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ON THE NON-VANISHING OF HECKE L-VALUES MODULO p

MING-LUN HSIEH

Abstract. In this article, we follow Hida’s approach to establish an analogue of Washington’s theorem on the non-vanishing modulo p of Hecke L-values for CM fields with anticyclotomic twists.

Contents

Introduction 1

1. Notation and definitions 3

2. Hilbert modular varieties and Hilbert modular forms 4

3. CM points 8

4. Construction of the Eisenstein series 10

5. Evaluation of Eisenstein series at CM points 18

6. Non-vanishing of Eisenstein series modulo p 20

References 24

Introduction

The purpose of this paper is to study the non-vanishing modulo p property of Hecke L-values for CM fields via arithmetic of Eisenstein series. Let F be a totally real field of degree d over Q and K be a totally imaginary quadratic extension of F . Let Σ be a CM type of K. Then we can attach the CM period Ω= (Ω∞,σ)σ ∈ (C×)Σ to a Néron differential on an abelian schemeA/Z of CM type (K, Σ). Let p > 2 be a rational prime and let ` 6= p be a rational prime and l be a prime of F above `. Let c be the nontrivial element in Gal(K/F ).

We fix an arithmetic Hecke character χ of K× with infinity type kΣ + κ(1 − c), where k is a positive integer and κ = Σσ∈Σκσσ with integers κσ≥ 0. For a multi-index κ =P

σ∈Σκσσ ∈ Z[Σ], we write Ωκ= Ωκ∞,σσ and aκ= aPσκσ for a ∈ C×.

Let Kln be the ray class field of conductor ln and let Kl = ∪nKln. Let Kl be the maximal pro-`

anticyclotomic extension of K in Kl and let Γ = Gal(Kl/K). Let Xl be the set of finite order characters of Γ. For every ν ∈ Xl , we consider the complex number

Lalg,l(0, χν) := πκΓΣ(kΣ + κ)L(l)(0, χν) ΩkΣ+2κ

, where ΓΣ(kΣ + κ) =Q

σ∈ΣΓ(k + κσ). It is known that Lalg,l(0, χν) ∈ Z(p) if p is uramified in F and prime to the conductor of χ . We are interested in the non-vanishing property of Lalg,l(0, χν) modulo p when ν varies in Xl . To be precise, we fix two embeddings ι: Q ,→ C and ιp: Q ,→ Cp once and for all and let m be the maximal ideal of Z(p) induced by ιp. We ask if the following non-vanishing modulo p property holds for (χ, l).

(NV) ι−1(Lalg,l(0, χν)) 6≡ 0 mod m for almost all ν ∈ Xl .

Here almost all means "except for finitely many ν ∈ Xl " if dimQ`Fl= 1 and "Zariski dense subset of Xl " if dimQ`Fl> 1 (See [Hid04a, p.737]).

This problem has been studied extensively by Hida for general CM fields in [Hid04a] and [Hid07] under the hypothesis that Σ is p-ordinary and by T. Finis in [Fin06] for imaginary quadratic fields under a different

Date: August 12, 2012.

2010 Mathematics Subject Classification. 11F67 11G15.

The author is partially supported by National Science Council grant 98-2115-M-002-017-MY2.

1

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hypothesis. Let τK/F be the quadratic character associated to K/F and DK/F be the different of K/F . Let C be the conductor of χ. The following theorem is proved by Hida in [Hid07].

Theorem. Suppose that Σ is p-ordinary and p > 2 is unramified in F . If (pl, C) = 1 and C is a product of split prime factors over F , then (NV) holds for (χ, l) unless the following three conditions are satisfied simultaneously:

(M1) K/F is unramified everywhere,

(M2) τK/F(c) has value −1, where c is the polarization ideal ofA/Z, (M3) For all ideal a of F prime to pC, χNF /Q(a) ≡ τK/F(a) (mod m).

We shall say χ is residually self-dual if the condition (M3) holds for χ. By [Hid10, Lemma 5.2], the hypotheses (M1-3) is equivalent to the condition (V): χ is residually self-dual, and the root number associated to χ is congruent to −1 modulo m.

We are mainly concerned about the (NV) property of self-dual characters. Recall that χ is self-dual if χ|A×

F = τK/F|·|A

F. Such characters are of its own interest because an important class of them arises from Hecke characters associated to CM abelian varieties over totally real fields (cf. [Shi98, Thm.20.15]). Note that as the conductor of self-dual characters by definition is divisible by ramified primes, these characters in general are not covered in Hida’s theorem unless K/F is unramified. Our main motivation for the (NV) property of self-dual characters is the application to Iwasawa main conjecture for CM fields (cf. [Hid07] and [Hsi11]). In our subsequent work [Hsi11], this property is used to show the non-vanishing modulo p of the period integral of certain theta functions which is related to Fourier-Jacobi coefficients of Eisenstein series on unitary groups of degree three. When K is an imaginary quadratic field and l splits in K, the problem of the non-vanishing modulo p of Hecke L-values associated to self-dual characters has been solved completely by T.

Finis in [Fin06] through direct study on the period integral of theta functions modulo p (self-dual characters are called anticyclotomic in [Fin06]).

We shall state our main result after preparing some notation. Write C = C+IR, where C+, I and R are a product of split, inert and ramified prime factors over F respectively. Let vpbe the p-adic valuation induced by ιp. For each v|C, let µpv) be the local invariant defined by

µpv) := inf

x∈K×v

vp(χ(x) − 1).

Note that µpv) agrees with the one defined in [Fin06] when χ is self-dual. Following Hida, we make the following hypotheses for (p, K, Σ):

p > 2 is unramified in F ; (unr)

Σ is p-ordinary.

(ord)

Our main result is as follows.

Theorem A. Let χ be a self-dual Hecke character of K× such that (L) µpv) = 0 for every v|C,

(R) The global root number W (χ) = 1, where χ:= χ|·|

1 2

AK, (C) R is square-free.

In addition to (unr), (ord), we further assume

• (pl, DK/FC) = 1,

• l splits in K.

Then (NV) holds for (χ, l).

Note that as χ is self-dual, the assumption (R) is equivalent to Hida’s condition (V). Indeed, the assumptions (L) and (R) are necessary for the (NV) property. The assumption (R) is due to the functional equation of the complex L-function L(s, χ), and the failure of (NV) without (L) has been observed by Gillard (cf. [Fin06, Theorem 1.1]). We remark that our result in particular can be applied to Hecke characters attached to certain CM elliptic curves over totally real fields. For example, let E be an elliptic curve over F with CM by an imaginary quadratic field M. Let K = F M and let χ be the Hecke character of K× such that L(s, χ−1) = L(E/F, s). Then it is well known that the assumptions (L) and (C) hold if (DK/F, #(O×M)) = 1

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and p > 3. In general, (C) is expected to be unnecessary. The very reason we impose them is due to the difficulty of the computation of certain Gauss sums Aβ(χ) = Aβs)|s=0 defined in (4.14). We leave the removal of (C) to our forthcoming paper [Hsi14, §6].

We also consider the case χ is not residually self-dual. In particular, this implies the failure of (V). We prove the following result in Corollary 6.5, which gives a partial generalization of Hida’s theorem.

Theorem B. Suppose that (unr), (ord) and (pl, DK/FC) = 1. Suppose further that the following conditions hold:

(L) µpv) = 0 for every v|C, (N) χ is not residually self-dual.

Then (NV) holds for (χ, l).

The proof is based on Hida’s ideas in [Hid04a], where Hida provided a general strategy to study the problem of the non-vanishing of Hecke L-values modulo p via a study on the Fourier coefficients of Eisenstein series.

The starting point of Hida is Damerell’s formula, which relates a sum of suitable Eisenstein series evaluated at CM points to Hecke L-values for CM fields. And then he proves a key result on Zariski density of CM points in Hilbert modular varieties modulo p, by which he is able to reduce the problem to non-vanishing of an Eisenstein series modulo p using a variant of Sinnot’s argument. The assumption that C is a product of split primes solely results from the difficulty of the calculation of Fourier coefficients of Eisenstein series.

Following Hida’s strategy, we first construct an Eisenstein measure which interpolates the Hecke L-values by the evaluation at CM points. The construction of our Eisenstein measure is from representation theoretic point of view, and Damerell’s formula is actually a period integral of Eisenstein series against a non-split torus. Fourier coefficients of our Eisenstein series are decomposed into a product of local Whittaker integrals.

Through an explicit calculation of these local integrals, we find that some Fourier coefficient is non-zero modulo p provided that certain epsilon dichotomy holds (See Proposition 6.7).

Here is the outline of this article. We fix notation and recall some basic facts about Hilbert modular varieties and CM points in the first three sections. We basically follow the exposition in [Hid04a] except that we use an adelic description of CM points. Readers who are familiar with [Hid04a] may begin with §4, which is the bulk of this paper. In §4, we give the construction of Eisenstein series and the calculation of some local Whittaker integrals. The formulas of the key integrals eAβ(χ) are summarized in Proposition 4.4 and Proposition 4.5.

The explicit calculation of the period integral of our Eisenstein series is carried out in §5. Finally we show some Fourier coefficient of our Eisenstein series is non-zero modulo p in §6.

Acknowledgments. The author would like to thank Prof. Hida for helpful email correspondence during prepa- ration of this article. Also the author would like to thank Prof. Sun, Hae-Sang for useful conversation during the stay in Korea Institute of Advanced Study in September 2009. Finally, the author is very grateful to the referee for many valuable suggestions on the improvements of our main results (especially on Lemma 6.4 and Corollary 6.5) in the previous version of this manuscript.

1. Notation and definitions

1.1. Throughout F is a totally real field of degree d over Q and K is a totally imaginary quadratic extension of F . Let c be the complex conjugation, the unique non-trivial element in Gal(K/F ). Let O (resp. R) be the ring of integer of F (resp. K). Let DF (resp. DF) be the different (resp. discriminant) of F /Q. Let DK/F be the different of K/F . For every fractional ideal b of O, set b= b−1D−1F . Denote by a = Hom(F , C) the set of archimedean places of F . Denote by h (resp. hK) the set of finite places of F (resp. K). We often write v for a place of F and w for the place of K above v. Denote by Fv the completion of F at v and by \$v a unifomrmizer of Fv. Let Kv = FvFK.

Fix two rational primes p 6= `. Let l be a prime of F above `. Let Σ be a fixed CM type of K as in the introduction. We shall identify Σ with a by the restriction to F . We assume (unr) and (ord) for (p, K, Σ) throughout this article. Let

Σp= {w ∈ hK| w|p and w is induced by ιp◦ σ for σ ∈ Σ} .

We recall that Σ is p-ordinary if Σp∩ Σpc = ∅ and Σp∪ Σpc = {w ∈ hK| w|p}. Note that (ord) implies that every prime of F above p splits in K.

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1.2. If L is a number field, AL is the adele of L and AL,f is the finite part of AL. The ring of integers of L is denoted by OL. For a ∈ AL, we put

ilL(a) := a(OL⊗ bZ) ∩ L.

Let ψQ be the standard additive character of AQ/Q such that ψQ(x) = exp(2πix), x∈ R. We define ψL: AL/L → C× by ψL(x) = ψQ◦ TL/Q(x). For β ∈ L, ψL,β(x) = ψL(βx). If L = F , we write ψ for ψF.

We choose once and for all an embedding ι : Q ,→ C and an isomorphism ι : C ' Cp, where Cp is the completion of an algebraic closure of Qp. Let ιp = ιι : Q ,→ Cp be their composition. We regard L as a subfield in C (resp. Cp) via ι(resp. ιp) and Hom(L,Q) = Hom(L, Cp).

Let Z be the ring of algebraic integers of Q and let Zpbe the p-adic completion of Z in Cpwith the maximal ideal mp. Let m = ι−1p (mp).

1.3. Let F be a local field. Denote by |·|F the absolute value of F . We often drop the subscript F if it is clear from the context. We fix the choice of our Haar measure dx on F . If F = R, dx is the Lebesgue measure on R. If F = C, dx is the twice the Lebesgue measure. If F is a non-archimedean local field, dx (resp.

d×x) is the Haar measure on F (resp. F×) normalized so that vol(OF, dx) = 1 (resp. vol(OF×, d×x) = 1). If µ : F×→ C× is a character of F×, define

a(µ) = infn ∈ Z≥0| µ|1+\$nvOv = 1 .

2. Hilbert modular varieties and Hilbert modular forms

2.1. We follow the exposition in [Hid04b, §4.2]. Let V = F e1⊕ F e2 be a two dimensional F -vector space and h , i : V ×V → F be the F -bilinear alternating pairing defined by he1, e2i = 1. LetL = Oe1⊕ Oe2 be the standard O-lattice in V . Let G = GL2 /F. We identify vectors in V with row vectors according to the basis e1, e2, so G has a natural right action on V .

For each finite place v of F , we put

Kv0= {g ∈ G(Fv) | (L ⊗OOv)g =L ⊗OOv} . Let K0 = Q

v∈hKv0 and Kp0 = Q

v|pKv0. For a prime-to-p` positive integer N , we define an open-compact subgroup U (N ) of G(AF ,f) by

(2.1) U (N ) := {g ∈ G(AF ,f) | g ≡ 1 (mod NL )} .

Let K be an open-compact subgroup of G(AF ,f) such that Kp= Kp0. We assume that K ⊃ U (N ) for some N as above and that K is sufficiently small so that the following condition holds:

(neat) K is neat and det(K) ∩ O×+ ⊂ (K ∩ O×)2.

2.2. Kottwitz models. We first review Kottwitz models of Hilbert modular varieties.

Definition 2.1 (S-quadruples). Let be a finite set of rational primes and let W()= Z()N], ζ = exp(2πiN ).

Define the fibered category A()K over SCH/W() as follows. Let S be a locally noethoerian connected W()- scheme and let s be a geometric point of S. Objects are abelian varieties with real multiplication (AVRM) over S of level K, i.e. a S-quadruple A = (A, ¯λ, ι, η())S consisting of the following data:

(1) A is an abelian scheme of dimension d over S.

(2) ι : O ,→ EndSA ⊗ZZ().

(3) λ is a prime-to- polarization of A over S and ¯λ is the O(),+-orbit of λ. Namely λ = O¯ (),+λ :=λ0∈ Hom(A, At) ⊗ZZ() | λ0= λ ◦ a, a ∈ O(),+ .

(4) η()= η()K()is a π1(S, s)-invariant K(p)-orbit of isomorphisms of OK-modules η():L ⊗ZA()f V()(As) := H1(As, A()f ). Here we define η()g for g ∈ G(A()F ,f) by η()g(x) = η()(g ∗ x).

Furthermore, (A, ¯λ, ι, η())S satisfies the following conditions:

• Lettdenote the Rosati involution induced by λ on EndSA ⊗ Z(). Then ι(b)t= ι(b), ∀ b ∈ O.

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• Let eλbe the Weil pairing induced by λ. Lifting the isomorphism Z/N Z ' Z/N Z(1) induced by ζN to an isomorphism ζ : bZ ' bZ(1), we can regard eλas an F -alternating form eλ: V()(As) × V()(As) → DF−1ZA()f . Let eη denote the F -alternating form on V()(As) induced by eη(x, x0) = hxη, x0ηi.

Then

eλ= u · eη for some u ∈ A()F ,f.

• As O ⊗ZOS-modules, we have an isomorphism Lie A ' O ⊗ZOS locally under Zariski topology of S.

For two S-quadruples A = (A, ¯λ, ι, η())S and A0= (A0, λ0, ι0, (η0)())S, we define the morphisms by HomA()K (A, A0) =n

φ ∈ HomO(A, A0) | φλ0= ¯λ, φ ◦ (η0)()= η()o .

We say A∼ A0 (resp. A ' A0) if there exists a prime-to- isogeny (resp. isomorphism) in HomA()

K

(A, A0).

We consider the cases when = ∅ and {p}. When  = ∅ is the empty set and W()= Q(ζN), we define the functor EK : SCH/Q(ζN)→ SET S by

EK(S) =A = (A, ¯λ, ι, η)S∈ AK(S) / ∼ .

By the theory of Shimura-Deligne, EK is represented by a quasi-projective scheme ShK over Q(ζN). We define the functor EK : SCH/Q→ SET S by

EK(S) =n

(A, ¯λ, ι, η) ∈ A()K (S) | η()(L ⊗ZZ) = Hb 1(As, bZ)o / ' . By the discussion in [Hid04b, p.136], we have EK

→ E K under the hypothesis (neat).

When = {p}, we write W for W(p) and define functor EK(p): SCH/W → SET S by EK(p)(S) =n

A = (A, ¯λ, ι, η(p))S∈ A(p)K(p)(S)o / ∼ .

In [Kot92], Kottwitz shows EK(p) is representable by a quasi-projective scheme Sh(p)K over W if K is neat.

Similarly we define the functor E(p)K : SCH/W → SET S by E(p)K (S) =n

(A, ¯λ, ι, η(p)) ∈ A(p)K (S) | η(p)(L ⊗ZZb(p)) = H1(As, bZ(p))o / ' . It is shown in [Hid04b, §4.2.1] that E(p)K → E K(p).

2.3. Igusa schemes.

Definition 2.2 (S-quintuples). Let n be a positive integer. We define the fibered category A(p)K,n whose objects are AVRM over an W-scheme of level Kn, i.e. a S-quintuple (A, j)S consisting of a S-quadruple A = (A, ¯λ, ι, η(p)) ∈ A(p)K(p)(S) and a monomorphism

j : O⊗ µpn,→ A[pn]

as O-group schemes over S. We call j a level-pn structure of A. Morphisms are HomA(p)

K,n

((A, j), (A0, j0)) =



φ ∈ HomA(p) K(p)

(A, A0) | φj = j0

 .

Define the functor I(p)K,n: SCH/W → SET S by I(p)K,n(S) =n

(A, j) = (A, ¯λ, ι, η(p), j)S ∈ A(p)K,n(S) | η(p)(L ⊗ZZb(p)) = T(p)(A)o / ' .

It is known that I(p)K,nare relatively representable over E(p)K (cf. [SGA64, Prop. 3.12]), so it is represented by a scheme over W, which we denote by IK,n.

For n ≥ n0> 0, the natural morphism πn,n0 : IK,n→ IK,n0 induced by the inclusion O⊗ µpn0 ,→ O⊗ µpn

is finite étale . The forgetful morphism π : IK,n → Sh(p)K defined by π : (A, j) 7→ A are étale for all n > 0.

Hence IK,n is smooth over Spec W. The image of π is the pre-image of ordinary abelian schemes in IK,n⊗ ¯Fp.

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2.4. Complex uniformization. We describe the complex points ShK(C). Put X+= {τ = (τσ)σ∈a∈ Ca| Im τσ> 0 for all σ ∈ a} .

Let F+ be the set of totally positive elements in F and let G(F )+ = {g ∈ G(F ) | det g ∈ F+}. Define the complex Hilbert modular variety by

M (X+, K) := G(F )+\X+×G(AF ,f)/K.

It is well known that M (X+, K)→ Sh K(C) by the theory of abelian varieties over C.

For τ = (τσ)σ∈a ∈ X+, we let pτ be the period map V ⊗QR→ C a defined by pτ(ae1+ be2) = aτ + b, a, b ∈ F ⊗QR = Ra. We can associate a AVRM to (τ, g) ∈ X+×G(AF ,f) as follows.

• The complex abelian varietyAg(τ ) = Ca/pτ(Lg), whereLg:= (L ⊗ZZ)gb −1∩ V .

• The F+-orbit of polarization h , ican onAg(τ ) is given by the Riemann form h , i ◦ p−1τ .

• The ιC: O ,→ EndAg(τ ) ⊗ZQ is induced from the pull back of the natural F -action on V via pτ.

• The level structure ηg:L ⊗ZAf

→ (g ∗ L ) ⊗ZAf = H1(Ag(τ ), Af) is defined by ηg(v) = vg−1. LetAg(τ ) denote the C-quadruple (Ag(τ ), h , ican, ι, Kηg). Then [(τ, g)] 7→ [Ag(τ )] gives rise to an isomorphism M (X+, K)→ Sh K(C).

Let z = {zσ}σ∈a be the standard complex coordinates of Ca and dz = {dzσ}σ∈a. Then O-action on dz is given by ιC(α)dzσ = σ(α)dzσ, σ ∈ a = Hom(F , C). Let z = zid be the coordinate corresponding to ι: F ,→ Q ,→ C. Then

(2.2) (O ⊗ZC)dz = H0(Ag(τ ), ΩAg(τ )/C).

2.5. Hilbert modular forms.

2.5.1. For τ ∈ C and g =a b c d



∈ GL2(R), we put

(2.3) J (g, τ ) = cτ + d.

For τ = (τσ)σ∈a∈ X+ and g= (gσ)σ∈a∈ G(F ⊗QR), we put J (g, τ ) = Y

σ∈a

J (gσ, τσ).

Definition 2.3. Denote by Mk(K, C) the space of holomorphic Hilbert modular form of parallel weight k and level K. Each f ∈ Mk(K1n, C) is a C-valued function f : X+×G(AF ,f) → C such that the function f (−, gf) : X+ → C is holomorphic for each gf ∈ G(AF ,f) and

f (α(τ, gf)u) = J (α, τ )f (τ, gf) for all u ∈ K1n and α ∈ G(F )+. 2.5.2. Fourier expansion. For every f ∈ Mk(K1n, C), we have the Fourier expansion

f (τ, gf) = X

β∈F+∪{0}

Wβ(f , gf)e2πiTF /Q(βτ ).

We call Wβ(f , gf) the β-th Fourier coefficient of f at gf.

For a semi-group L in F , let L+= F+∩ L and L≥0 = L+∪ {0}. If B is a ring, we denote by BJLK the set of all formal series

X

β∈L

aβqβ, aβ∈ B.

Let a, b ∈ (A(pN )F ,f )× and let a = ilF(a) and b = ilF(b). The q-expansion of f at the cusp (a, b) is given by

(2.4) f |(a,b)(q) = X

β∈(N−1ab)≥0

Wβ(f ,b−1 0

0 a



)qβ∈ CJ(N

−1ab)≥0K.

If B is a W-algebra in C, we put

Mk(K, B) =f ∈ Mk(K, C) | f |(a,b)(q) ∈ BJ(N

−1ab)≥0K at all cusps (a, b) .

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2.5.3. Tate objects. Let S be a set of d-linear Q-independent elements in Hom(F, Q) such that l(F+) > 0 for l ∈ S . If L is a lattice in F and n a positive integer, let LS ,n = {x ∈ L | l(x) > −n for all l ∈S } and put B((L;S )) = lim

n→∞BJLS ,nK. To a pair (a, b) of two prime-to-pN fractional ideals , we can attach the Tate AVRM T atea,b(q) = aZ Gm/qb over Z((ab;S )) with O-action ιcan. As described in [Kat78], T atea,b(q) has a canonical ab−1-polarization λcan and also carries ωcana canonical O ⊗ Z((ab;S ))-generator of ΩT atea,b induced by the isomorphism Lie(T atea,b(q)/Z((ab;S ))) = aZLie(Gm) ' a⊗ Z((ab;S )). Let La,b=L ·b

a−1



= be1⊕ ae2. Then we have a level N -structure ηcan : N−1La,b/La,b

→ T ate a,b(q)[N ] over Z[ζN]((N−1ab;S )) induced by the fixed primitive N-th root of unity ζN. We write T atea,b for the Tate Z((ab;S ))-quadruple (T atea,b(q), λcan, ιcan, η(p)can) at (a, b). In addition, since a is prime to p, we let ηp,can0 : OZµpn = aZ µpn ,→ T atea,b(q) be the canonical level pn-structure induced by the natural inclusion aZµpn ,→ aZGm.

2.5.4. Geometric modular forms. We collect here definitions and basic facts of geometric modular forms. For the precise theory, we refer to [Kat78] or [Hid04b]. Let T = ResO/ZGm and κ ∈ Hom(T, Gm). Let B be a Z(p)-algebra. Consider [A] = [(A, λ, ι, η(p))] ∈ EK(C) for a B-algebra C with a differential form ω generating H0(A, ΩA/C) over O ⊗ZC. A geometric modular form f over B of weight κ and level K is a functorial rule of assigning a value f (A, ω) ∈ C satisfying the following axioms.

(G1) f (A, ω) = f (A0, ω0) ∈ C if (A, ω) ' (A0, ω0) over C, (G2) For a B-algebra homomorphism ϕ : C → C0, we have

f ((A, ω) ⊗CC0) = ϕ(f (A, ω)), (G3) f (A, aω) = κ(a−1)f (A, ω) for all a ∈ T (C) = (O ⊗ZC)×, (G4) f (T atea,b, ωcan) ∈ B[ζN]J(N

−1ab)≥0K at all cusps (a, b).

For a positive integer k, we regard k ∈ Hom(T, Gm) as the character t 7→ NF /Q(t)k. We denote by Mk(K, B) the space of geometric modular forms over B of weight k and level K.

For each f ∈ Mk(K, C), we regard f as a holomorphic Hilbert modular form of weight k and level K by f (τ, gf) = f (Ag(τ ), h , ican, ιC, ηg, 2πidz),

where dz is the differential form in (2.2). By GAGA principle, this gives rise to an isomorphism Mk(K, C)→ Mk(K, C). As discussed in [Kat78, §1.7], the evaluation f (T atea,b, ωcan) is independent of the auxiliary choice ofS in the construction of the Tate object. Moreover, we have the following important identity which bridges holomorphic modular forms and geometric modular forms.

f |(a,b)(q) = f (T atea,b, ωcan) ∈ CJ(N

−1ab)≥0K.

By q-expansion principle, if B is W-algebra in C, then Mk(K, B) = Mk(K, B).

2.5.5. p-adic modular forms. Let B be a p-adic ring in Cp. Let V (K, B) be the space of Katz p-adic modular forms over B defined by

V (K, B) := lim

←−m

lim−→

n

H0(IK,n/B/pmB, OIK,n).

In other words, Katz p-adic modular forms are formal functions on Igusa towers.

Let C be a B/pmB-algebra. For each C-point [(A, j)] ∈ lim

−→mlim

←−nIK,n(C), the level p-structure j induces an isomorphism j: OZLie( bGm/C) = OZC→ Lie(A). Let dt/t be the canonical invariant differential form of bGm. Then jdt/t := dt/t ◦ jis a generator of H0(A, ΩA) as a O ⊗ZC-module. We thus have a natural injection

(2.5)

Mk(K, B) ,→ V (K, B)

f 7→ bf (A, j) := f (A, jdt/t)

which preserves the q-expansions in the sense that bf |(a,b)(q) := bf (T atea,b, η0p,can) = f |(a,b)(q). We will call bf the p-adic avatar of f .

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2.6. Hecke action. Let h ∈ G(A(p)F ,f) and let hK := hKh−1. We define a morphism |h : E(p)

hK

→ E K(p) by A = (A, ¯λ, ι, η(p)) 7→ A |h = (A, ¯λ, ι, hη(p)).

Then |h induces an W-isomorphism Sh(p)K → Sh (p)K

h, and |h thus acts on spaces of modular forms. In particular, for F ∈ V (K, W), we define F |h ∈ V (hK, W) by

F |h(A) = F (A |h).

Let K0(l) := {g ∈ K | e2g ∈ Oe2(mod lL )}. Define the Ul-operator on V (K0(l), W) by F |Ul= X

u∈O/lO

F |\$l u

0 1

 .

Using the description of complex points of Sh(p)K (C) in §2.4, it is not difficult to verify by definition that for (τ, g) ∈ X+×G(AF ,f) two pairs (Ag(τ ) |h, ω) and (Agh(τ ), ω) of C-quadruples and invariant differential forms are Z(p)-isogenous, so we have the isomorphism:

(2.6) Mk(K, C)→ M k(hK, C)

f 7→ f |h(τ, g) = f (τ, gh).

3. CM points

3.1. In this section, we give an adelic description of CM points in Hilbert modular varieties. Fix a prime-to-p integral ideal C of R such that (pl, CDK/F) = 1. Write C = C+C, where C= IR, I (resp. R) is a product of inert (resp. ramified) primes in K/F and C+ = FFc is a product of split primes in K/F such that (F, Fc) = 1 and F ⊂ Fcc. Recall that we have assumed (unr) and (ord) in the introduction. Let Σ be a p-ordinary CM type of K and identify Σ with a by the restriction to F . We choose ϑ ∈ K such that

(d1) ϑc= −ϑ and Im σ(ϑ) > 0 for all σ ∈ Σ, (d2) c(R) := DF−1(2ϑDK/F−1 ) is prime to pDK/FlCCc.

Let ϑΣ := (σ(ϑ))σ∈Σ∈ X+. Let D = −ϑ2∈ F+ and define ρ : K ,→ M2(F ) by ρ(aϑ + b) = b −Da

a b

 .

Consider the isomorphism qϑ : K→ F 2 = V defined by qϑ(aϑ + b) = ae1+ be2. It is clear that (0, 1)ρ(α) = qϑ(α) and qϑ(xα) = qϑ(x)ρ(α) for α, x ∈ K. Let C(Σ) be the K-module whose underlying space is CΣ with the K-action given α(xσ) = (σ(α)xσ). Then we have a canonical isomorphism K ⊗QR = C(Σ), and pϑ:= qϑ−1: V ⊗QR→ K ⊗ QR = C(Σ) is the period map associated to ϑΣ.

3.2. A good level structure.

3.2.1. For each v|pFFc, we decompose v = ww into two places w and w of K with w|FΣp. Here w|FΣpmeans w|F or w ∈ Σp. Let ew (resp. ew) be the idempotent associated to w (resp. w). Then {ew, ew} gives an Ov-basis of Rv. Let ϑw∈ Fv such that ϑ = −ϑwew+ ϑwew.

For inert or ramified place v and w the place of K above v, we fix a Ov-basis {1, θv} such that θv is a uniformizer if v is ramified and θ = −θ if v - 2. Let δv:= θv− θv be a fixed generator of the relative different DKw/Fv.

Fix a finite idele dF = (dFv) ∈ AF ,f such that ilF(dF) = DF. By (d2), we may choose dFv = 2ϑδv−1 if v|DK/FI (resp. dFv = −2ϑwif w|FΣp).

3.2.2. We shall choose a basis {e1,v, e2,v} of R ⊗O Ov for each finite place v 6= l of F . If v - plCCc, we choose {e1,v, e2,v} in R ⊗ Ov such that R ⊗O Ov = Ove1,v ⊕ Ove2,v. It is clear that {e1,v, e2,v} can be taken to be {ϑ, 1} except for finitely many v. If v|pFFc, let {e1,v, e2,v} = {ew, dFv· ew} with w|FΣp. If v is inert or ramified, let {e1,v, e2,v} = {θv, dFv · 1}. For every integer n ≥ 0, we let Rn = O + lnR, and let n

e(n)1,l, e(n)2,lo

:= {−1, −dFl\$lnθl} be a basis of RnOOl.

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For v ∈ h, let ςv (resp. ςl(n)) be the element in GL2(Fv) such that eiςv−1 = qϑ(ei,v) (resp. eil(n))−1 = qϑ(e(n)i,l )). For v = σ ∈ a, let ςv =Im σ(ϑ) 0

0 1



. Define ς(l) =Q

v6=lςv ∈ GL2(A(l)F) and ς(n) = ς(l)×ςl(n) ∈ GL2(AF). Let ςf and ςf(n) be the finite components of ς and ς(n) respectively. By the definition of ς(n), we have

(L ⊗ZZ) · (ςb f(n))−1 = qϑ(l−nRnZZ).b

The matrix representation of ςv according to the basis {e1, e2} for v|plDK/FCCc is given as follows:

(3.1)

ςv=dFv −2−1tv

0 d−1F

v



, tv= θv+ θv if v|DK/FI,

ςv=

" d

Fv

212

dFv

−2ϑw

−1 w

#

=−ϑw12 1 −1

w



if v|pFFc and w|FΣp,

ςl(n)=−bl 1 al 0

 dFl\$ln 0

0 1



l= alϑ + bl, al∈ O×l , bl∈ Ol).

3.3. For every a ∈ A×K,f, we let

An(a)/C:=Aρ(a)ς(n)Σ) = (Aρ(g)ς(n)Σ), h , ican, ιcan, η(a)) ∈ ShK(C)

be the C-quadruple associated to (ϑΣ, ρ(a)ςf(n)) as in §2.4. Then An(a)/C is an abelian variety with CM by K. Let W be the p-adic completion of the maximal unramified extension of Zp in Cp. By the general theory of CM abelian varieties, the C-quadruple An(a)/C descends to a W -quadruple An(a). Moreover, since K is p-ordinary, An(a) ⊗Wpis an ordinary abelian variety, hence the level p-structure η(a)p over C descends to a level p-structure over W . Thus we obtain a map xn : A×K,f → lim←−mIK,m(W ) ⊂ IK,∞(W ), which factors through CK:= A×K,f/K×the idele class group of K. The collection of points Cl:= tn=1xn(CK) in IK,∞(W ) is called CM points in Hilbert modular varieties.

3.4. Polarization ideal. The alternating pairing h , i : K×K :→ F defined by hx, yi = (c(x)y − xc(y))/2ϑ induces an isomorphism R ∧OR = c(R)−1DF−1 for the fractional ideal c(R) = DF−1(2ϑDK/F−1 ). Then c(R) is the polarization of CM points x0(1). From the equation

D−1F det(ςf) = ∧2L ςf−1= ∧2R = c(R)−1D−1F ,

we find that c(R) = (det(ςf)). Moreover, for a ∈ A×K, the polarization ideal of x0(a) is c(a) := c(R)NK/F(a), a= ilK(a).

3.5. Measures associated to Ul-eigenforms.

3.5.1. We briefly recall Hida’s construction of the measure associated to an Ul-eigenform in [Hid04a, §3].

Define the compact subgroup Un = (C1)Σ×(Rn⊗ bZ)× in A×K = (C×)Σ×A×K,f, where C1 is the unit circle in C×. Let Cln = K×A×F\A×K/Un and let [·]n : A×K → Cln be the quotient map. Let Cl = lim

←−nCln. For a ∈ A×K, we let [a] := lim←−n[a]n ∈ Cl be the holomorphic image in Cl. Henceforth, every ν ∈ Xl will be regarded implicitly as a p-adic character of Cl by geometrically normalized reciprocity law.

LetE ∈ V (K0(l),O) for some finite extension O of Zp and let χ be the p-adic avatar of χ. Assuming theb following:

(i) E is a Ul-eigenform with the eigenvalue al(E) ∈ Z×p; (ii) E(xn(ta)) =χb−1(a)E(xn(t)), a ∈ Un· A×F,

Hida in [Hid04a, (3.9)] associates a Zp-valued measure ϕE on Cl to the Ul-eigenform E such that for a function φ : Cln→ Zp, we have

(3.2)

Z

Cl

φdϕE:= al(E)−n· X

[t]n∈Cln

E(xn(t))χ(t)φ([t]b n).

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3.5.2. Let ∆ be the torsion subgroup of Cl. Let Clalg be the subgroup of Cl generated by [a] for a ∈ (A(l)K)× and ∆alg = Clalg∩ ∆. We choose a set of representatives B = {b} of ∆/∆alg in ∆ and a set of representatives R = {r} of ∆alg in (A(pl)K,f)×. Thus ∆ = B[R] = {b[r]}b∈B,r∈R. For a ∈ (A(pl)K,f)×, we define

E|[a] := E|ρς(a), ρς(a) := ς−1ρ(a)ς ∈ G(A(pl)F ,f).

By definition,E|[a](xn(t)) =E(xn(ta)). Following Hida (cf. [Hid07, (4.4) p.25]), we put

(3.3) ER =X

r∈R

χ(r)b E|[r].

In [Hid04a], Hida reduces the non-vanishing of L-values to the non-vanishing of Eisenstein series by proving the following theorem.

Theorem 3.1 (Theorem 3.2 and Theorem 3.3 [Hid04a]). Suppose the following conditions in addition to (unr) and (ord):

(H) Write the order of the Sylow `-subgroup of F[χ]× as `r(χ). Then there exists a strict ideal class c ∈ ClF such that c = c(a) for some R-ideal a and for every u ∈ O prime to l, we can find β ≡ u mod lr(χ)with aβ(ER, c) 6≡ 0 (mod mp),

where aβ(ER, c) is the β-th Fourier coefficient of ER at the cusp (O, c−1). Then Z

Cl

νdE 6≡ 0 (mod mp) for almost all ν ∈ Xl .

Remark. As pointed by the referee, if l has degree one over Q, the above theorem is Theorem 3.2 [Hid04a]. In general, the theorem holds under the assumption (h) in Theorem 3.3 loc.cit. , which is slightly weaker than (H) (See the discussion [Hid04a, p.778]).

4. Construction of the Eisenstein series

4.1. Let χ be a Hecke character of K× with infinity type kΣ + κ(1 − c), where k ≥ 1 is an integer and κ =P κσσ ∈ Z[Σ], κσ ≥ 0. Let c(χ) be the conductor of χ. We assume that C = c(χ)S, where S is only divisible by primes split in K/F and (c(χ)l, S) = 1. Put

χ= χ|·|A12

K and χ+= χ|A× F. Let K0 :=Q

v∈aSO(2, R) be a maximal compact subgroup of G(F ⊗QR). For s ∈ C, we let I(s, χ+) denote the space consisting of smooth and K0 -finite functions φ : G(AF) → C such that

φ(a b 0 d



g) = χ−1+ (d) a d

s AF

φ(g).

Conventionally, functions in I(s, χ+) are called sections. Let B be the upper triangular subgroup of G. The adelic Eisenstein series associated to a section φ ∈ I(s, χ+) is defined by

EA(g, φ) = X

γ∈B(F )\G(F )

φ(γg).

The series EA(g, φ) is absolutely convergent for Re s  0.

4.2. Fourier coefficients of Eisenstein series. Put w =0 −1

1 0



. Let v be a place of F and let Iv(s, χ+) be the local constitute of I(s, χ+) at v. For φv∈ Iv(s, χ+) and β ∈ Fv, we recall that the β-th local Whittaker integral Wβv, gv) is defined by

Wβv, gv) = Z

Fv

φv(w1 xv

0 1



gv)ψ(−βxv)dxv, and the intertwining operator Mw is defined by

Mwφv(gv) = Z

Fv

φv(w1 xv

0 1

 gv)dxv.

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By definition, Mwφv(gv) is the 0-th local Whittaker integral. It is well known that local Whittaker integrals converge absolutely for Re s  0, and have meromorphic continuation to all s ∈ C.

If φ = ⊗vφv is a decomposable section, then it is well known that EA(g, φ) has the following Fourier expansion:

(4.1)

EA(g, φ) = φ(g) + Mwφ(g) +X

β∈F

Wβ(EA, g), where

Mwφ(g) = 1

p|DF|R ·Y

v

Mwφv(gv) ; Wβ(EA, g) = 1

p|DF|R ·Y

v

Wβv, gv).

The sum φ(g) + Mwφ(g) is called the constant term of EA(g, φ).

4.3. The choice of local sections and Fourier coefficients. In this subsection, we will choose for each place v a good local section φχ,s,v in Iv(s, χ+) and calculate its local β-th Fourier coefficient for β ∈ Fv×. 4.3.1. We first introduce some notation and definitions. Let S =v ∈ h | v - lCCcDK/F . Let v be a place of F . Let L/Fv be a finite extension and let dL be a generator of the absolute different DL of L. Let ψL := ψ ◦ TL/Fv. Given a character µ : L× → C, we recall that the epsilon factor (s, µ, ψL) in [Tat79] is defined by

(s, µ, ψL) = |c|sL Z

c−1O×L

µ−1(x)ψL(x)dLx (c = dL\$La(µ)).

Here dLx is the Haar measure on L self-dual with respect to ψL. If ϕ is a Bruhat-Schwartz function on L, the zeta integral Z(s, µ, ϕ) is given by

Z(s, µ, ϕ) = Z

L

ϕ(x)µ(x) |x|sLd×x (s ∈ C).

The local root number W (µ) is defined by

W (µ) := (1 2, µ, ψL)

(cf. [MS00, p.281 (3.8)]). It is well known that |W (µ)|C= 1 if µ is unitary.

To simplify the notation, we let F = Fv (resp. E = K ⊗FFv) and let dF = dFv be the fixed generator of the absolute different DF in §3.2.1. Write χ (resp. χ+, χ) for χv (resp. χ+,v, χv). If v ∈ h, we let Ov= OF

(resp. Rv = R ⊗OOv) and let \$ = \$v be a uniformizer of F . For a set Y , denote by IY the characteristic function of Y .

4.3.2. v is archimedean. Let v = σ ∈ Σ and F = R. For g ∈ G(F ) = GL2(R), we put δ(g) = |det(g)| ·

J (g, i)J (g, i)

−1

. Define the section φhk,s,σ∈ Iv(s, χ+) of weight k by

(4.2) φhk,s,σ(g) := J (g, i)−kδ(g)s. The intertwining operator Mwφk,s,σ is given by

(4.3) Mwφhk,s,σ(g) = ik(2π)Γ(k + 2s − 1)

Γ(k + s)Γ(s) · J (g, i)kdet(g)−kδ(g)1−s. For (x, y) ∈ R×R+ and β ∈ R×, it is well known that

(4.4) Wβhk,s,σ,y x 0 1



)|s=0= (2πi)k

Γ(k) σ(β)k−1exp(2πiσ(β)(x + iy)) · IR+(σ(β)).

Define the section φn.h.k,κσ,s,σ∈ I(s, χ+) of weight k + 2κσ by

(4.5) φn.h.k,κσ,s,σ(g) := J (g, i)−k−κσJ (g, i)κσδ(g)s. Let V+ be the weight raising differential operator in [JL70, p.165] given by

V+ =1 0 0 −1



⊗ 1 +0 1 1 0



⊗ i ∈ Lie(GL2(R)) ⊗RC.

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Denote by V+κσ the operator (V+)κσ acting on Iv(s, χ+). By [JL70, Lemma 5.6 (iii)], we have (4.6) V+κσσφhk,s,σ= 2κσΓ(k + κσ+ 2s)

Γ(k + 2s) φn.h.k,κ

σ,s,σ.

4.3.3. v ∈ S. In this case, χ is unramified. Define φχ,s,v(g) to be the spherical Godement section in Iv(s, χ+).

To be precise, put

φχ,s,v(g) =fΦv(g) := |det g|s Z

F×

Φv((0, t)g)χ+(t) |t|2sd×t, where Φv = IOv⊕Ov.

It is well known that the local Whittaker integral is (4.7) Wβχ,s,v,cv

1



)|s=0=1 − χ(\$)v(βcv)+1

1 − χ(\$) · |DF|−1· IOv(βcv), and the intertwining operator is given by

(4.8) Mwφχ,s,v(cv

1



) = Lv(2s − 1, χ+) |cv|1−sχ−1+ (cv).

4.3.4. v|FFc. If v|FFc is split in K, write v = ww with w|F and χv = (χw, χw). Then a(χw) ≥ a(χw). We shall define our local section at v to be the Godement section associated to certain Bruhat-Schwartz functions.

We first introduce some Bruhat-Schwartz functions. For a character µ : F×→ C×, we define ϕµ(x) = IOv×(x)µ(x) (x ∈ F ).

Define ϕw= ϕχw and

ϕw= (ϕχ−1

w if χwis ramified, IOv if χwis unramified.

Let Φv(x, y) = ϕw(x)ϕbw(y), whereϕbwis the Fourier transform of ϕw defined by

ϕbw(y) = Z

F

ϕw(x)ψ(yx)dx.

Define φχ,s,v∈ Iv(s, χ+) by

(4.9) φχ,s,v(g) = fΦv(g) := |det g|s Z

F×

Φv((0, t)g)χ+(t) |t|2sd×t.

A straightforward calculation shows that the local Whittaker integral is

(4.10)

Wβχ,s,v, 1) = Z

F×

ϕw(x)ϕbw(−βx−1) · χ+(x) |x|2s−1d×x

= Z

F×

ϕw(x)ϕw(βx−1) · |DF|−1· χ+(x) |x|2s−1d×x

= χ+(β)ϕw(β) |β|2s−1· |DF|−1, and the intertwining operator is given by

(4.11) Mwφχ,s,v(1) = 0.

4.3.5. v = l. Let φχ,v,s∈ Iv(s, χ+) be the unique N (Ov)-invariant section supported in the big cell B(F )wN (Ov) and φχ,v,s(w) = 1. One checks easily that φχ,s,v|Ul given by

φχ,s,v|Ul(g) = X

u∈Ov/lOv

φχ,s,v(g\$ u 0 1

 )

is also supported in the big cell and is invariant by N (Ov). In particular, φχ,s,v is an Ul-eigenform, and the eigenvalue is χ−1+ (\$l).

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