact Zp-module M.) This is the -primitive Selmer group. The Selmer groups are


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independent of the choices of the decomposition groups. If contains all theplaces at which T is ramified and all the places over p, then

Sel

FT,(Tv)v|p= kerH1GF,, T A Av|p H1Iv, T / T v A A.We also put

XF

T,(Tv)v|p := HomASelFT,(Tv)v|p, A.

We will just write XF (T ) or SelF (T ) when the Tvs are clear from the context

or are not important.Given T,

{Tv

}v

|p and as above for a given F, for any extension E/F

(not necessarily finite) we put SelE (T ) := SelEE (T,(Tw)w|p) and XE (T ) :=X

EE (T,(Tw)w|p), where E is the set of places ofE over those in and if

w|v|p then Tw = gwTv for gw GF such that g1w GE,w gw GF,v . This isindependent of the choices ofgws. We have

SelE (T ) = limFFE

SelF(T ) and XE (T ) = lim

FFEXF(T ),

where F

runs over the finite extensions ofF contained in E.When is empty or contains only primes over p we drop it from the no-

tation. The corresponding Selmer group is called the primitive Selmer group.

3.1.4 Passing from F to F+

Assume that F is a CM number field and let F+ be its maximal totally realsubfield (so c restricts to the nontrivial element of Gal(F/F+)). Let T be asabove with the additional assumption that the GF action on T is the restriction

of a GF+-action. Let v|p be a place ofF+. Ifv is inert in F then we assumethat GF,v GF+,v. If v splits in F then we fix a splitting v = wwc andassume that GF,w = GF,v and GF,wc = cGF+,vc1. For each place v|p ofF+, we assume we are given a GF+,v-stable A-summand Tv T. Ifv splitsin F, v = wwc, then we let Tw = Tv and Twc = cTv .

Let + be a finite set of finite places of F+ and let be the set ofplaces ofF over those in +. We may then define SelF (T ), Sel

+F+ (T ), and

Sel+

F+ (T F), where F is the quadratic character of GF+ correspond-

ing to the extension F /F+. Since p > 2 the usual action of Gal(F/F+) onH1(F,T A A) yields a decompositionSelF (T ) = SelF (T )+ SelF (T ),

where the superscript denotes that c acts as 1.


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Lemma 3.1 The restriction map from GF+ to GF yields isomorphisms

Sel+

F+ (T )SelF (T )+, Sel

+F+ (T F)

SelF (T ).

Proof This follows easily from Shapiros lemma, the inflation-restrictionsequence, and the fact that Zp[Gal(F/F+)] = Zp Zp(F) as a GF+-module.

3.1.5 Fitting ideals and characteristic ideals

Let A be a Noetherian ring. We write FittA(X) for the Fitting ideal in A ofa finitely generated A-module X (and therefore of finite presentation). Thisis the ideal generated by the determinant of the r

r-minors of the matrix

giving the first arrow in a given presentation ofX:

As Ar X 0.

In particular, ifX is not a torsion A-module then FittA(X) = 0.Fitting ideals behave well with respect to base change. For any Noetherian

A-algebra B , FittB (X A B) = FittA(X)B. In particular, ifI A is an ideal,then

FittA/I(X/IX) = FittA(X) mod I.To define the notion of characteristic ideal we need to recall a few facts

about divisorial ideals. Recall first that a divisorial ideal is an ideal whichis equal to the intersection of all principal ideals containing it. In particularany principal ideal is divisorial. Let us assume now that A is a Noetheriannormal domain. For any prime ideal Q A of height one, denote by ordQthe essential valuation attached to Q. Then any divisorial ideal I is of theform

I = x A : ordQ(x) mQ Q of height one,where the mQ are non-negative integers almost all equal to zero. The mQs areuniquely determined and we set ordQ(I ) := mQ (AQ is a DVR and ordQ(I )is the valuation of any generator ofI AQ). IfI and J are two divisorial ideals,then the following are equivalent:

(i) ordQ(I ) ordQ(J ) for all prime ideals Q of height one(ii) I J.In particular, if I is divisorial and x A, we have (x) I if and only ifordQ(I ) ordQ(x) for all Q of height one. The characteristic ideal of anA-module X is the divisorial ideal CharA(X) defined by

CharA(X) :=

x A : ordQ(x) Q(X) Q of height one

,


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where Q(X) is the AQ-length of the Q-localization XQ (possibly infinite).One checks easily that

CharAA/FittA(X)= CharA(X).Unlike Fitting ideals, characteristic ideals do not behave well under basechange in general. This is particularly true ifX contains a nontrivial pseudo-null submodule. However, since inclusion of divisorial ideals is easier to rec-ognize, most of the time we will work with characteristic ideals. In the casesof interest to us, for the purposes of base change we are able to make do witha weaker statement (see Corollary 3.8).

The following easy lemma will be useful in identifying Fitting and charac-teristic ideals.

Lemma 3.2 LetA be a ring, a A a proper ideal contained in the Jacobsonradical of A, and assume that A/a is a domain. LetL A be such that itsreduction L modulo a is non-zero. LetI (L) be an ideal and letI its imagein A/a. IfL I, then I = (L).

Proof We need to show that L I. As in the statement of the lemma, wedenote the image of reduction modulo a by a bar. By assumption, thereexist s

I such that

= L. On the other hand, since I

(L), there exists

A such that = L. Therefore L = L, and hence = 1 since A/a isa domain and L is non-zero. As a is contained in the radical of A, it thenfollows that is a unit in A, so L= 1 I.

3.1.6 Fitting and characteristic ideals of Selmer groups

Lemma 3.3 Let F be a number field and S a finite set of finite places of F.Let A be a profinite Zp-algebra and let M be a finitely generated A-module

equipped with a continuous action of GF,S. Then H1(GF,S, M

A A) is

co-finitely generated overA.

Recall that an A-module X is co-finitely generated if HomA(X,A) isfinitely generated. A consequence of this lemma is that the dual Selmergroups XF (T,(Tv)v|p) defined before are finitely generated over A.

Proof See the proposition in Sect. 4 of [16] where it is essentially deducedfrom the arguments used to prove Proposition 3.7 below.

For a number field F and T and A as before, we set

FtF,A(T ) := FittA

XF (T )

and ChF,A(T ) := CharA

XF (T )

.

Of course, we have only defined ChF,A(T ) ifA is Noetherian and normal.


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3.2 Iwasawa theory of Selmer groups

We now develop the Iwasawa theory of Selmer group over the fields Q,K

, and K

.

3.2.1 Iwasawa algebras

Let Q := Zp[[Q]], K := Zp[[K]], and K := Zp[[K ]]. The projection+K

Q determines an isomorphism +KQ.

Let K : GK K K be the canonical character. We similarly definecharacters K, : GK ,K and Q : GQ . Note that

Q mod (1 + p)m= mm.For a profinite Zp-algebra A we set Q,A := A Zp Q, where Zp de-

notes the tensor product in the category of profinite Zp-modules (and contin-uous morphisms); in particulary Q,A = A[[Q]]. We similarly define K,Aand

K,A.

3.2.2 Selmer groups as modules over Iwasawa algebras

Let A be a profinite Zp-algebra and let T be a free A-module of finite rankequipped with a continuous A-linear action ofGQ. We assume given a Gp-stable A-free direct summand Tp ofT.

Shapiros lemma provides the following.

Proposition 3.4 LetF = Q orK. There is a canonical isomorphism ofF,A-modules

SelF(T ) = Sel

FT A F,A1F .

The right-hand side is defined by viewing TA F,A as a F,A[GF]-module.When F = K the same isomorphism holds with F replaced by K, F,Aby

K,A, andF by K,.

Proof This should be well-known, but for the readers convenience we pro-vide a proof. For this we use Shapiros lemma as recalled in Sect. 3.1.1, notingfirst that

limn

HomZ

ZGal(Fn/F )

, A

= limn

Homcont(n, Qp/Zp)

= Homcont

limn

n, Qp/Zp

= F,A

1F

,


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where n := A[Gal(Fn/F )] and the last identification is as F,A[GF]-modules. Appealing to Shapiros lemma, it follows that

H1F, T A A

= limn H1

Fn, T A A

= lim

n

H1

F, HomZ

ZGal(Fn/F )

, T A A

= lim

n

H1

F, T A HomZ

ZGal(Fn/F )

, A

= lim

n

H1

F, T A n= H1F, T A F,A1F .

That this identifies SelF(T ) with SelF (T A F,A(1F )) then follows fromthe analysis in Sect. 3.1.2. The same arguments apply to the situation whereF =K and F =K.

Remark 3.5 This recovers [16, Prop. 3.2].

As a direct consequence of the preceding proposition and Lemma 3.3, thedual group XQ,A(T ) = XQ,Q (T A Q,A(

1Q )) is finitely generated over

Q,A

. We then put

FtQ,A(T ) := FtQ,Q,A

T A Q,A1Q

and

ChQ,A(T ) := ChQ,Q,A

T A Q,A1Q

.

These belong to Q,A. We similarly define FtK,A(T ), ChK,A(T ) K,A

and FtK

,A(T ), Ch

K,A

(T )

K,A.

Combining Proposition 3.4 with Lemma 3.1 yields the following.

Lemma 3.6 There are +K,A-isomorphisms

SelK

+(T ) = SelQ(T ) SelQ(T K)

and

XK

+(T )

=XQ

(T )

XQ

(T

K).

3.2.3 Dual Selmer groups as torsion modules

Suppose that A is a domain and finite over Zp . Let (T,Tp) be as before andassume that T is geometric (in the sense of Fontaine-Mazur) and pure with


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regular Hodge-Tate weights and such that the rank ofTp is equal to the rankof the +1-eigenspace for the action of the complex conjugation c. Then it isconjectured (by Greenberg, Bloch-Kato, Fontaine-Perrin-Riou) that XQ(T )(resp. XK

(T )) is torsion over Q,A (resp. K,A). When T is one dimen-

sional this fact is a simple consequence of class field theory. In general, thisseems to be a deep fact. It has been proved by Kato [36] when T is the Galoismodule associated with an elliptic cuspidal eigenform f of weight k 2 andp Nf is a prime at which f is ordinary and Tp is the rank-one unramifiedGp-subrepresentation; Katos proof uses an Euler system constructed fromSiegel units and the K-theory of modular curves.

3.2.4 Control of Selmer groups

Let A be a profinite Zp-algebra. By elementary properties of Pontrjagin du-ality, for any ideal a A we have a canonical isomorphismA[a] = (A/a)

and hence for any free A-module M a canonical identification M/a A/a(A/a) = MA A[a]. This implies that for any (T,Tp) as in Sect. 3.2.2 wehave a canonical map

Sel

F

(T /aT )

Sel

F

(T )

[a].

Proposition 3.7 Suppose the action of Ip on T /Tp factors through the im-

age of Ip in Q and that {p} contains all primes at which T is rami-fied. Let F = Q, K, orK+, and suppose also that there is no nontrivialA-subquotient ofT on which GF acts trivially. Then the above map inducesisomorphisms

SelF (T /aT )= SelF (T )[a] and XF (T /aT ) = XF (T )/aXF (T ).

Proof Let S= {p}. Let x1, . . . , xk be a system ofA-generators ofa. Weprove by induction on j that H1(GF,S, T A[x1, . . . , xj]) H1(GF,S,T A)[x1, . . . , xj] is an isomorphism. Assume this is known for x1, . . . , xjreplaced with x1, . . . , xj1. Consider the exact sequence

0 A[x1, . . . , xj] Bxj xjB 0

with B = A[x1, . . . , xj1]. After tensoring with T, from the associated longexact cohomology sequence and noting that H0(G

F,S, T

Ax

jB)

=0 by the

hypotheses on T, we find that there is an exact sequence

0 H1GF,S, T A A[x1, . . . , xj] H1(GF,S, T A B)H1(GF,S, T A xjB).


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From the long exact cohomology sequence associated with 0 xjB B B/xjB 0 and the hypotheses on T, we get an exact sequence

0

H1(GF,S, T

A xjB)

H1(GF,S, T

A B).

Since the composition is just multiplication by xj, it follows thatker = H1(GF,S, T A B)[xj]. By the induction hypothesis, we thus have

H1

GF,S, T A A[x1, . . . , xj]= H1GF,S, T A A[x1, . . . , xj],

and therefore H1(GF,S, T A A[a]) = H1(GF,S, T A A)[a].Let w|p be a place of F and let Iw = gwIpg1w GQ . By our

hypothesis on the action of inertia at p, Iw acts trivially on T /Tw=T /gwT. Therefore H1(Iw, T / T w A A[a]) = HomZ(Iw, T / T w A A[a])

and H1(Iw, T / T w A A) = HomZ(Iw, T / T w A A), so H1(Iw, T / T w AA[a]) H1(Iw, T / T w A A).

The proposition now follows from the commutativity of the diagram

H1(GF,S, T A A[a])res

w|p H1(Iw, T / T w A A[a])

H1(GF,S, T A A)[a]res

w|p H1(Iw, T / T w A A)

where the vertical arrows are the maps from the preceding paragraphs.

Corollary 3.8 Let F = Q, K, orK+. With the hypotheses and notationof the preceding proposition,

(i) FtF,A/a(T /aT ) = FtF,A(T ) mod a;(ii) if A and A/a are Noetherian normal domains then (f ) mod a di-vides ChF,A/a(T /a) for any principal ideal (f ) ChF,A; in particu-

lar, if A is a unique factorization domain then ChF,A(T ) mod a divides

ChF,A/a(T /a).

Proof Part (i) follows from basic properties of Fitting ideals (cf. Sect. 3.1.5),and part (ii) follows from the fact that the characteristic ideal is the small-est divisorial ideal containing the Fitting ideal (and that principal ideals aredivisorial).

3.2.5 Descent from K to K+

Let I K be the kernel of the surjection K Q induced by thecanonical projection K Q; we also write I for the kernel of the map


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K

Zp induced by the trivial map K 1. Note that the inclusion+K,A K,A identifies +K,A with K,A/IK,A.For (T,Tp) and A as in Sect. 3.2.2 there is a canonical map

SelK+(T ) SelK(T )Iof+

K,A-modules.

Proposition 3.9 Under the hypotheses of Proposition 3.7, the above map isan isomorphism and induces an isomorphism

XK(T )/IXK(T )

XK

+(T )

of +K,A-modules. Furthermore, if A is an unique factorization domain then

ChK(T ) modI divides Ch

K+

(T ).

Proof The canonical map SelK

+(T ) SelK(T )[I] equals the composi-

tion map

SelK

+

(T )

Sel

K+T A

K,A

1K,

I

= SelK T A K,A

1K I

= SelK(T )I,where the first isomorphism comes from Proposition 3.7 and the second andthird from Proposition 3.4. The claim about characteristic ideals then followsfrom part (ii) of Corollary 3.8.

3.2.6 Specializing the cyclotomic variable

Specializing the cyclotomic variable is more subtle in general; control canfail when the associated p-adic L-function has a trivial zero. The follow-ing proposition establishes a control statement for a situation in which thereshould be no trivial zeros.

Let (T,Tp) and A be as in Sect. 3.2.2. Let IQ be the kernel of the surjectionQ Zp induced by the trivial homomorphism Q 1.

Proposition 3.10 Suppose there is no nontrivial A-subquotient of T onwhich GQ acts trivially. Assume

{p}

contains all prime at which T is

ramified. Then there is an exact sequence

0 SelQ (T ) SelQ(T )Q

H0Ip, T / T p A Q,A1Q Q Q/IQGp .


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In particular, if(H0(Ip, T / T p A Q,A(1Q )) Q Q/IQ)Gp = 0, then re-striction yields an isomorphism

SelQ (T )

SelQ

(T )Q

and even an isomorphism

SelK (T )Sel

K+

(T )+K

if there is no non-trivial A-subquotient ofT on which GK acts trivially.

Proof Let S= {p}. Arguing as in the proof of Proposition 3.7 establishes

H1GQ,S, T A A H1GQ,S, T A Q,A1Q [IQ].On the other hand, the exact sequence

0 A Q,A(1) Q,A 0

yields an exact sequence

H0Ip, T / T p A Q,A1Q Q Q/IQGp H1Ip, T / T p A AGp H1Ip, T / T p A Q,A1Q Gp .

We deduce from this that there is an exact sequence

SelQ ( T ) SelQ

T A Q,A1Q[IQ]

H0

Ip, T / T p A Q,A

1Q Q Q/IQ

Gp ,

where the first map is induced from the inclusion A Q,A and, by Proposi-tion 3.4, is identified with the restriction map SelQ (T ) SelQ(T )Q . Thesearguments are easily adapted to apply to SelK (T ) SelK+(T )

+K .

3.2.7 Relaxing the ramification

Let T and A be as in Sect. 3.2.2.

Lemma 3.11 Suppose F is a number field an vp a place of F at which Tis unramified. Suppose also thatA is Noetherian and normal. Then

CharA

H1

Iv, T A AGF,v= detA1 q1v frob1v |T.


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Here qv is the order of the residue field ofv. The lemma is immediate fromthe hypotheses on T and v. As an immediate consequence of the lemma wehave

Corollary 3.12 Suppose F is a number field and S is a finite set of placesof F not dividing p and such that T is unramified at all v S. Let beany finite set of places containing S. Suppose also that A is Noetherian andnormal. Then

ChF,A(T ) Ch/SF,A (T )

vSdetA

1 q1v frob1v |T

.

The relationship between the Selmer groups SelQ

(T ) as varies has

been analyzed by Greenberg and Vatsal in more detail in the second sectionof [20]. Before explaining their results, we introduce some notation. Let Abe a profinite Zp-algebra. For any finite set of primes and any compactA[GF]-module M, we put

H1(F,M) :=

vF,vp,vH1(GF,v,M),

where F is the set of places ofF over those in .

Let T and A be as in Sect. 3.2.2. Let be a prime. As explained inSect. 3.1.2, there is an isomorphism

H1

GQ,, HomZ

ZGal(Qn/Q)

, T A A

w|

H1

GQn,w, T A A

.

Taking the inductive limit over n yields an isomorphism

H1GQ,, T A Q,A

1Q

w| H1

GQ,w, T A A

.In particular, there is a Q,A-isomorphism

H1

Q, T A Q,A

1Q H1Q, T A A.

Lemma 3.13 Let be a finite set of primes.

(i) H1(Q, T A A) is a finitely generated torsion Q,A-module havingno nontrivial pseudo-null Q,A-submodules.

(ii) If A is Noetherian and normal, then the characteristic ideal ofH1(Q, T A A) is the ideal generated by

PT ,

1Q(frob)

,


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where PT ,(X) := detFA (1 X frob1 ; (T A FA)I ) with FA the totalring of fractions ofA.

Proof The first part follows from a simple computation and actually holds

even if we assume only that T is finitely generated over A. The second partfollows from a simple adaptation of the proof of Proposition 2.4 in [20].

Proposition 3.14 Let (T,Tp) and A be as in Sect. 3.2.2. Suppose A has adecreasing sequence of ideals A I1 I2 such that

n=1 In = 0 and

each A/In is a free Zp-module of finite rank. Suppose also thatXQ,A(T,Tp)is a torsion Q,A-module and there are no nontrivial A-submodules ofT

onwhich GQ acts trivially. Suppose {p} contains all primes at which T isramified. For any finite sets of primes

there is an exact sequence of

Q,A-modules

0 SelQ(T,Tp) SelQ(T,Tp)res H1/

Q, T A A

0,and hence a dual exact sequence of Q,A-modules

0 H1/

Q, T A A

XQ(T,Tp) X

Q(T,Tp) 0.

Proof When A is the ring of integers of a finite extension ofQp, this is justCorollary 2.3 of [20], proved by a Poitou-Tate duality argument. The generalcase follows from this one. Clearly we just need to show that SelQ(T,Tp)

resH1

/(Q, T A A) is surjective. But by Proposition 3.7, SelQ(T/In) =SelQ(T )[In], and these are torsion Q,A/In -modules. Appealing to the caseof the proposition proved in [20] (with Zp in place ofA) gives a surjection

SelQ

(T )

[In

]

res

H1Q, T A A

[In

] 0from which the desired surjection follows upon taking the direct limitover n.

3.3 Selmer groups and modular forms

In this section we introduce the Selmer groups for ordinary modular forms.We begin by recalling some of the standard definitions and results for modularforms.

3.3.1 Elliptic modular forms

For positive integers N and k we let Sk (N) and Mk (N) denote the space ofcusp forms and modular forms, respectively, of weight k for the congruence


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subgroups 1(N). For a Dirichlet character modulo N we let Sk (N,) andMk(N,) be the respective subspaces of Sk(N ) and Mk(N ) of forms withNebentypus . For f Mk(N) we write its Fourier expansion (q-expansion)at the infinite cusp as

f () =

n=0a(n, f )qn, q = e2 i,

where is an element of the upper half-plane h. For a subring A C (resp.a subring A C containing the values of ) we let Sk (N; A) and Mk(N; A)(resp. Sk(N, ; A) and Mk (N,; A)) be the submodules consisting of formswith q-expansion coefficients in A. Recalling that we have identified Qp with

C this defines modules Sk(N; A), Mk(N; A), Sk(N, ; A), and Mk(N,; A)for subrings A Qp .

For a holomorphic function f : h C and = a bc d

SL2(R), we definef|k as usual by (f|k )() := (c+d)kf ( a+bc+d).(Soiff Sk (N,), thenf|k = (d)f for all 0(N ).)

Recall that there is an action of the Hecke algebra h(N) of level N onthe spaces Sk(N; A), Mk(N; A), Sk (N,; A), and Mk(N,; A). The ringh(N) is generated over Z by the so-denoted T (n)-operators and the dia-

mond operators c for (c,N) = 1; when n is a prime T (n) is the dou-ble coset 1(N ) 1 00 n 1(N ) and c = c where c SL2(Z) is such thatc

c1 0

0 c

mod N. Iff Sk(N, ; A) then f|kc = f|kc = (c)f.

3.3.2 Eigenforms

Let f Sk(N ) be a normalized eigenform for the action ofh(N). Here nor-malized means a(1, f ) = 1 (so f|kT (n) = a(n,f)f). Let Nf be the conduc-

tor off (i.e., the level of the associated newform) and let f be the Neben-typus off (the unique character mod Nf such that f|kc = f(c)f for allc Z such that (c,N) = 1). The coefficients a(n,f) generate a finite exten-sion of Q that we denote by Q(f ). We let Z(f ) be the ring of integers ofQ(f ). Let f : h(N) Z(f ) be the homomorphism associated with f; fis characterized by f(T (n)) = a(n,f).

3.3.3 Periods of eigenforms

Let f Sk (N), k 2, be a normalized eigenform. We recall the definition ofthe periods off that we will use to define the p-adic L-function off.

Recall that the Eichler-Shimura map

Per: Sk(N ) H1

1(N), Symk2C2


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is defined by putting Per(g) equal to the class of the cocycle

( )

g(z)

zk1, zk1, . . . , 1

dz,

where the integration is over any path between and (). This map is h(N)-invariant.

Let Z(f )(p) := Q(f ) 1p (Zp) and let

M(f)(p) := H1

1(N), Symk2Z(f )(p)[f].

Suppose N = Nf. Then M(f)(p) is free of rank 2 over Z(f )(p), andvia the inclusion Z(f )(p) C we can view M(f)(p) as a submodule ofH1(1(N), Symk2(C2)) that spans the two-dimensional C-spaceH1(1(N), Symk2(C2))[f]. We fix a Z(f )(p)-basis (+, ) such that() = , where is the involution associated with the conjugation ac-tion of

1 00 1

on H1(1(N), Symk2(C2)). We define the periods f C

off by

Per(f ) = +f + + f .

These periods are well-defined up to units in Z(f )(p) .

3.3.4 Galois representations for eigenforms

Let f Sk(N ), k 2, be a normalized eigenform. We fix L Qp a finiteextension of Qp containing Q(f ). Let OL be the ring of integers of L andFL its residue field. As proved by Eichler, Shimura, and Deligne, there existsa continuous semisimple Galois representation (f, Vf) over L with Vf a

two-dimensional L-space andf : GQ GLL(Vf)

a continuous homomorphism, characterized by being unramified at primespNf and the property that

tr f(frob) = a(, f ), pN .

This representation is irreducible and satisfies

det f = f 1k.

In particular, det f(c) = 1.


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By the continuity of f there exist GQ-stable OL-lattices in Vf. LetTf Vf be such a lattice and let Tf := Tf OL FL. Then Tf has an in-duced continuous FL-linear GQ-action; we denote the corresponding homo-morphism GQ

GLFL (Tf) by

f. We distinguish the following case:

the representation f is irreducible. (irred)

When this is the case for one choice ofTf then it is true for all choices ofTf and the resulting FL-representations are isomorphic.

3.3.5 p-ordinary eigenforms

Recall that an eigenform of level divisible by p is said to be ordinary at p (or

just ordinary since p is fixed) if

a(p,f) is a unit in Zp. (ord)

We will say that an ordinary eigenform f of level Npr , p N, is a p-stabilized newform if either Nf = Npr or Nf = N and r = 1.

When the condition (ord) holds, the Galois representation (f, Vf) re-stricted to Gp contains a Gp-stable L-line V

+f Vf such that the action

of Gp

on V+f

is given by the unramified character whose value on frobp

is

a(p,f). Then Ip acts on the quotient V

f := Vf/V+f by f 1k . Let p de-note the OL -valued character giving the action ofGp on V

f . We distinguish

the following situation:

+p and p are distinct modulo the maximal ideal ofOL. (dist)

3.3.6 Selmer groups attached to ordinary eigenforms

Let f Sk(N ), k 2, be an ordinary normalized eigenform. Let L Qp bea finite extension ofQp containing Q(f ). Let (f, Vf) be the Galois repre-sentation associated with f as above. Fix Tf Vf a GQ-stable OL-latticeand let T+f := Tf V+f . Let T := Tf(det 1f ) and T+ := T+f (det 1f ). LetF = Q, K, Q, K, or K and let be a continuous OL -valued characterofGF. We define the Selmer groups and dual Selmer groups associated withf and as follows. For any finite set of primes we set

Sel

F,L(f,) := Sel

FT , T+ andXF,L(f,) := XF

T , T+ .

A priori these groups depend on the choice ofTf. However, if(irred) holds,then any two such lattices are homothetic and so their corresponding Selmer


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groups are isomorphic as F,OL -modules. In general, for different choices oflattices these modules only differ in their support at primes above p.

For F = Q, K, or K we put

ChF,L(f,) := ChF,OLT , T+ andFtF,L(f,) := FtF,OL

T , T+ .

When {p} or is trivial, then we drop it from our notation.The following is an important fact about these Selmer groups. Its proof is

due to Kato (see [36, Thm. 17.4(1)]).

Theorem 3.15 (Kato) For f as above with p Nf, the dual Selmer groupsXQ(f,) are torsion Q,OL -modules.

3.3.7 Some p-adic deformations of characters

For F = Q or K let F : AF/F F be F := F recF. We simi-larly define

K: AK/K Zp[[Gal(K/K)]] = ,K via the projection

K

Gal(K/K).For an indeterminate W we let W

:=Zp

[[W

]], and for any Zp-algebra A

we let W,A := A[[W]]. We can identify W with Q (resp. +K) by identi-fying (resp. +) with 1 + W. Via this identification Q (resp. +K ) definesa homomorphism W : A/Q W (resp. +W : AK/K W) and Qdefines a homomorphism W : GQ W.

If : W Qp is a Zp-homomorphism such that (1 + W ) = (1 + p)k with k an integer and a primitive pr th root of unity, thenxkxkp ( W)(x) = k(x)|x|kQ, where is the unique idele classcharacter ofp-power order and conductor such that ,p(1

+p)

=. Also,

W = k k .3.3.8 Arithmetic homomorphisms

Let A be a finite integral extension of Zp in Qp . Given a topological A-algebra R we let XR,A := HomcontA-alg(R, Qp). IfA = Zp then we just writeXR for XR,A . Given r R and XR we put r() := (r). We put XW,A :=XW,A ,A and XF,A

:=XF,A ,A for F

=Q or K.

Recall that XW is called arithmetic if(1 + W ) = (1 + p)k2 forsome p-power root of unity and some integer k. The integer k is called theweight of and denoted k . We let t > 0 be the integer such that is aprimitive pt1th root of unity; we sometimes call this the level of. We let := 1 with 1 as in Sect. 3.3.7.


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Let I be a local reduced finite integral extension ofW,A. We let

XaI,A := { XI,A : |W is arithmetic, k|W 2}.

Given Xa

I,A we write k , t , and for the weight, level, and character of|W.The identification ofQ and

+K

with W defines a notion of arithmetichomomorphisms in XQ and X+

K

, with corresponding weights k , lev-els t , and characters .

3.3.9 Hida families

Let be an even Dirichlet character modulo Np, p N. Let I be a lo-

cal reduced finite integral extension of W,Zp[]. Recall that an I-adic el-liptic modular form of tame level N and character is a q-expansionf=n=0 a(n)qn I[[q]] such that for all XaI,Zp[],

f =

n=0

a(n)

qn Mk

Npt , k2; (I)

.

Iff is always an eigenform (resp. cusp form, p-stabilized newform) then we

say f is an I-adic eigenform (resp. cusp form, newform). Similarly, if f isalways ordinary then we say fis an ordinary I-adic modular form.

Given an I-adic form fwe will write f for the associated tame character ;this is called the Nebentypus off. We will also write a(n, f) for the coefficientofqn in the series defining f.

3.3.10 Selmer groups for Hida families

Let fbe an I-adic cusp eigenform (so in particular, I is a local reduced finiteintegral extension ofW,Zp[f]). Assume that

f satisfies (irred) for some (hence all) XaI,Zp[f]. (irred)f

From the theory of pseudo-representations it then follows that there exists acontinuous I-linear Galois representation (f, Tf) with Tf a free I-module ofrank two and f : GQ GLI(Tf) a continuous representation characterizedby the property that f is unramified at all primes Np and satisfies

trace f(frob) = a(, f), Np,

and

det f= f11W .


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The induced GQ-action on Tf I (I) is isomorphic to f (possibly afterextension of scalars).

Let Vf = Tf I FI, where FI is the ring of fractions of I. If f is ordi-nary, then one can use the corresponding property for each f to deduce

the existence of a FI-line V+f Vf which is stable under the action ofGp and on which Gp acts via the unramified character f characterized byf(frobp) = a(p, f). The action ofGp on Vf := Vf/V+f is then via 1f det f.The condition (dist) for some (hence all) f is equivalent to

2f f1 mod mI, (dist)fwhere mI is the maximal ideal ofI. If(dist)f holds, then T

+f := Tf V+f is a

free I-summand ofTf of rank one.Assuming that (dist)f holds, for any finite set of primes and for F =

Q, K, or K we put

SelF (f) := SelF

Tfdet 1f

, T+f

det 1f

and

XF (f) := SelF

Tfdet 1f

, T+f

det 1f

.

We also put

ChF (f)

:=Ch

F,ITfdet 1f , T+f det 1f andFtF (f) := FtF,I

Tfdet 1f

, T+f

det 1f

,

recalling that ChF (f) has only been defined if I is normal (so K,I is aNoetherian normal domain).

3.3.11 Relating SelF (f) to SelF,L(f)

Assume (irred)f and (dist)f hold. Let XaI,Z[f] be an arithmetic homo-morphism. Let L Qp be any finite extension of Qp containing (I) (andhence Q(f )). There is a GQ-isomorphism Tf = TfI, OL which when re-stricted to Gp determines a Gp-isomorphism T

+f

= T+f I,p OL. Here wehave written I, to emphasize that OL is being considered as an I-modulevia . Let p := ker . If {p} contains all the primes dividing Np, thenfor F = Q, K, or K+ there are isomorphisms

SelF (f)[p] I, OL = SelF

Tf/pTf

det 1f

I, OL = SelF,L(f)

(3.4)and

XF,L(f )= XF

Tf/p Tf

det 1f

I, OL = XF (f)/p XF (f)I, OL.(3.5)


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IfF = Q (resp. F =K or K+) these are isomorphisms ofQ,I-modules(resp K,I- or

+K,I-modules). In (3.4) and (3.5) respectively, the first and

second identifications follow from Proposition 3.7.

Lemma 3.16 Let f be an I-adic ordinary eigenform for which (dist)f and(irred)f hold. Then forF = Q orK, XF(f) is a torsion F,I-module.

Proof It is sufficient to prove the lemma under the hypothesis that containsall the primes dividing Np.

Let XaI,Zp[f] be such that p Nf (there are infinitely many such )and L Qp containing (I). As XQ,L(f ) is a torsion Q,OL -module, it

follows from (3.5) that XQ(f) is a torsion Q,I-module. The case F = Kfollows from the case F = Q and Proposition 3.9 and Lemma 3.6.

3.3.12 Relating SelQ,L(f ) to SelQ,L(f )

Let f Sk (N), k 2, be an ordinary normalized eigenform. Let L Qpbe a finite extension ofQp containing Q(f ). Assume that (irred) and (ord)hold for f and let be a finite set of primes containing all those dividingpNf. Let be a pth-power root of unity and 0 m k 2 an integer. Let : Q,OL OL[] the homomorphism sending to (1+p)m and let p bethe kernel of. Let Tf Vf be a GQ-stable OL-lattice and let T = (Tf OLOL[])(det 1f mm) and T+ = (T+f OL OL[])(det 1f mm).Then by Proposition 3.10 the cokernel of the restriction map

SelQ,L[]

f,

mm

= SelQ

T , T+

SelQ(f ) OL OL[][p] = SelQ(T )Qinjects into (H0(Ip, T OL[] OL[](

1Q ) OL[] OL[])Gp , which van-

ishes unless m = 1 and = 1, in which case it is isomorphic to1

a(p,f )1 OL/OL. This proves the following lemma.

Lemma 3.17 With the notation and hypotheses as above

(i) Ifm = 1 or = 1 thenSelQ,L[]

f,

mm= SelQ,L(f ) OL OL[][p], (3.6)

and therefore


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XQ,L[]

f, mm

= XQ,L(f ) OL OL[]/p

XQ,L(f ) OL OL[]

.

(3.7)

(ii) Ifm = 1 and = 1 then there are exact sequences

0 SelQ,L(f ) SelQ,L(f )[p] 1

a(p,f) 1 OL/OL (3.8)

and

OL/a(p,f) 1OL XQ,L(f )/p XQ,L(f ) XQ,L(f ) 0.(3.9)3.3.13 No pseudo-null submodules

Some of the most arithmetically interesting consequences of main conjecturesoccur when the dual Selmer groups do not have non-zero pseudo-null sub-modules (non-zero submodules whose localizations at each height one primeare zero). Greenberg [18] has identified conditions under which this can hold

for very general Selmer groups. The results in this section should be viewedas special cases of Greenbergs.Let f Sk(N ), k 2, be an ordinary p-stabilized eigenform. Let L Qp

be a finite extension ofQp containing Q(f ) and let OL be the ring of integersofL. Assume that (irred) and (ord) hold for f.Let be a finite set of primesofQ containing all those that divide pNf.

Let Tf Vf be a GQ-stable OL-lattice and T+f Tf the unrami-fied rank one Gp-stable OL-summand. Let T := Tf(det 1f ) and T+ :=T+f (det

1f ). Put M := T OL Q,OL (

1Q ), M+ := T+ OL Q,OL (

1Q ),

and M := M/M+. Given an element 0 = x Q,OL we put Tx :=HomOL (L/OL, M[x]), Tx := HomOL (L/OL, M[x]), Vx :=HomOL (L,M[x]), and Vx := Hom(L,M[x]). Then Vx /Tx

M[x] andVx /Tx

M[x] are isomorphisms ofG := GQ,-modules.

Lemma 3.18 Letx = (1 + p)m Q,OL with 0 = m Z. Then

H1Ip, MGp [x] = H1Ip, M[x]Gp = L/OL.Proof Since m = 0, H1(Ip, M[x]) = H1(Ip, M)[x]. Let m be the maximalideal of Q,OL . By local duality H

2(Qp, M[m]) is dual to

H0(Qp, M[m](1)), and the latter is zero since M[m](1) is ramified at


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p. It follows that H1(Qp, M[x]) is divisible (since m= (x,L) with L auniformizer ofL). By the local Euler characteristic formula

dimL H1

Qp, V

x = 1 + dimL H0

Qp, V

x + dimL H0

Qp, V

x

(1)= 1,from which it follows that H1(Qp, M[x]) = L/OL. Since M[x]Ip is finite,it then follows that H1(Ip, M[x])Gp = L/OL.

Lemma 3.19 Suppose XQ,L(f ) is a torsion Q,OL -module. Then for allbut finitely many x = (1 + p)m Q,OL , m Z:

(i) H1(G, M[x])H1(Ip, M[x])Gp;(ii) X

1(,Tx (1)) = 0;

(iii) H2(G, M[x]) =X2(,M[x]) = 0;(iv) H1(G,M)/xH1(G, M) = 0.

Recall thatXi (, ) := ker{Hi (G, )

v H1(Qv, )}. Also, Tx :=

HomOL (Tx , OL).

Proof By Tate-Poitou duality

dimL H1(G, Vx )= dimLX1

, Vx (1)

+ dimL H0(G, Vx ) dimL H0G, Vx (1) dimL H0(R, Vx ) +

v

dimL H0Qv , Vx (1)+ dimL Vx .

As H0(G, Vx ) = 0 = H0(G, Vx (1)), dimL H0(R, Vx ) = 1, andH0(Qv, V

x (1)) = 0 for all but finitely many x, it follows that

dimL H1(G, Vx ) = dimLX1, Vx (1)+ 1 (3.10)for all but finitely many x. On the other hand, letting H1ord(G, Vx ) be thekernel of the restriction map H1(G, Vx ) H1(Ip, Vx )Gp and Imx its im-age, we also have that

dimL H1(G, Vx ) = dimL H1ord(G, Vx ) + dimLImx .

Since XQ,L

(f ) is a torsion Q,OL -module, Sel

Q,L(f )

[x

] =ker

{H1(G,

M[x]) H1(Ip, M)Gp [x]} is finite for all but finitely many x, and henceH1ord(G, Vx ) = 0 for all but finitely many x. In particular, for all but finitelymany x

dimL H1(G, Vx ) = dimLImx . (3.11)


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From Lemma 3.18 it follows that dimLImx 1. Combining this with (3.10)and (3.11) yields

dimL H1(G, Vx ) = dimLImx = 1 and dimLX1, V

x (1)= 0for all but finitely many x. Part (i) of the lemma then follows from

Lemma 3.18 and the first equality. Part (ii) follows from the second equalityand the fact that X1(,Tx (1)) is torsion-free (which follows from (irred))with OL-rank equal to dimLX1(,Vx (1)).

Since H2(Qv, M[x]) is dual to H1(Qv, Tx (1)) and the latter is zerofor all but finitely many x, it follows that for all but finitely many x,H2(G, M[x]) =X2(,M[x]). But by global duality X2(,M[x]) isdual toX1(,Tx (1)), which is zero for all but finitely many x by (ii). Thisproves (iii). Part (iv) follows easily from part (iii).

Proposition 3.20 Let f Sk(N ), k 2, be an ordinary p-stabilized eigen-form. Let L Qp be a finite extension of Qp conta

i Was Aw a Main Conjecture

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