L’INSTITUTFOURIER DE ANNALES

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A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Masataka CHIDA

Selmer groups and central values ofL-functions for modular forms Tome 67, n^o3 (2017), p. 1231-1276.

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SELMER GROUPS AND CENTRAL VALUES OF L-FUNCTIONS FOR MODULAR FORMS

by Masataka CHIDA

Abstract. — In this article, we construct an Euler system using CM cycles on Kuga–Sato varieties over Shimura curves and show a relation with the central values of Rankin–SelbergL-functions for elliptic modular forms and ring class characters of an imaginary quadratic field. As an application, we prove that the non-vanishing of the central values of Rankin–SelbergL-functions implies the finiteness of Selmer groups associated to the corresponding Galois representation of modular forms under some assumptions.

Résumé. — Dans cet article, nous construisons un système d’Euler en utilisant les cycles CM sur les variétés de Kuga–Sato au-dessus de courbes de Shimura, et montrons une relation avec les valeurs centrales de fonctionsLde Rankin–Selberg associées aux formes modulaires de poids 2 et aux caractères de classes d’un corps quadratique imaginaire. Comme application, nous prouvons que la non-annulation des valeurs centrales de fonctions Lde Rankin–Selberg implique la finitude des groupes de Selmer associés à la représentation galoisienne de la forme modulaire sous certaines hypothèses.

1. Introduction

Let`be a prime and fix an embeddingι`:Q→C`, whereC`=Qc`. Let N be a positive integer andkan even positive integer. Let

f =

∞

X

n=1

an(f)e^2πinz∈Sk(Γ₀(N))^new

be a normalized cuspidal eigenform. Denote by E = Q`({an(f)}n) the Hecke field off over Q` and fix a uniformizerλ of the ring of integersO ofE. Denote the residue field of E byF. Let

ρf : Gal(Q/Q)→GL2(E)

Keywords:Modular forms, Selmer groups, Bloch–Kato conjecture.

Math. classification:11F67, 11R23.

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be the Galois representation associated to f. We put ρ^∗_f =ρ_f ⊗E(^2−k₂ ) and denote by V_f the representation space of ρ^∗_f. Fix a Gal(Q/Q)-stable O-latticeTf and setAf =Vf/Tf. LetLbe an abelian extension ofQand χ a character of the Galois group Gal(L/Q). By the Bloch–Kato conjecture [5], it is expected that the central value of theL-function off twisted by the characterχis related to the order of theχ-part of the Selmer group Sel(L, Af)(see [22, §14] for the definition of the χ-part). Kato [22] proved that the non-vanishing of the central valueL(f, χ, k/2) implies the finiteness of theχ-part of the Selmer group Sel(L, A_f). Moreover, Kato showed a result on the upper bound of the size of the Selmer group in terms of the special values ofL-functions using the Euler system of Beilinson–Kato elements in K2 of modular curves. For an elliptic curve over Q and an imaginary quadratic filed K, similar results in the anticyclotomic setting are considered by Bertolini–Darmon [2] and Longo–Vigni [26] using the Euler system constructed from CM points on Shimura curves. These results were generalized to modular abelian varieties over totally real fields by Longo [25] and Nekovář [30]. In this paper, we will consider the generalization of these results for the central values ofL-function associated to higher weight modular forms twisted by ring class characters over an imaginary quadratic fieldK.

We fix an imaginary quadratic fieldKof discriminantDK <0 satisfying (N, DK) = 1 and denote the integer ring ofKbyOK. ThenKdetermines a factorizationN =N⁺N⁻, whereN⁺ is divisible only by primes which splits inKandN⁻is divisible only by primes which are inert inK. Assume that

(ST) N⁻ is a square-free product of an odd number of inert primes.

Letk>4 be an even integer.⁽¹⁾ Fix an integermsuch that (N DK, m) = 1 and letK_mbe the ring class field ofKof conductorm. Letχbe a character of the Galois groupGm = Gal(Km/K). Then we can define the Rankin–

SelbergL-functionL(f /K, χ, s) associated tof andχ. We define a complex number Ω_f,N⁻ by

Ω_f,N−= 4^k−1π^k||f||_Γ₀_(N₎ ξ_f(N⁺, N⁻) ,

where||f||Γ₀(N)is the Petersson norm off andξ_f(N⁺, N⁻) is the congruence number of f among cusp forms in Sk(Γ₀(N))^N⁻^−new(see §1 for details). Then it is known that the value L(f /K,χ,k/2)

Ω_f,N− belongs toE(χ). Then

(1)Although our proof also works fork= 2, similar results for the weight 2 case are already known by [2], [26], etc. Therefore we will focus on the higher weight case.

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Bloch–Kato conjecture predicts a relation between the value L(f /K,χ,k/2) Ω_f,N−

and the size of theχ-part of Selmer group Sel(K_m, A_f).

We consider the following condition.

Hypothesis (CR⁺).

(1) ` > k+ 1 and#(F^×`)^k−1>5,

(2) The restriction of the residual Galois representationρ_f ofρf to the absolute Galois group ofQ(

q

(−1)^`−1² `)is absolutely irreducible, (3) ρ_f is ramified at q if either (i) q | N⁻ and q² ≡ 1 (mod`) or

(ii)q||N⁺and q≡1 (mod`),

(4) ρ_f restricted to the inertia group ofQq is irreducible ifq²|N and q≡ −1 (mod`).

Our main result is the following theorem.

Theorem 1.1. — Letχbe a ring class character of conductorm. Sup- pose thatf is a cuspidal newform. Assume the following conditions:

(1) `does not divideN DK[Km:K],

(2) the residual Galois representationρ_f satisfies the condition(CR⁺).

Ifc = ordλ

L(f /K,χ,k/2) Ω_f,N−

is finite, then we have λ^c² ·Sel(Km, Af)^χ = 0.

In particular, ifL(f /K, χ, k/2)is non-zero, then theχ-part of the Selmer groupSel(K_m, A_f)is finite.

Remark 1.2.

(1) The assumption (ST) implies that f is not a CM form. Hence the residual Galois representationρ_f =ρ_f,λ satisfies the condition (CR⁺) for all but finitely manyλ.

(2) Let Ω^can_f be Hida’s canonical period defined by Ω^can_f =4^k−1π^k||f||Γ₀(N)

ηf(N) ,

whereηf(N) is the congruence number off among cusp forms in Sk(Γ₀(N)). Under the hypothesis (CR⁺), one can show that

Ω_f,N− =u·Ω^can_f for someu∈ O^×,

if we further assume thatρ_f is ramified at all primes dividingN⁻. A similar result is given as a corollary of the anticyclotomic Iwasawa main conjecture studied in [9] under the ordinary condition. In this paper, we remove the ordinary condition. Also we should mention that Kings–

Loeffler–Zerbes [23, 24] proved that the non-vanishing ofL-values implies

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the finiteness of Selmer groups in the case of Rankin–Selberg product of two modular forms in many cases, and their result contains the case of

“modular forms twisted by ray class characters” (not necessarily ring class characters). However their result does not give a precise quantitative bound of the size of Selmer groups. Moreover we remark that our methods are amenable to being generalized to Hilbert modular forms. This is a big difference with Kato’s Euler system used in [22] or the Euler system of Rankin–Eisenstein classes used in [23, 24].

To prove our main theorem, we develop an analogue of the method of Bertolini–Darmon [2] on the Euler system obtained from CM points on Shimura curves. In [9], we used an Euler system obtained from CM points on Shimura curves and congruences between modular forms of higher weight and modular forms of weight two in the ordinary case. However, in the non-ordinary case it seems difficult to use such congruences. Therefore we choose to use CM cycles on Kuga–Sato varieties over Shimura curves instead of CM points. For the construction of our Euler system, we also use a level raising result (Theorem 6.3) for higher weight modular forms;

the assumption (CR⁺) is necessary to obtain this level raising result. More precisely, under the assumption (CR⁺) we have a freeness result (Propo- sition 6.1) of the space of definite quaternionic modular forms as Hecke modules that is used as an important step in the proof of the level raising result. The freeness result is a generalization of [8, Proposition 6.8] to the

“low weight crystalline case” which is closely related to “R=T” theorems and our case was considered by Taylor [36]. Then one can construct an Euler system using CM cycles and a level raising argument.

Moreover we show a relation between the Euler system and central values of Rankin–SelbergL-functions (Theorem 8.4), the so-called “first explicit reciprocity law” by Bertolini–Darmon. In the case of weight 2, the explicit reciprocity law is proved by Kummer theory and the theory of`-adic uniformization of Shimura curves. To show the explicit reciprocity law in the higher weight case, it is necessary to compute the image of CM cycles under the`-adic Abel–Jacobi map which is defined by Hochschild–Serre spectral sequence. Since it is difficult to compute the image of CM cycles directly, we give a different description of the image of CM cycles using the theory of vanishing cycles and the theory of`-adic uniformization of Shimura curves.

This is a main ingredient of our proof.

This article is organized as follows. First, we review the theory of modular forms on quaternion algebras and the special value formula of Waldspurger in §2. Moreover, we recall basic facts on Galois cohomology and Selmer

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groups in §3. In §4, we review the theory of vanishing cycles which is used in §5 and §6. In §5, we prepare some fundamental results on the cohomology of Shimura curves. In §6, we show a level raising result for higher weight modular forms and prove a key result to compute the image of CM cycles under the`-adic Abel–Jacobi map. In §7 and §8, we construct special cohomology classes using CM cycles on Kuga–Sato varieties and give a proof of the main theorem.

Acknowledgments. — The author expresses sincere gratitude to Jan Nekovář for helpful discussions, comments and encouragement. The author also thanks Yoichi Mieda greatly for his advice on the description of the`- adic Abel–Jacobi map using the monodromy pairing. He thanks Naoki Imai who answered for some questions kindly. He also thanks Ming-Lun Hsieh for helpful discussions on the special value formula for Rankin–SelbergL- functions and the freeness result for Hecke modules. The author thanks the referee for helpful comments and suggestions. The author is partly supported by JSPS KAKENHI Grant Number 23740015 and the research grant of Hakubi project of Kyoto University.

2. Theta elements and the special value formula In this section, we recall the construction of the theta element and the relation with central values of anticyclotomicL-functions for modular forms following [8, §§2, 3 and 4].

Fix an embedding ι_∞:Q,→Cand an isomorphism ι:C→^∼ Cp for each rational primep, whereCp is thep-adic completion of an algebraic closure ofQp. LetZb := lim

←−Z/mZbe the finite completion of Z. For aZ-algebra A, we denoteA⊗_ZZb byA.b

LetKbe an imaginary quadratic field of discriminant−DK<0 and let δ=√

−DK. Denote byz7→z¯the complex conjugate onK. Defineθ by θ=D⁰+δ

2 , D⁰ =

(DK if 2-DK, DK/2 if 2|DK.

Fix positive integers N⁺ that are only divisible by prime split in K and N⁻ that are only divisible by primes inert in K. We assume that N⁻ is the square-free product of an odd number of primes. LetB be the definite quaternion over Q which is ramified at the prime factors ofN⁻ and the archimedean place. We can regardK as a subalgebra ofB (see [8, §2.2]).

Write T and N for the reduced trace and norm of B respectively. Let

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G=B^× be the multiplicative algebraic group ofB overQand letZ=Q^× be the center ofG. Let ` - N⁻ be a rational prime. Letm be a positive integer such that (m, N⁺N⁻`) = 1. We choose a basis ofB =K⊕K·J overKsuch that

• J²=β∈Q^× withβ <0 andJ t=tJ for allt∈K,

• β∈(Z^×q)²for allq|N⁺ andβ ∈Z^×q forq|DK. Fix a square root√

β∈Qofβ. We fix an isomorphismi^(N⁻⁾=Q

q-N⁻i_q : Bb^(N⁻⁾∼=M₂(A^(N

−)

f ) as follows. For each finite place q|m`N⁺, the isomor- phismiq :Bq ∼=M2(Qq) is defined by

iq(θ) =

T(θ) −N(θ)

1 0

, iq(J) =p β·

−1 T(θ)

0 1

(p

β ∈Z^×q).

For each finite place q - N⁺N⁻`m, choose an isomorphism iq : Bq :=

B⊗_QQq ∼=M2(Qq) such that

i_q(O_K⊗Zq)⊂M₂(Zq).

From now on, we shall identifyBq and G(Qq) with M2(Qq) and GL2(Qq) viai_q forq-N⁻. Finally, we define

i_K :B ,→M₂(K), a+bJ7→i_K(a+bJ) :=

a bβ b a

(a, b∈K) and leti_C:B→M2(C) be the composition i_C=ι_∞◦iK.

Fix a decomposition N⁺OK =N⁺N⁺ once and for all. For each finite placeq, we defineςq∈G(Qq) as follows:

ς_q=











1 ifq-N⁺m,

δ⁻¹ θ θ 1 1

!

ifq=q¯qis split withq|N⁺, qⁿ 0

0 1

!

ifq|m andqis inert inK (n= ordq(m)), 1 q⁻ⁿ

0 1

!

ifq|m andqsplits inK (n= ordq(m)).

Definexm:A^×K →G(A) by

xm(a) :=a·ς (ς :=Y

q

ςq).

This collection{x_m(a)}_a∈

A^×_K of points is called Gross points of conductor massociated toK.

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LetOK,m=Z+mOKbe the order ofKof conductorm. For each positive integer M prime to N⁻, we denote by RM the Eichler order of level M with respect to the isomorphisms{i_q :B_q 'M₂(Qq)}_q_-_N−. Then one can see that the inclusion mapK ,→B is an optimal embedding ofOK,m into the Eichler order B ∩ςRbM(ς)⁻¹ (i.e. (B∩ςRbM(ς)⁻¹)∩K = OK,m) if ordq(M)6ordq(m) for all primesq|m.

Let k > 2 be an even integer. For a ring A, we denote by Lk(A) = Sym^k−2(A²) the set of homogeneous polynomials in two variables of degree k−2 with coefficients inA. We write

Lk(A) = M

−^k₂<r<^k₂

A·vr (vr:=X^k−2² ^−rY^k−2² ^+r).

Also we let ρ_k : GL₂(A) → Aut_AL_k(A) be the unitary representation defined by

ρk(g)P(X, Y) = det(g)⁻^k−2² ·P((X, Y)g) (P(X, Y)∈Lk(A)).

IfA is aZ(`)-algebra with ` > k−2, we define a perfect pairing h,i_k :Lk(A)×Lk(A)→A

by hX

i

aivi,X

j

bjvji_k = X

−k/2<r<k/2

arb−r·(−1)^k−2² ^+rΓ(k/2 +r)Γ(k/2−r) Γ(k−1) . ForP, P⁰∈Lk(A), this pairing satisfies

hρk(g)P , ρk(g)P⁰i_k =hP, P⁰i_k. Via the embeddingi_C, we obtain a representation

ρ_k,∞:G(R) = (B⊗_QR)^× −→ⁱ^C GL₂(C)→Aut_CL_k(C).

ThenC·v_ris the eigenspace on whichρ_k,∞(t) acts by (t/t)^rfort∈(K⊗_Q C)^×. IfAis aK-algebra andU ⊂G(Af) is an open compact subgroup, we denote byS^B_k(U, A) the space of modular forms of weight k defined over A, consisting of functionsf :G(Af)→Lk(A) such that

f(αgu) =ρk,∞(α)f(g) for allα∈G(Q) andu∈U.

SetS^B_k(A) := lim

−→^US^B_k(U, A). LetA(G) be the space of automorphic forms onG(A). For v∈Lk(C) and f ∈S^B_k(C), we define a function Ψ(v⊗f) : G(Q)\G(A)→Cby

Ψ(v⊗f)(g) :=hρk,∞(g∞)v, f(gf)i.

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Then the mapv⊗f 7→Ψ(v⊗f) gives rise to aG(A)-equivariant morphism Lk(C)⊗S^B_k(C) → A(G). Let ω be a unitary Hecke character of Q. We writeS^B_k(U, ω,C) ={f ∈S^B_k(U,C)|f(zg) =ω(z)f(g) for allz ∈Z(A)}.

LetA^B_k(U, ω,C) be the space of automorphic forms on G(A) of weightk and central characterω, consisting of functions Ψ(f⊗v) :G(A)→Cfor f ∈S_k^B(U, ω,C) andv∈Lk(C). For each positive integerM, we put

S^B_k(M,C) =S^B_k(Rb^×_M,1,C), A^B_k(M,C) =A^B_k(Rb^×_M,1,C), where1is the trivial character.

Letπbe a unitary cuspidal automorphic representation on GL2(A) with trivial central character andπ⁰ the unitary irreducible cuspidal automorphic representation on G(A) with trivial central character attached to π via the Jacquet–Langlands correspondence [18]. Let π⁰_fin denote the finite constituent of π⁰. Let R := RN⁺ be an Eichler order of level N⁺. The multiplicity one theorem together with our assumptions implies that π_fin⁰ can be realized as a uniqueG(Af)-submodule S^B_k(π_fin⁰ ) of S^B_k(C) and S^B_k(N⁺,C)[π_fin⁰ ] := S^B_k(π⁰_fin)∩S^B_k(N⁺,C) is one dimensional. We fix a nonzero newform f_π0 ∈ S^B_k(N⁺,C)[π⁰_fin]. Define the automorphic form ϕπ⁰ ∈ A^B_k(N⁺,C) by

ϕπ⁰ := Ψ(v^∗₀⊗fπ⁰) (v^∗₀=D

k−2 2

K ·v0).

Define the local Atkin–Lehner element τ_q^N⁺ ∈ G(Qq) by τ_q^N⁺ = J for q|∞N⁻,τ_q^N⁺ = 1 for finite placeq- N and τ_q^N⁺ =

0 1

−N⁺ 0

if q|N⁺. Let τ^N⁺ := Q

qτ_q^N⁺ ∈ G(A). Let Cl(R) be a set of representatives of B^×\Bb^×/Rb^×Qb^×inBb^×=G(Af). Define the inner product offπ⁰ with itself by

hfπ⁰, f_π⁰i_R:= X

g∈Cl(R)

1

#Γg

· hfπ⁰(g), f_π⁰(gτ^N⁺)i_k, where Γ_g:= (B^×∩gRb^×g⁻¹Qb^×)/Q^×.

Let ` - N⁻ be a rational prime. Let λ and l be the primes of Q and K induced byι` respectively. We recall the description of `-adic modular forms on B^×. Let A be a OK_l-algebra. For an open compact subgroup U ⊂Rb^×, we define the space of`-adic modular forms of weightkand level U by

S_k^B(U, A) :=n

fb:Bb →Lk(A)

fb(αgu) =ρk(u⁻¹_` )f(g), αb ∈B^×, u∈UQb o

.

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Also we writeS_k^B(N⁺, A) :=S_k^B(Rb^×, A). We leti_K_l:B ,→M2(K_l) be the compositioniK_l :=ι`◦iK. Defineρk,`:B_`^×→AutLk(C`) by

ρ_k,`(g) :=ρ_k(i_K_l(g)).

By definition, ρk,` is compatible with ρk,∞ in the sense that ρk,`(g) = ρ_k,∞(g) for everyg∈B^×, and one can check that

ρk,`(g) =ρk(γli`(g)γ_l⁻¹),whereγl:=

√ β −√

βθ

−1 θ

∈GL2(Kl).

Here i_` : B_` ' M₂(Q`) is the fixed isomorphism. If ` is invertible in A, there is an isomorphism:

S^B_k(N⁺, A)∼=S_k^B(N⁺, A), f 7→fb(g) :=ρ_k(γ_l⁻¹)ρ_k,`(g⁻¹_` )f(g).

LetQ(f) be the finite extension ofQgenerated by the Fourier coefficients of the newformf =fπ ∈S_k^new(Γ0(N)). Let O ⊂C` be the completion of the ring of integers ofQ(f) with respect toλ⁰=λ∩Q(f). Fix a uniformizer λin O. TheO-moduleS_k^B(N⁺,O)[π_fin⁰ ] :=S_k^B(N⁺,O)∩ S_k^B(N⁺,C`)[π_fin⁰ ] has rank one. We sayfπ⁰ ∈S^B_k(N⁺,C)[π⁰_fin] isλ-adically normalized iffcπ⁰

is a generator ofS_k^B(N⁺,O)[π⁰_fin] overO. This is equivalent to the following condition:

fc_π⁰(g₀)6≡0 (modλ) for some g₀∈G(Af).

Now we define the theta elements. For a positive integer m, let G_m = K^×\Kˆ^×/Ob^×_K,m be the Picard group of the order OK,m. We identify Gm

with the Galois group of the ring class fieldKmof conductormoverKvia geometrically normalized reciprocity law.

Denote by [·]m: Kb^× → Gm, a7→[a]m the natural projection map. We consider the automorphic formϕ_π⁰ = Ψ(v^∗₀⊗fπ⁰). It is easy to see that the function

ϕb_π⁰ :Kb^×→C, a7→ϕb_π⁰(a) :=ϕ_π⁰(x_m(a))

factors through Gm, so we can extend ϕb_π⁰ linearly to be a function ϕb_π⁰ : C[Gm] → C. Let Pm := [1]m ∈ Gm be the distinguished Gross point of conductorm. We put

ϕbπ⁰(σ(Pm)) =ϕπ⁰(xm(a)) ifσ= [a]m∈ Gm. We define the theta element Θ(fπ⁰)∈C[Gm] by

Θ(fπ⁰) := X

σ∈G_m

ϕbπ⁰(σ(Pm))·σ.

Then we have the following special value formula.

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Proposition 2.1. — Letχbe a character ofGm. Then we have

χ(Θ(f_π⁰)²) = Γ(k/2)²·L(f_π/K, χ, k/2)

Ω_π,N⁻ ·(−1)^k²·m·D^k−1_K ·(#O_K/2)² 2

×p

−DK

−1·χ(N⁺), where

Ω_π,N− =4^k−1π^k||f_π||_Γ₀_(N) hfπ⁰, fπ⁰i_R is theλ-adically normalized period forf.

Proof. — This formula is a special case of Hung’s result [15, Proposi- tion 5.3]. Also see [8, Proposition 4.3] for the case whereχis an unramified

character.

3. Selmer groups for modular forms 3.1. Definition of Selmer groups

First we recall the definition of Selmer groups following Bloch–Kato [5].

Letf be a cuspidal Hecke eigenform of weight k with respect to Γ0(N).

LetQ(f) denote the Hecke field generated by the eigenvalues {a_q(f)} of the Hecke operators{Tq}. Letλ be the prime ofQ(f) above the prime ` induced by the fixed embeddingι`. DenoteE=Q(f)λ. Also we denote the integer ring ofE byO and the uniformizer byλand write On =O/λⁿO.

Then there exist a 2-dimensional Galois representation ρf =ρf,λ:G_Q= Gal(Q/Q)→GL2(E)

such that det(1−ρ_f(Frob_q)·X) = 1−a_q(f)X+q^k−1X² for any prime q satisfyingq-`N. LetVf be the representation space of ρf⊗ε

2−k 2

` , where ε_`: Gal(Q/Q)→Z^×` is the `-adic cyclotomic character. We choose aG_Q- stableO-latticeTf in Vf, and denote Af =Vf/Tf. Then there is an exact sequence 0→T_f →ⁱ V_f →^pr A_f →0.

For a finite extension F/Qp, Bloch–Kato [5] defined the finite part of Galois cohomology groups by

H_f¹(F, Vf) :=

(Ker

H¹(F, V_f)→H¹(F^ur, V_f)

`6=p, Ker

H¹(F, Vf)→H¹(F, Vf⊗_Q_pBcris)

`=p,

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where B_cris is the p-adic period ring defined by Fontaine and F^ur is the maximal unramified extension ofF. Also we denote

H_f¹(F, T_f) =i⁻¹(H_f¹(F, V_f)) and

H_f¹(F, Af) = Imh

H_f¹(F, Vf),→H¹(F, Vf)−→^pr H¹(F, Af)i .

For a number fieldF, we define theλ-part of the Selmer group off by Sel(F, Af) = Ker

"

H¹(F, Af)→Y

v

H¹(Fv, Af) H_f¹(F_v, A_f)

# .

We also define

H_f¹(F, Vf) = Ker

"

H¹(F, Vf)→Y

v

H¹(Fv, Vf) H_f¹(Fv, Vf)

# .

Moreover we setAf,n = Af[λⁿ] = Ker[Af λⁿ

→ Af] and Tf,n = Tf/λⁿTf. Then there exists a Galois-equivariant bilinear pairing Tf ×Tf → O(1) such that the induced pairings onTf,n ∼=Af,n are non-degenerate for all n. For details, see Nekovář [27, Proposition 3.1].

Proposition 3.1. — The pairing above induces the local Tate pairing h,i_v:H¹(Fv, Tf)×H¹(Fv, Af)→H²(Fv, E/O(1))∼=E/O, h,i_v:H¹(F_v, T_f,n)×H¹(F_v, A_f,n)→H²(F_v,On(1))∼=On, for each place v of F. The local Tate pairing is perfect and satisfies the following properties.

(1) The pairing h , i_v makes H_f¹(Fv, Tf) and H_f¹(Fv, Af) into exact annihilators of each other at any placev.

(2) Ifxandy belong toH¹(F, Af,n), then X

v

hx, yi_v = 0,

where the sum is over all placesv ofF but is a finite sum.

Proof. — See Besser [4, Proposition 2.2].

Definition 3.2. — For each placev, we define H_f¹(Fv, Af,n)to be the preimage ofH_f¹(Fv, Af)inH¹(Fv, Af,n). Then we let

Sel(F, Af,n) = Ker

"

H¹(F, Af,n)→Y

v

H¹(Fv, Af,n) H_f¹(F_v, A_f,n)

# .

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Also we defineH_f¹(F_v, T_f,n)to be the image ofH_f¹(F_v, T_f)inH¹(F_v, T_f,n).

Moreover we define the singular part of local cohomology group H_sing¹ (F_v, T_f,n)to be the quotient

H_sing¹ (F_v, T_f,n) = H¹(F_v, T_f,n) H_f¹(Fv, Tf,n). Ifv does not divideN, then we have

H_sing¹ (Fv, Tf,n) =H¹(F^ur, Tf,n)^G^Fv.

By Proposition 3.1, H_f¹(Fv, Af,n) and H_sing¹ (Fv, Tf,n) are the Pontryagin dual of each other.

For each primeqand G_Q-moduleM, we denote H_f¹(Fq, M) =M

v|q

H_f¹(Fv, M) and

H_sing¹ (Fq, M) =M

v|q

H_sing¹ (Fv, M).

Lemma 3.3. — Letqbe a prime which splits inK. Then H_sing¹ (K_m,q, T_f,n) = 0

for sufficiently largem.

Proof. — Proceed as in the proof of [2, Lemma 2.4].

3.2. The Euler system argument

Here we give a generalization of the Euler system argument introduced by Bertolini–Darmon [2] to the case of higher weight modular forms.

Definition 3.4. — A primepis said to be n-admissible if (1) pdoes not divideN `[Km:K].

(2) pis inert inK.

(3) λdoes not dividep²−1.

(4) λⁿ divides p^k² +p^k−2² −ε·a_p(f), whereε=±1.

Lemma 3.5. — Letp be ann-admissible prime. Then H_f¹(Km,p, Af,n) and H_sing¹ (Km,p, Tf,n) are both isomorphic to On[Gm]. In particular, the χ-parts of these groups are both isomorphic toOn.

Proof. — This is a direct generalization of [2, Lemma 2.6].

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Define the map ∂_p to be the composition

H¹(Km, Af,n)→H¹(Km,p, Af,n)→H_sing¹ (Km,p, Af,n).

If∂p(κ) = 0 forκ∈H¹(Km,p, Af,n) (resp. H¹(Km,p, Tf,n)), let vp(κ)∈H_f¹(Km,p, Af,n) (resp. H_f¹(Km,p, Tf,n)) denote the natural image ofκunder∂_p.

Theorem 3.6 ([9, Theorem 6.3]). — Let s∈H¹(K_m, A_f,n) be a nonzero element. Then there exist infinitely manyn-admissible primespsuch that∂p(s) = 0andvp(s)6= 0.

Definition 3.7. — For a prime p, we define the compactified Selmer groupH_p¹(K_m, T_f,n)to be

H_p¹(Km, Tf,n) = Ker



H¹(Km, Tf,n)→Y

v-p

H¹(Km,v, Tf,n) H_f¹(K_m,v, T_f,n)



.

Recall that λis the fixed uniformizer ofO.

Theorem 3.8. — Let t be a positive integer. Suppose that for all but finitely many n-admissible primesp there exists an elementκp ∈ H_p¹(K_m, T_f,n+t)^χsuch thatλ^t−1·∂_p(κ_p)6= 0. Thenλⁿ·Sel(K_m,A_f,n+t)^χ= 0 Proof. — Assume that there exists an element s in Sel(Km, Af,n+t)^χ satisfyingλⁿs6= 0. By Theorem 3.6 and the assumption, we can take an n+t-admissible primepsatisfying the following properties simultaneously:

(1) vp(λⁿs)6= 0 and ∂p(λⁿs) = 0.

(2) there exists an elementκ_p∈H_p¹(K_m, T_f,n+t)^χ such that λ^t−1∂_p(κ_p)6= 0.

By the properties of the local Tate pairing, we have X

q

hλ^t−1∂_q(κ_p), v_q(λⁿs)i_q = 0.

SinceH_f¹(K_m,q, A_f,n)^χandH_f¹(K_m,q, T_f,n)^χ are annihilators of each other, we have

hλ^t−1∂_q(κ_p), v_q(λⁿs)i_q= 0 forq6=p.

Thereforehλ^t−1∂_p(κ_p),resp(λⁿs)i_q= 0 by Proposition 3.1 (2). Since the local Tate pairing is perfect, the assumptionλ^t−1∂_p(κ_p)6= 0 impliesv_p(λⁿs) =

0. This gives a contradiction.

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4. Review of vanishing cycles

In §5 and §6 we will use the theory of vanishing cycles in several important steps. Therefore, in this section we briefly recall the theory of vanishing cycles following the exposition in Rajaei [33].

4.1. Vanishing cycles

LetRbe a characteristic 0 henselian discrete valuation ring with residue field k of characteristic p. Fix a uniformizer $ in R. Denote the fraction field ofR by K and the maximal unramified extension of K byK^ur. Let X → S = SpecR be a proper and generically smooth curve and F a constructible torsion sheaf onX whose torsion is prime top. Leti:Xk → X, j :XK →X, i:X_k →X_O_Kur and j :X_K →X_O_Kur be the canonical maps. By the proper base change theorem and the Leray spectral sequence forj, we have

RΓ(X_K, j^∗F) =RΓ(X_O_Kur, Rj_∗j^∗F)→^' RΓ(X_k, i^∗Rj_∗j^∗F).

Then the adjunction morphism givesφ : i_∗F→i^∗Rj_∗j^∗F. We define the vanishing cycles by

RΦF:= Cone(φ), and the nearby cycles by

RΨF:=i^∗Rj_∗j^∗F. Then we have a distinguished triangle

→i^∗F→RΨF→RΦF⁺¹→.

For i > 0, we have RⁱΦF = RⁱΨF. Let Σ be the set of singular points ofX_k. Assume that a neighbourhood of each singular pointxis (locally) isomorphic to the subscheme of A²S = S[t₁, t₂] with equation t₁t₂ = a_x (denoteex:=v(ax)>0). When the special fiberX_kis reduced, Deligne [11]

proved the sheaves RⁱΦF vanish for i 6= 1 and R¹ΦF is supported at Σ, and the specialization mapH¹(X_k,F)→H¹(X_K,F) is injective. Now we have the specialization sequence

0−→H¹(X_k, i^∗F)(1)−→H¹(X_K,F)(1)−→^β M

x∈Σ

(R¹ΦF)_x(1)

−→H²(X_k, i^∗F)(1)^sp(1)−→H²(X_K,F)(1)−→0.

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Then we define the character group for the sheafFonX by X(F) = Ker

"

M

x∈Σ

R¹ΦFx(1)→Ker(sp(1))

# ,

so that there is a short exact sequence

(4.1) 0−→H¹(X_k,F)(1)−→H¹(X_K,F)(1)−→X(F)−→0.

Forx∈Σ, let (X_k)xbe the henselization ofX_k atxandBxthe set of two branches ofX_k atx(i.e. the irreducible components of (X_k)_x). Forx∈Σ, we define the modulesZ(x) andZ⁰(x) by

Z(x) := Cokerh

Z−→^diagZ^B^x i

and

Z⁰(x) := Kerh

Z^B^x −→^sum Z i

.

Choose an ordering forBx for each x∈Σ and define a basis of Z⁰(x) by δ_x⁰ := (1,−1). Denote the dual basis by δx ∈ Z(x). We set Λ = Z`. For x∈Σ, one hasH_xⁱ(X_k, RΨΛ) = 0 fori6= 1,2 and the trace map gives an isomorphism H_x²(X_k, RΨΛ)−→^' Λ(−1) and H_x¹(X_k, RΨΛ) −→^' Z(x)⊗Λ.

Moreover we haveR¹ΦΛx

−→' Z⁰(x)⊗Λ. Therefore we have the perfect pairing

(R¹ΦΛ)x×H_x¹(X_k, RΨΛ)−→H_x²(X_k, RΨΛ)−→^' Λ(−1).

This pairing gives the cospecialization map 0−→H⁰(Xe_k, RΨΛ)−→H⁰(Xe_k, i^∗Λ)−→M

x∈Σ

H_x¹(X_k, RΨΛ)

β⁰

−→H¹(X_K,Λ)−→H¹(X_k, i^∗Λ)−→0, whereXe_k →X_k is the normalization map.

4.2. Monodromy pairing

Let ` be a prime different from p and let I be the inertia subgroup of Gal(K/K). We consider the map t_` : I → Z`(1) which is defined by σ7→σ($^1/`)/$^1/`, where$ is the uniformizer ofR. Forσ∈Iandx∈Σ, we define the variation map

Var(σ)x : (R¹ΦΛ)x→H_x¹(X_k, RΨΛ)

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bya7→ −ext_`(σ)(aδ_x)δ_x, and define the monodromy logarithm N_x: (R¹ΦΛ)_x(1)→H_x¹(X_k, RΨΛ)

byN_x(t_`(σ)a) = Var(σ)_x(a) for a∈(R¹ΦΛ)_x and σ∈I. Then we have a commutative diagram

(R¹ΦΛ)x(1) −−−−→^' Z⁰(x)⊗Λ



 y^N^x



 y^φ^x H_x¹(X_k, RΨΛ) −−−−→^' Z(x)⊗Λ,

where the right vertical map φx is given by δ⁰_x 7→ −exδx. Moreover we define the monodromy operatorN by the following diagram:

H¹(X_K,Λ)(1) =H¹(X_k, RΨΛ)(1) −−−−→^β L

x∈Σ

(R¹ΦF)x(1)



 y^N



 y

L_N

x

H¹(X_K,Λ) =H¹(X_k, RΨΛ) ^β

0

←−−−− L

x∈Σ

H_x¹(X_k, RΨΛ).

Then we have an explicit description of the mapN.

Theorem 4.1 (Picard–Lefschetz formula [10]). — With notation as above, we have the following formula:

N(t`(σ)a) = (σ−1)a fora∈H¹(X_K,Λ)andσ∈I.

LetB be the set of irreducible components ofX_k. Define the modulesX andXb by the exact sequences

0−→X−→M

x∈Σ

Z⁰(x)−→Z^B −→Z−→0 and

0−→Z−→Z^B−→M

x∈Σ

Z(x)−→Xb −→0.

Then the monodromy pairing

u : X⊗X→Z is given by the diagram

X −−−−→ L

x∈ΣZ⁰(x)



 y

u∗



 y

L

x∈Σφx

Xb ←−−−− L

x∈ΣZ(x).

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Also we have

X⊗Λ = Im

"

H¹(X_K,Λ)(1)→M

x∈Σ

(R¹ΦΛ)x(1)

#

and

Xb⊗Λ = Coker

"

H⁰(Xe_k,Λ)→M

x∈Σ

H_x¹(X_k, RΨΛ)

# .

Therefore we obtain the diagram

H¹(X_K,Λ)(1) −−−−→ X⊗Λ −−−−→ L

x∈Σ

(R¹ΦΛ)_x(1)



 y^N



 y^u^∗^⊗Λ



 y

LNx

H¹(X_K,Λ) ←−−−− Xb⊗Λ ←−−−− L

x∈Σ

H_x¹(X_k, RΨΛ).

Note that the cokernel ofu∗is the group of connected components.

Let F be a locally constant Z^`-sheaf on X. The cospecialization exact sequence is

0−→H⁰(Xe_k, RΨF)−→H⁰(Xe_k, i^∗F)−→M

x∈Σ

H_x¹(X_k, RΨ(F))

β⁰

−→H¹(X_K,F)−→H¹(X_k, i^∗F)−→0.

Now we define the cocharacter group by Xb(F) := Im(β⁰).

Then we have a canonical isomorphism (R¹ΦF)_x '(R¹ΦΛ)_x⊗Fx and a natural map H_x¹(X_k, RΨΛ)⊗Fx → H_x¹(X_k, RΨ(F)). These maps give a generalization of the monodromy pairing

λ : X(F)→Xb(F) by composition of the maps

H¹(X_K,F)(1) −−−−→ X(F) −−−−→ L

x∈Σ

(R¹ΦF)x(1)



 y^N



 y^λ



 y

L_N

x⊗1

H¹(X_K,F) ←−−−− Xb(F) ←−−−− L

x∈Σ

H_x¹(X_k, RΨF).

Then the monodromy operator N is described by the Picard–Lefschetz formula:

N(t`(σ)a) = (σ−1)a fora∈H¹(X_K,F) andσ∈I.