Critical p-adic L-functions and applications to CM forms Goa,...

Critical p-adic L-functions and applications toCM formsGoa, India

Joel Bellaıche

August 16, 2010

Objectives

Objectives: 1.– To give an analytic construction of the p-adicL-function of a modular form in the missing cases (critical cases)(cf. critical p-adic L-function on my webpage or Arxiv)

2.– To compute and study these L-functions, especially in the CMcase (work in progress)

A. Previous constructions of p-adic L-functions of modularforms, and their reformulation by Stevens

We start with a modular form

f (z) =∞∑

n=0

anqn, q = e2iπz

of weight k + 2 (k ≥ 0), level Γ1(N), nebentypus ε. We assumethat f is normalized (a1 = 1), and an eigenform for the Heckeoperators (Tl for l 6 |N and the diamond operators).Let p be a prime not dividing N.To define a p-adic L-function for f , we need first to refine (orp-stabilize) it. Define α and β the roots of

X 2 − apX + pk+1ε(p).

One has 0 ≤ vp(α), vp(β) ≤ k + 1 and vp(α) + vp(β) = k + 1.

A. Previous constructions of p-adic L-functions of modularforms, and their reformulation by StevensWe start with a modular form

f (z) =∞∑

n=0

anqn, q = e2iπz

of weight k + 2 (k ≥ 0), level Γ1(N), nebentypus ε. We assumethat f is normalized (a1 = 1), and an eigenform for the Heckeoperators (Tl for l 6 |N and the diamond operators).

Let p be a prime not dividing N.To define a p-adic L-function for f , we need first to refine (orp-stabilize) it. Define α and β the roots of

X 2 − apX + pk+1ε(p).


A. Previous constructions of p-adic L-functions of modularforms, and their reformulation by StevensWe start with a modular form

f (z) =∞∑

n=0

anqn, q = e2iπz

of weight k + 2 (k ≥ 0), level Γ1(N), nebentypus ε. We assumethat f is normalized (a1 = 1), and an eigenform for the Heckeoperators (Tl for l 6 |N and the diamond operators).Let p be a prime not dividing N.To define a p-adic L-function for f , we need first to refine (orp-stabilize) it. Define α and β the roots of

X 2 − apX + pk+1ε(p).


There are two forms of level Γ := Γ1(N)∩ Γ0(p) attached to f thatare eigenforms for all Hecke operators (Tl for l 6 |Np, Up, and thediamond operators):

fα(z) = f (z)− βf (pz), Upfα = αfα

fβ(z) = f (z)− αf (pz), Upfβ = βfβ

To refine f is to choose one of those forms. The forms fα and fβare called refinements of f .

Theorem (Mazur, Swinnerton-Dyer, Manin, Amice-Velu, Visik(70’s))

Let f be as above, and assume thata.– f is cuspidal.

b.– vp(α) < k + 1 (non-critical slope)Then there exists a unique analytic p-adic L-function Lp(fα, σ),σ ∈ W(Cp) = Hom(Z∗p,C∗p) such that

(INTERPOLATION) For every σ of the form σ(t) = ψ(t)t i , ψfinite image, 0 ≤ i ≤ k,

Lp(fα, σ) = ep(f , α, σ)L(f , ψ−1, i + 1)

Period.

(GROWTH) The function Lp(fα) has order at most vp(α), that is

” |Lp(fα, .)| = O(| logvp(α)p |) ”.


Let f be as above, and assume thata.– f is cuspidal.b.– vp(α) < k + 1 (non-critical slope)

Then there exists a unique analytic p-adic L-function Lp(fα, σ),σ ∈ W(Cp) = Hom(Z∗p,C∗p) such that


Lp(fα, σ) = ep(f , α, σ)L(f , ψ−1, i + 1)

Period.


” |Lp(fα, .)| = O(| logvp(α)p |) ”.


Let f be as above, and assume thata.– f is cuspidal.b.– vp(α) < k + 1 (non-critical slope)Then there exists a unique analytic p-adic L-function Lp(fα, σ),σ ∈ W(Cp) = Hom(Z∗p,C∗p) such that


Lp(fα, σ) = ep(f , α, σ)L(f , ψ−1, i + 1)

Period.


” |Lp(fα, .)| = O(| logvp(α)p |) ”.

I will now explain Stevens’ beautiful and conceptual proof of thistheorem.

Classical Modular Symbols:Let ∆0 be the group of divisors of degree 0 on P1(Q). GL2(Q)acts on it.Let Γ be a congruence subgroup, and V a Γ-module.We define the abelian group of V -valued modular symbols by

SymbΓ(V ) := HomΓ(∆0,V ).

This is a functor in V , obviously left-exact.Suppose that V is acted upon by a subgroup (or submonoid) H ofGL2(Q) larger than Γ. Then SymbΓ(V ) inherits an action of theHecke operators [ΓhΓ], h ∈ H, since it is a group of Γ-invariants inan H-module.

Theorem (Ash-Stevens (80’s))

For every Γ-module V , SymbΓ(V ) ' H1c (Γ,V ) functorially in V

and as Hecke-modules.

Idea of proof: long exact sequence of cohomology for X (Γ),Y (Γ);observe that X (Γ)− Y (Γ) = P1(Q)/Γ

I will now explain Stevens’ beautiful and conceptual proof of thistheorem.Classical Modular Symbols:

Let ∆0 be the group of divisors of degree 0 on P1(Q). GL2(Q)acts on it.Let Γ be a congruence subgroup, and V a Γ-module.We define the abelian group of V -valued modular symbols by







I will now explain Stevens’ beautiful and conceptual proof of thistheorem.Classical Modular Symbols:Let ∆0 be the group of divisors of degree 0 on P1(Q). GL2(Q)acts on it.

Let Γ be a congruence subgroup, and V a Γ-module.We define the abelian group of V -valued modular symbols by







I will now explain Stevens’ beautiful and conceptual proof of thistheorem.Classical Modular Symbols:Let ∆0 be the group of divisors of degree 0 on P1(Q). GL2(Q)acts on it.Let Γ be a congruence subgroup, and V a Γ-module.We define the abelian group of V -valued modular symbols by


This is a functor in V , obviously left-exact.

Suppose that V is acted upon by a subgroup (or submonoid) H ofGL2(Q) larger than Γ. Then SymbΓ(V ) inherits an action of theHecke operators [ΓhΓ], h ∈ H, since it is a group of Γ-invariants inan H-module.





I will now explain Stevens’ beautiful and conceptual proof of thistheorem.Classical Modular Symbols:Let ∆0 be the group of divisors of degree 0 on P1(Q). GL2(Q)acts on it.Let Γ be a congruence subgroup, and V a Γ-module.We define the abelian group of V -valued modular symbols by







classical modular symbols

Now we begin to feed the modular-symbols machine withinteresting Γ-modules V . Let k ≥ 0.Pk = space of polynomials of degree k or less, with its action ofGL2(Q). Vk = dual of Pk .

The space SymbΓ(Vk) has actions of Hecke operators, and if(1 00 −1

)normalizes Γ (which we shall assume), an involution ι

given by this matrix.

Theorem (Eichler-Shimura, Manin-Shokurov)

SymbΓ(Vk) = Sk+2(Γ) ⊕ Mk+2(Γ)= Sk+2(Γ) ⊕ Sk+2(Γ) ⊕ Ek+2(Γ)

Symb+Γ (Vk) = Sk+2(Γ) ⊕ E+

k+2(Γ)

Symb−Γ (Vk) = Sk+2(Γ) ⊕ E−k+2(Γ).


Now we begin to feed the modular-symbols machine withinteresting Γ-modules V . Let k ≥ 0.Pk = space of polynomials of degree k or less, with its action ofGL2(Q). Vk = dual of Pk .The space SymbΓ(Vk) has actions of Hecke operators, and if(

1 00 −1






k+2(Γ)

Symb−Γ (Vk) = Sk+2(Γ) ⊕ E−k+2(Γ).



1 00 −1




SymbΓ(Vk) = Sk+2(Γ) ⊕ Mk+2(Γ)

= Sk+2(Γ) ⊕ Sk+2(Γ) ⊕ Ek+2(Γ)Symb+

Γ (Vk) = Sk+2(Γ) ⊕ E+k+2(Γ)

Symb−Γ (Vk) = Sk+2(Γ) ⊕ E−k+2(Γ).



1 00 −1






k+2(Γ)

Symb−Γ (Vk) = Sk+2(Γ) ⊕ E−k+2(Γ).



SymbΓ(Vk) = Sk+2(Γ) ⊕ Sk+2(Γ) ⊕ Ek+2(Γ)Symb+

Γ (Vk) = Sk+2(Γ) ⊕ E+k+2(Γ)

Symb−Γ (Vk) = Sk+2(Γ) ⊕ E−k+2(Γ).

(One idea from the proof: construct Sk+2(Γ)→ SymbΓ(Vk),f 7→ φf with

φf ({a} − {b}) =

∫ b

af (z)P(z)dz .

The RHS is a linear form of P(z) ∈ Pk , so is in Vk . Themodularity of f implies that φf is a modular symbol. Note: thisRHS is sums of special values of L(f , ψ, i + 1) for variousfinite-order characters ψ and integers 0 ≤ i ≤ k .)

Stevens’ Overconvergent Modular Symbols

Define A as the Frechet space of analytic functionsg(x) =

∑anxn, an ∈ Qp that converge on a ball of center 0 and

some radius > 1 (depending of g). For every k ∈ Z, one can definean action of the monoid

S0(p) =

{γ =

(a bc d

)∈ M2(Zp) ∩GL2(Q), p 6 |a, p|c

},

on A, by setting

(γ ·k g)(x) = (a + cx)kg

(b + dx

a + cx

).

Ak = A with this action. Dk = dual of Ak .For Γ ⊂ Γ0(p) ⊂ S0(p), the space of overconvergent modularsymbols is

SymbΓ(Dk).

This space has an action of the Hecke operators (Tl , Up,diamonds, ι). It is very big, but

Stevens’ Overconvergent Modular SymbolsDefine A as the Frechet space of analytic functionsg(x) =


some radius > 1 (depending of g).

For every k ∈ Z, one can definean action of the monoid

S0(p) =

{γ =

(a bc d

)∈ M2(Zp) ∩GL2(Q), p 6 |a, p|c

},

on A, by setting

(γ ·k g)(x) = (a + cx)kg

(b + dx

a + cx

).


SymbΓ(Dk).





S0(p) =

{γ =

(a bc d

)∈ M2(Zp) ∩GL2(Q), p 6 |a, p|c

},

on A, by setting

(γ ·k g)(x) = (a + cx)kg

(b + dx

a + cx

).


SymbΓ(Dk).





S0(p) =

{γ =

(a bc d

)∈ M2(Zp) ∩GL2(Q), p 6 |a, p|c

},

on A, by setting

(γ ·k g)(x) = (a + cx)kg

(b + dx

a + cx

).

Ak = A with this action.

Dk = dual of Ak .For Γ ⊂ Γ0(p) ⊂ S0(p), the space of overconvergent modularsymbols is

SymbΓ(Dk).





S0(p) =

{γ =

(a bc d

)∈ M2(Zp) ∩GL2(Q), p 6 |a, p|c

},

on A, by setting

(γ ·k g)(x) = (a + cx)kg

(b + dx

a + cx

).

Ak = A with this action. Dk = dual of Ak .

For Γ ⊂ Γ0(p) ⊂ S0(p), the space of overconvergent modularsymbols is

SymbΓ(Dk).





S0(p) =

{γ =

(a bc d

)∈ M2(Zp) ∩GL2(Q), p 6 |a, p|c

},

on A, by setting

(γ ·k g)(x) = (a + cx)kg

(b + dx

a + cx

).


SymbΓ(Dk).


overconvergent modular symbols

Theorem (Stevens (90’s))

For any h ≥ 0,SymbΓ(Dk)slope≤h

is finite dimensional.

(Idea: compacity of Up)For k ≥ 0, obvious exact sequence of Γ-modules:

0 −→ Pk −→ Ak

dk+1

dxk+1−→ A−2−k −→ 0.

This is even an exact sequence of Σ0(p)-module if we twist theaction on A−2−k by detk+1.) Dual exact sequence:

0 −→ D−2−k −→ Dk −→ Vk −→ 0.






0 −→ Pk −→ Ak

dk+1

dxk+1−→ A−2−k −→ 0.

This is even an exact sequence of Σ0(p)-module if we twist theaction on A−2−k by detk+1.)

Dual exact sequence:

0 −→ D−2−k −→ Dk −→ Vk −→ 0.






0 −→ Pk −→ Ak

dk+1

dxk+1−→ A−2−k −→ 0.

This is even an exact sequence of Σ0(p)-module if we twist theaction on A−2−k by detk+1.) Dual exact sequence:

0 −→ D−2−k −→ Dk −→ Vk −→ 0.


Theorem (Stevens-Pollack)

The induced

0→ SymbΓ(D−2−k)→ SymbΓ(Dk)→ SymbΓ(Vk)→ 0.

is still exact. This is an Hecke-sequence if we twist the action ofTl , Up on SymbΓ(D−2−k) by (1 + l)k+1, pk+1.

Idea of proof: Ash-Stevens, long exact sequence of cohomology,and computation of H2

c (Γ,D−2−k), which is 0.

Corollary (Stevens’ control theorem (90’s))

The induced map

SymbΓ(Dk)slope<k+1 −→ SymbΓ(Vk)slope<k+1

is an isomorphism.

End of Stevens’ Construction:

Start with f cuspidal, vp(α) < k + 1. Choose a sign ±.Let φfα ∈ Symb±Γ (Vk) be the modular symbol corresponding to fαby Eichler-Shimura.Lift φfα uniquely to Φfα ∈ Symb±Γ (Dk) by the control theorem(possible since vp(α) < k + 1).Set µfα = Φfα({0} − {∞}) ∈ D.Define L±p (fα,−) as the Mellin transform of the destribition µfα .One gets this way one-half of the p-adic L-function (the values oncharacters σ such that σ(−1) = ±1). One uses the other sign ∓for the other half.Proving the interpolation property is a simple computation usingthe way φfα is defined. The growth property results easily from Φfα

being an eigenform for Up of slope vp(α).Remark: One can do this with f Eisenstein as well (still withvp(α) < k + 1, so vp(α) = 0), at least for one choice of ±. Get anon-zero µfα , but the p-adic L-function is 0.

End of Stevens’ Construction:Start with f cuspidal, vp(α) < k + 1. Choose a sign ±.

Let φfα ∈ Symb±Γ (Vk) be the modular symbol corresponding to fαby Eichler-Shimura.Lift φfα uniquely to Φfα ∈ Symb±Γ (Dk) by the control theorem(possible since vp(α) < k + 1).Set µfα = Φfα({0} − {∞}) ∈ D.Define L±p (fα,−) as the Mellin transform of the destribition µfα .One gets this way one-half of the p-adic L-function (the values oncharacters σ such that σ(−1) = ±1). One uses the other sign ∓for the other half.Proving the interpolation property is a simple computation usingthe way φfα is defined. The growth property results easily from Φfα


End of Stevens’ Construction:Start with f cuspidal, vp(α) < k + 1. Choose a sign ±.Let φfα ∈ Symb±Γ (Vk) be the modular symbol corresponding to fαby Eichler-Shimura.

Lift φfα uniquely to Φfα ∈ Symb±Γ (Dk) by the control theorem(possible since vp(α) < k + 1).Set µfα = Φfα({0} − {∞}) ∈ D.Define L±p (fα,−) as the Mellin transform of the destribition µfα .One gets this way one-half of the p-adic L-function (the values oncharacters σ such that σ(−1) = ±1). One uses the other sign ∓for the other half.Proving the interpolation property is a simple computation usingthe way φfα is defined. The growth property results easily from Φfα


End of Stevens’ Construction:Start with f cuspidal, vp(α) < k + 1. Choose a sign ±.Let φfα ∈ Symb±Γ (Vk) be the modular symbol corresponding to fαby Eichler-Shimura.Lift φfα uniquely to Φfα ∈ Symb±Γ (Dk) by the control theorem(possible since vp(α) < k + 1).

Set µfα = Φfα({0} − {∞}) ∈ D.Define L±p (fα,−) as the Mellin transform of the destribition µfα .One gets this way one-half of the p-adic L-function (the values oncharacters σ such that σ(−1) = ±1). One uses the other sign ∓for the other half.Proving the interpolation property is a simple computation usingthe way φfα is defined. The growth property results easily from Φfα


End of Stevens’ Construction:Start with f cuspidal, vp(α) < k + 1. Choose a sign ±.Let φfα ∈ Symb±Γ (Vk) be the modular symbol corresponding to fαby Eichler-Shimura.Lift φfα uniquely to Φfα ∈ Symb±Γ (Dk) by the control theorem(possible since vp(α) < k + 1).Set µfα = Φfα({0} − {∞}) ∈ D.

Define L±p (fα,−) as the Mellin transform of the destribition µfα .One gets this way one-half of the p-adic L-function (the values oncharacters σ such that σ(−1) = ±1). One uses the other sign ∓for the other half.Proving the interpolation property is a simple computation usingthe way φfα is defined. The growth property results easily from Φfα


End of Stevens’ Construction:Start with f cuspidal, vp(α) < k + 1. Choose a sign ±.Let φfα ∈ Symb±Γ (Vk) be the modular symbol corresponding to fαby Eichler-Shimura.Lift φfα uniquely to Φfα ∈ Symb±Γ (Dk) by the control theorem(possible since vp(α) < k + 1).Set µfα = Φfα({0} − {∞}) ∈ D.Define L±p (fα,−) as the Mellin transform of the destribition µfα .

One gets this way one-half of the p-adic L-function (the values oncharacters σ such that σ(−1) = ±1). One uses the other sign ∓for the other half.Proving the interpolation property is a simple computation usingthe way φfα is defined. The growth property results easily from Φfα


End of Stevens’ Construction:Start with f cuspidal, vp(α) < k + 1. Choose a sign ±.Let φfα ∈ Symb±Γ (Vk) be the modular symbol corresponding to fαby Eichler-Shimura.Lift φfα uniquely to Φfα ∈ Symb±Γ (Dk) by the control theorem(possible since vp(α) < k + 1).Set µfα = Φfα({0} − {∞}) ∈ D.Define L±p (fα,−) as the Mellin transform of the destribition µfα .One gets this way one-half of the p-adic L-function (the values oncharacters σ such that σ(−1) = ±1). One uses the other sign ∓for the other half.

Proving the interpolation property is a simple computation usingthe way φfα is defined. The growth property results easily from Φfα


End of Stevens’ Construction:Start with f cuspidal, vp(α) < k + 1. Choose a sign ±.Let φfα ∈ Symb±Γ (Vk) be the modular symbol corresponding to fαby Eichler-Shimura.Lift φfα uniquely to Φfα ∈ Symb±Γ (Dk) by the control theorem(possible since vp(α) < k + 1).Set µfα = Φfα({0} − {∞}) ∈ D.Define L±p (fα,−) as the Mellin transform of the destribition µfα .One gets this way one-half of the p-adic L-function (the values oncharacters σ such that σ(−1) = ±1). One uses the other sign ∓for the other half.Proving the interpolation property is a simple computation usingthe way φfα is defined. The growth property results easily from Φfα

being an eigenform for Up of slope vp(α).

Remark: One can do this with f Eisenstein as well (still withvp(α) < k + 1, so vp(α) = 0), at least for one choice of ±. Get anon-zero µfα , but the p-adic L-function is 0.

End of Stevens’ Construction:Start with f cuspidal, vp(α) < k + 1. Choose a sign ±.Let φfα ∈ Symb±Γ (Vk) be the modular symbol corresponding to fαby Eichler-Shimura.Lift φfα uniquely to Φfα ∈ Symb±Γ (Dk) by the control theorem(possible since vp(α) < k + 1).Set µfα = Φfα({0} − {∞}) ∈ D.Define L±p (fα,−) as the Mellin transform of the destribition µfα .One gets this way one-half of the p-adic L-function (the values oncharacters σ such that σ(−1) = ±1). One uses the other sign ∓for the other half.Proving the interpolation property is a simple computation usingthe way φfα is defined. The growth property results easily from Φfα


B. The missing p-adic L-functions

Recall the map θ : M†−k(Γ)→ M†k+2(γ) which is (q ddq )k+1 on

q-developements.

Theorem (Stevens-Pollack (00’s))

If vp(α) = k + 1 but fα is not in the image of θ, one can still liftφfα to Φfα uniquely and define the p-adic L-function.

Such an fα is called non-critical (a slightly weaker condition thannon-critical slope). What happens in the critical case?Notations: if M is a space on which the Hecke operators acts,M[fα]= eigenspace for the Hecke operators with same eigenvaluesas fα.M(fα) = generalized eigenspace for the Hecke operators with sameeigenvalues as fα.

Conjecture (Stevens-Pollack)

Assume f is cuspidal, new. Then Symb±Γ (Dk)[fα] has dimension 1.

B. The missing p-adic L-functionsRecall the map θ : M†−k(Γ)→ M†k+2(γ) which is (q d

dq )k+1 onq-developements.










Such an fα is called non-critical (a slightly weaker condition thannon-critical slope). What happens in the critical case?

Notations: if M is a space on which the Hecke operators acts,M[fα]= eigenspace for the Hecke operators with same eigenvaluesas fα.M(fα) = generalized eigenspace for the Hecke operators with sameeigenvalues as fα.





TheoremThis is true for f cuspidal, and also for f Eisenstein if vp(α) > 0.More precisely, if e is the degree of the eigencurve at fα over W,the algebra Hfα generated by the Hecke operators inEnd(Symb±Γ (Dk)(fα)) is isomorphic to Qp[T ]/T e , and

Symb±Γ (Dk)(fα) is free of rank 1 over Hfα

This result offers a natural definition of the p-adic L-function of fαin all cases, including the critical ones: start with a generator Φfα

of Symb±Γ (Dk)[fα], take the distribution µfα := Φfα({0} − {∞}),and define L(fα, σ) as the Mellin transform of µfα .

This definition coincide with the classical one in the cuspidal,non-critical slope case, and with the definition of Stevens-Pollackin the cuspidal, non-critical case.

This p-adic L-function has all expected properties: it has order ofgrowth at most k + 1, it satisfies a functional equation, etc. Alsoone can construct a two-variables p-adic L-function, the firstvariable being a point of the eigencurve (in a neighborhood of fα),the second being σ, that interpolates the one-variable p-adicL-functions, critical or not.

This definition coincide with the classical one in the cuspidal,non-critical slope case, and with the definition of Stevens-Pollackin the cuspidal, non-critical case.This p-adic L-function has all expected properties: it has order ofgrowth at most k + 1, it satisfies a functional equation, etc. Alsoone can construct a two-variables p-adic L-function, the firstvariable being a point of the eigencurve (in a neighborhood of fα),the second being σ, that interpolates the one-variable p-adicL-functions, critical or not.

We now give a sketch of the proof of this theorem.

Construction of the eigencurve through modular symbols(developing unwritten ideas of Stevens):Fix W an affinoid subset of W. I construct an adequate moduleDW of distributions over W , with action of S0(p) such that for allh ≥ 01.– SymbΓ(DW )slope≤h is a finite projective W -module.2.– If k ∈ Z, k ∈W , the fiber at k of that module isSymbΓ(Dk)slope≤h, except when k = 0, h ≥ 1.

We can now construct the eigencurve CSymb. We need onlyconstruct the open part of the eigencuve that lies over W and onwhich the slope vp(Up) ≤ h, which we call C≤h

Symb,W . We define itas the rigid spectrum of the affinoid algebra generated by theHecke operators in EndW (SymbΓ(DW )slope≤h).Since this algebra is an algebra over W , and is torsion-free,C≤h

Symb,W has a flat map toward W . To construct the fulleigencurve CSymb, glue those open parts according to the standardprocedure, axiomatized by Buzzard.

We now give a sketch of the proof of this theorem.Construction of the eigencurve through modular symbols(developing unwritten ideas of Stevens):

Fix W an affinoid subset of W. I construct an adequate moduleDW of distributions over W , with action of S0(p) such that for allh ≥ 01.– SymbΓ(DW )slope≤h is a finite projective W -module.2.– If k ∈ Z, k ∈W , the fiber at k of that module isSymbΓ(Dk)slope≤h, except when k = 0, h ≥ 1.




We now give a sketch of the proof of this theorem.Construction of the eigencurve through modular symbols(developing unwritten ideas of Stevens):Fix W an affinoid subset of W. I construct an adequate moduleDW of distributions over W , with action of S0(p) such that for allh ≥ 0

1.– SymbΓ(DW )slope≤h is a finite projective W -module.2.– If k ∈ Z, k ∈W , the fiber at k of that module isSymbΓ(Dk)slope≤h, except when k = 0, h ≥ 1.




We now give a sketch of the proof of this theorem.Construction of the eigencurve through modular symbols(developing unwritten ideas of Stevens):Fix W an affinoid subset of W. I construct an adequate moduleDW of distributions over W , with action of S0(p) such that for allh ≥ 01.– SymbΓ(DW )slope≤h is a finite projective W -module.

2.– If k ∈ Z, k ∈W , the fiber at k of that module isSymbΓ(Dk)slope≤h, except when k = 0, h ≥ 1.




We now give a sketch of the proof of this theorem.Construction of the eigencurve through modular symbols(developing unwritten ideas of Stevens):Fix W an affinoid subset of W. I construct an adequate moduleDW of distributions over W , with action of S0(p) such that for allh ≥ 01.– SymbΓ(DW )slope≤h is a finite projective W -module.2.– If k ∈ Z, k ∈W , the fiber at k of that module isSymbΓ(Dk)slope≤h, except when k = 0, h ≥ 1.






Symb,W . We define itas the rigid spectrum of the affinoid algebra generated by theHecke operators in EndW (SymbΓ(DW )slope≤h).

Since this algebra is an algebra over W , and is torsion-free,C≤h


Using a theorem of Chenevier, one can prove:

TheoremThere is a canonical isomorphism between the usual eigengurve Cand CSymb.

Recall from Fabrizio’s talk that he has constructed with Iovita andStevens a natural map

SymbΓ(Dk)slope≤h → (M†k+2(Γ)⊕ S†k+2(Γ))slope≤h,

and that they hope/conjecture that for every k ∈ Z it is surjective,or even an isomorphism. I know nothing of this map, but with theideas above, I can prove:

TheoremIf k ∈ Z, k 6= 0, the source and target of this map have the samesemi-simplication as H-modules, and in particular have the samedimension. If k = 0, the semi-simplification of the source is a lineplus the semi-simplification of the target.

Remark: I have no idea why this should be true withoutsemi-simplification in general (especially when k = −1).





and that they hope/conjecture that for every k ∈ Z it is surjective,or even an isomorphism.

I know nothing of this map, but with theideas above, I can prove:







and that they hope/conjecture that for every k ∈ Z it is surjective,or even an isomorphism. I know nothing of this map, but with theideas above, I can prove:



Geometry of the eigencurve:

Theorem(Bellaıche-Chenevier (2003), Chenevier (2004)) Let f be as intheorem. The eigencurve is smooth at fα.

Ingredients of proof: Galois deformations, Galois cohomology,pseudocharacters.

Remark: this is false if k = −1, i.e. f of weight 1. (work inprogress with Mladen Dimitrov).

End of the proof

We study the module Symb±Γ (DW ) as a module over the

eigencurve Cslope≤hSymb,W in a neighborhood of fα (we choose h > vp(α))

LemmaThis module is free of rank one in a neighborhood of fα.

Ingredients : commutative algebra using the smoothness; theory ofnewforms and new components over the eigencurve.

Corollary

The fiber over the weight space of the eigencurve at fα is thespectrum of the algebra Hf ,α.

Results from the definitions and a commutation of End with”taking the fiber” which is not tautological – really needs thelemma. Also, needs to deal with the exceptional case when k = 0.

Lemma (Riemann)

One can find a uniformizer T on the eigencurve at fα, and auniformiser Y of W at k, such that locally near fα, the eigencurvehas equation T e − Y

End of the proofWe study the module Symb±Γ (DW ) as a module over the

eigencurve Cslope≤hSymb,W in a neighborhood of fα (we choose h > vp(α))


Ingredients : commutative algebra using the smoothness; theory ofnewforms and new components over the eigencurve.

Corollary


Results from the definitions and a commutation of End with”taking the fiber” which is not tautological – really needs thelemma. Also, needs to deal with the exceptional case when k = 0.

Lemma (Riemann)



Corollary


Lemma (Riemann)


TheoremHf ,α is isomorphic to Qp[T ]/T e and Symb±Γ (Dk)(fα) is free ofrank 1 on it.

C. Study of critical p-adic L-functions

Note that for a refined form fα to be of critical slope, we needvp(β) = 0 since vp(α) + vp(β) = k + 1, so we need f ordinary atp. Below we shall assume that vp(α) ≥ vp(β).

Classification of forms of critical slope:Case 1: f Eisenstein. Such a form is always ordinary. Thenvp(α) = k + 1, and fα is of critical slope, and actually is critical(that is not in the image of θ).Case 2: f cuspidal CM by a imaginary quadratic field K . Theordinarity of f forces p to be split in K . In this case again, fα iscritical.Case 3: f cuspidal non CM, p-ordinary. Then fα is of critical slope,and is conjectured to be non-critical.

C. Study of critical p-adic L-functionsNote that for a refined form fα to be of critical slope, we needvp(β) = 0 since vp(α) + vp(β) = k + 1, so we need f ordinary atp. Below we shall assume that vp(α) ≥ vp(β).



Classification of forms of critical slope:

Case 1: f Eisenstein. Such a form is always ordinary. Thenvp(α) = k + 1, and fα is of critical slope, and actually is critical(that is not in the image of θ).Case 2: f cuspidal CM by a imaginary quadratic field K . Theordinarity of f forces p to be split in K . In this case again, fα iscritical.Case 3: f cuspidal non CM, p-ordinary. Then fα is of critical slope,and is conjectured to be non-critical.


Classification of forms of critical slope:Case 1: f Eisenstein. Such a form is always ordinary. Thenvp(α) = k + 1, and fα is of critical slope, and actually is critical(that is not in the image of θ).

Case 2: f cuspidal CM by a imaginary quadratic field K . Theordinarity of f forces p to be split in K . In this case again, fα iscritical.Case 3: f cuspidal non CM, p-ordinary. Then fα is of critical slope,and is conjectured to be non-critical.


Classification of forms of critical slope:Case 1: f Eisenstein. Such a form is always ordinary. Thenvp(α) = k + 1, and fα is of critical slope, and actually is critical(that is not in the image of θ).Case 2: f cuspidal CM by a imaginary quadratic field K . Theordinarity of f forces p to be split in K . In this case again, fα iscritical.

Case 3: f cuspidal non CM, p-ordinary. Then fα is of critical slope,and is conjectured to be non-critical.

The case of CM formsTake f CM by K , and assume that p splits in K . Let g ∈M†−k(Γ)be the companion form of fα :

θg = fα.

It is an ordinary overconvergent modular form, whose p-adicL-function is defined since the nineties by work ofGreenberg-Stevens and Kitagawa. Since g is CM, one can also useKatz’ construction.

To describe our formula, we need to introduce the analytic

function log[k]p on the weight space W(Cp) = Hom(Z∗p,C∗p). Let

us choose a generator γ of 1 + pZp. Set s = σ(γ), so s is in theopen ball of center 1 and radius 1 in Cp. Set

log[k]p (σ) =

∏ki=0 logp(γ−js)

(logp γ)k+1

It is easy to see that log[k]p is independent of the choice of γ. This

function vanishes at order 1 on every character of the formσ(t) = ψ(t)t i , ψ of finite image, 0 ≤ i ≤ k.


θg = fα.

It is an ordinary overconvergent modular form, whose p-adicL-function is defined since the nineties by work ofGreenberg-Stevens and Kitagawa. Since g is CM, one can also useKatz’ construction.To describe our formula, we need to introduce the analytic

function log[k]p on the weight space W(Cp) = Hom(Z∗p,C∗p).

Letus choose a generator γ of 1 + pZp. Set s = σ(γ), so s is in theopen ball of center 1 and radius 1 in Cp. Set

log[k]p (σ) =


(logp γ)k+1




θg = fα.

It is an ordinary overconvergent modular form, whose p-adicL-function is defined since the nineties by work ofGreenberg-Stevens and Kitagawa. Since g is CM, one can also useKatz’ construction.To describe our formula, we need to introduce the analytic

function log[k]p on the weight space W(Cp) = Hom(Z∗p,C∗p). Let

us choose a generator γ of 1 + pZp. Set s = σ(γ), so s is in theopen ball of center 1 and radius 1 in Cp. Set

log[k]p (σ) =


(logp γ)k+1



Theorem

Lp(fα, σ) = log[k](σ)Lp(g , σ/tk).

Corollary

Lp(fα, σ) vanishes on every special character. It is a function oforder exactly k + 1.

The case of an Eisenstein seriesIf f is an Eisenstein series, and vp(α) = k + 1, one expects a

somewhat similar formula for Lp(fα,−), namely a product of log[k]p

and two p-adic Dirichlet L-functions.

This was conjectured in a special case back in the nineties byStevens and Pasol (for a critical p-adic L-function that whoseexistence was only conjectural). This is much harder that in theCM case, probably because of the poles that may have the p-adicDirichlet L-functions.What I can prove is just

TheoremIf f is an Eisenstein series, and vp(α) = k + 1, then Lp(fα, σ) isnon zero.

This is in contrast with the case of Eisenstein series with theirordinary refinements.Recently a student of Stevens has made an important progresstoward this conjecture. It is possible that using this progress,Stevens and Pollack will be able to prove the conjecture soon.



and two p-adic Dirichlet L-functions.This was conjectured in a special case back in the nineties byStevens and Pasol (for a critical p-adic L-function that whoseexistence was only conjectural). This is much harder that in theCM case, probably because of the poles that may have the p-adicDirichlet L-functions.

What I can prove is just





and two p-adic Dirichlet L-functions.This was conjectured in a special case back in the nineties byStevens and Pasol (for a critical p-adic L-function that whoseexistence was only conjectural). This is much harder that in theCM case, probably because of the poles that may have the p-adicDirichlet L-functions.What I can prove is just







This is in contrast with the case of Eisenstein series with theirordinary refinements.

Recently a student of Stevens has made an important progresstoward this conjecture. It is possible that using this progress,Stevens and Pollack will be able to prove the conjecture soon.

Other constructions of p-adic L-functionWhat we constructed was the analytic p-adic L-function of fα.One can also attach to fα an algebraic p-adic L-function:

We consider ”the” Galois representation ρf of GQ attached to f ,together with its refinement attached to α, that is the line Lα inDcrys((ρf )|GQp

) on which the crystalline Frobenius acts by α.Note: in the case of an Eisenstein f , the natural representation ρf

to consider here, and in many other situations, is not the obvioussemi-simple sum of two characters, but the only indecomposable,crystalline at p representation with the same trace.

Other constructions of p-adic L-functionWhat we constructed was the analytic p-adic L-function of fα.One can also attach to fα an algebraic p-adic L-function:We consider ”the” Galois representation ρf of GQ attached to f ,together with its refinement attached to α, that is the line Lα inDcrys((ρf )|GQp

) on which the crystalline Frobenius acts by α.

Note: in the case of an Eisenstein f , the natural representation ρf


Other constructions of p-adic L-functionWhat we constructed was the analytic p-adic L-function of fα.One can also attach to fα an algebraic p-adic L-function:We consider ”the” Galois representation ρf of GQ attached to f ,together with its refinement attached to α, that is the line Lα inDcrys((ρf )|GQp

) on which the crystalline Frobenius acts by α.Note: in the case of an Eisenstein f , the natural representation ρf


To (ρf , Lα) one can attach an algebraic p-adic L-function.Greenberg was the first to define it in a big generality, but thisgenerality is not sufficient here as for our critical fα, (ρf , L) doesnot satisfy Panchiskin’s condition.Instead we can use either Perrin-Riou’s very abstract construction,or Pottharst’s more recent (2010) construction using(φ, Γ)-modules. It is not yet known if the two algebraic p-adicL-functions are the same.

It is natural to make an ”Iwasawa mainconjecture”, stating the equality, up to a function with no zerosand no poles, of the analytic p-adic L-function of fα and thealgebraic p-adic L-function of (ρf , Lα).

TheoremAssuming the conjectural computation of the p-adic L-function ofan Eisenstein series, the Iwasawa’s conjecture (version Pottharst) istrue for Eisenstein series.

Ingredients of the proof: use the Mazur-Wiles main conjecture forthe Dirichlet characters, and account for the extra zeros of log[k]

using Colmez’ comutations on the cohomology of (φ, Γ)-module.

To (ρf , Lα) one can attach an algebraic p-adic L-function.Greenberg was the first to define it in a big generality, but thisgenerality is not sufficient here as for our critical fα, (ρf , L) doesnot satisfy Panchiskin’s condition.Instead we can use either Perrin-Riou’s very abstract construction,or Pottharst’s more recent (2010) construction using(φ, Γ)-modules. It is not yet known if the two algebraic p-adicL-functions are the same. It is natural to make an ”Iwasawa mainconjecture”, stating the equality, up to a function with no zerosand no poles, of the analytic p-adic L-function of fα and thealgebraic p-adic L-function of (ρf , Lα).

TheoremAssuming the conjectural computation of the p-adic L-function ofan Eisenstein series, the Iwasawa’s conjecture (version Pottharst) istrue for Eisenstein series.

Ingredients of the proof: use the Mazur-Wiles main conjecture forthe Dirichlet characters, and account for the extra zeros of log[k]

using Colmez’ comutations on the cohomology of (φ, Γ)-module.

This theorem confirms that Potthart’s definition of the algebraicL-function is ”correct”. (or perhaps confirms that my definition ofthe analytic L-function is, that’s a matter of point of view)

I ignoreif the same results is true with Perrin-Riou’s definition of thep-adic L-function. I actually even ignore if this definition makesense for a reducible Galois representation like ρf .There is also a construction of a p-adic L-function of the thirdtype, which is in some sense intermediate between the analyticaland algebraic p-adic L-function, which is often easier to compareto both. I mean the construction with an Euler system like Rubin’sin the CM case an Kato’s in general. This has led to Kato’s proofof one-half of the main conjecture in the ordinary case.In the non-critical slope case, one knows that Kato’s p-adicL-function is the same as the classical analytic p-adic L-function ofMazur et al. This results would be easily to generalized to anyrefined form fα if we know that Kato’s construction can be put infamily, and that Kato’s p-adic L-function of fα is non zero. Ibelieve the first point is a work in progress of a student of Colmez.The second point seems harder, especially in the critical case.

This theorem confirms that Potthart’s definition of the algebraicL-function is ”correct”. (or perhaps confirms that my definition ofthe analytic L-function is, that’s a matter of point of view) I ignoreif the same results is true with Perrin-Riou’s definition of thep-adic L-function. I actually even ignore if this definition makesense for a reducible Galois representation like ρf .

There is also a construction of a p-adic L-function of the thirdtype, which is in some sense intermediate between the analyticaland algebraic p-adic L-function, which is often easier to compareto both. I mean the construction with an Euler system like Rubin’sin the CM case an Kato’s in general. This has led to Kato’s proofof one-half of the main conjecture in the ordinary case.In the non-critical slope case, one knows that Kato’s p-adicL-function is the same as the classical analytic p-adic L-function ofMazur et al. This results would be easily to generalized to anyrefined form fα if we know that Kato’s construction can be put infamily, and that Kato’s p-adic L-function of fα is non zero. Ibelieve the first point is a work in progress of a student of Colmez.The second point seems harder, especially in the critical case.

This theorem confirms that Potthart’s definition of the algebraicL-function is ”correct”. (or perhaps confirms that my definition ofthe analytic L-function is, that’s a matter of point of view) I ignoreif the same results is true with Perrin-Riou’s definition of thep-adic L-function. I actually even ignore if this definition makesense for a reducible Galois representation like ρf .There is also a construction of a p-adic L-function of the thirdtype, which is in some sense intermediate between the analyticaland algebraic p-adic L-function, which is often easier to compareto both. I mean the construction with an Euler system like Rubin’sin the CM case an Kato’s in general. This has led to Kato’s proofof one-half of the main conjecture in the ordinary case.

In the non-critical slope case, one knows that Kato’s p-adicL-function is the same as the classical analytic p-adic L-function ofMazur et al. This results would be easily to generalized to anyrefined form fα if we know that Kato’s construction can be put infamily, and that Kato’s p-adic L-function of fα is non zero. Ibelieve the first point is a work in progress of a student of Colmez.The second point seems harder, especially in the critical case.

This theorem confirms that Potthart’s definition of the algebraicL-function is ”correct”. (or perhaps confirms that my definition ofthe analytic L-function is, that’s a matter of point of view) I ignoreif the same results is true with Perrin-Riou’s definition of thep-adic L-function. I actually even ignore if this definition makesense for a reducible Galois representation like ρf .There is also a construction of a p-adic L-function of the thirdtype, which is in some sense intermediate between the analyticaland algebraic p-adic L-function, which is often easier to compareto both. I mean the construction with an Euler system like Rubin’sin the CM case an Kato’s in general. This has led to Kato’s proofof one-half of the main conjecture in the ordinary case.In the non-critical slope case, one knows that Kato’s p-adicL-function is the same as the classical analytic p-adic L-function ofMazur et al. This results would be easily to generalized to anyrefined form fα if we know that Kato’s construction can be put infamily, and that Kato’s p-adic L-function of fα is non zero. Ibelieve the first point is a work in progress of a student of Colmez.The second point seems harder, especially in the critical case.

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