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 Ω^{κ}_{∞}= Ω^{κ}_{∞,σ}^{σ} and
a^{κ}= a^{P}^{σ}^{κ}^{σ} for a ∈ C^{×}.

Let K_{l}n be the ray class field of conductor l^{n} and let K_{l}^{∞} = ∪_{n}K_{l}n. Let K^{−}_{l}∞ be the maximal pro-`

anticyclotomic extension of K in K_{l}^{∞} and let Γ^{−} = Gal(K^{−}_{l}∞/K). Let X^{−}_{l} be the set of finite order characters
of Γ^{−}. For every ν ∈ X^{−}_{l} , we consider the complex number

L^{alg,l}(0, χν) := π^{κ}ΓΣ(kΣ + κ)L^{(l)}(0, χν)
Ω^{kΣ+2κ}∞

, where ΓΣ(kΣ + κ) =Q

σ∈ΣΓ(k + κσ). It is known that L^{alg,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 L^{alg,l}(0, χν) modulo p when ν varies
in X^{−}_{l} . 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}_{∞}(L^{alg,l}(0, χν)) 6≡ 0 mod m for almost all ν ∈ X^{−}_{l} .

Here almost all means "except for finitely many ν ∈ X^{−}_{l} " if dim_{Q}_{`}F_{l}= 1 and "Zariski dense subset of X^{−}_{l} " 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

hypothesis. Let τ_{K/F} be the quadratic character associated to K/F and D_{K/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) τ_{K/F}(c) has value −1, where c is the polarization ideal ofA_{/Z},
(M3) For all ideal a of F prime to pC, χN_{F /Q}(a) ≡ τ_{K/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 = τ_{K/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 v_{p}be the p-adic valuation induced
by ιp. For each v|C^{−}, let µp(χv) be the local invariant defined by

µp(χv) := inf

x∈K^{×}_{v}

vp(χ(x) − 1).

Note that µp(χv) 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) µp(χv) = 0 for every v|C^{−},

(R) The global root number W (χ^{∗}) = 1, where χ^{∗}:= χ|·|^{−}

1 2

A_{K},
(C) R is square-free.

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

• (pl, D_{K/F}C) = 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 (D_{K/F}, #(O^{×}_{M})) = 1

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) = Aβ(χs)|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, D_{K/F}C) = 1. Suppose further that the following conditions
hold:

(L) µp(χv) = 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 eA_{β}(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 D_{F} (resp. D_{F}) be the different (resp. discriminant) of F /Q. Let D_{K/F} be
the different of K/F . For every fractional ideal b of O, set b^{∗}= b^{−1}D^{−1}_{F} . Denote by a = Hom(F , C) the set
of archimedean places of F . Denote by h (resp. h_{K}) 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 F_{v} the completion of F at v and by $v a
unifomrmizer of Fv. Let Kv = Fv⊗_{F}K.

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

Σ_{p}= {w ∈ h_{K}| w|p and w is induced by ι_{p}◦ σ for σ ∈ Σ} .

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

1.2. If L is a number field, A_{L} is the adele of L and A_{L,f} is the finite part of A_{L}. 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◦ T_{L/Q}(x). For β ∈ L, ψL,β(x) = ψL(βx). If L = F , we write ψ for ψ_{F}.

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 Q_{p}. Let ι_{p} = ιι_{∞} : Q ,→ C_{p} be their composition. We regard L as a
subfield in C (resp. C_{p}) via ι_{∞}(resp. ι_{p}) and Hom(L,Q) = Hom(L, C_{p}).

Let Z be the ring of algebraic integers of Q and let Z_{p}be the p-adic completion of Z in C_{p}with the maximal
ideal mp. Let m = ι^{−1}_{p} (m_{p}).

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(O_{F}^{×}, d^{×}x) = 1). If
µ : F^{×}→ C^{×} is a character of F^{×}, define

a(µ) = infn ∈ Z_{≥0}| µ|1+$^{n}_{v}O_{v} = 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⊕ O^{∗}e2 be
the standard O-lattice in V . Let G = GL_{2 /F}. We identify vectors in V with row vectors according to the
basis e_{1}, e_{2}, so G has a natural right action on V .

For each finite place v of F , we put

K_{v}^{0}= {g ∈ G(F_{v}) | (L ⊗OO_{v})g =L ⊗OO_{v}} .
Let K^{0} = Q

v∈hK_{v}^{0} and K_{p}^{0} = Q

v|pK_{v}^{0}. For a prime-to-p` positive integer N , we define an open-compact
subgroup U (N ) of G(A_{F ,f}) by

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

Let K be an open-compact subgroup of G(AF ,f) such that Kp= K_{p}^{0}. 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_{()}[ζN], ζ = exp(^{2πi}_{N} ).

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 ,→ End_{S}A ⊗_{Z}Z_{()}.

(3) λ is a prime-to- polarization of A over S and ¯λ is the O_{(),+}-orbit of λ. Namely
λ = O¯ _{(),+}λ :=λ^{0}∈ Hom(A, A^{t}) ⊗ZZ_{()} | λ^{0}= 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:

• Let^{t}denote the Rosati involution induced by λ on End_{S}A ⊗ Z_{()}. Then ι(b)^{t}= ι(b), ∀ b ∈ O.

• Let e^{λ}be the Weil pairing induced by λ. Lifting the isomorphism Z/N Z ' Z/N Z(1) induced by ζ_{N} to
an isomorphism ζ : bZ ' bZ(1), we can regard e^{λ}as an F -alternating form e^{λ}: V^{()}(As) × V^{()}(As) →
D_{F}^{−1}⊗ZA^{()}_{f} . Let e^{η} denote the F -alternating form on V^{()}(As) induced by e^{η}(x, x^{0}) = hxη, x^{0}ηi.

Then

e^{λ}= u · e^{η} for some u ∈ A^{()}_{F ,f}.

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

For two S-quadruples A = (A, ¯λ, ι, η^{()})_{S} and A^{0}= (A^{0}, λ^{0}, ι^{0}, (η^{0})^{()})_{S}, we define the morphisms by
HomA^{()}_{K} (A, A^{0}) =n

φ ∈ Hom_{O}(A, A^{0}) | φ^{∗}λ^{0}= ¯λ, φ ◦ (η^{0})^{()}= η^{()}o
.

We say A∼ A^{0} (resp. A ' A^{0}) if there exists a prime-to- isogeny (resp. isomorphism) in Hom_{A}^{()}

K

(A, A^{0}).

We consider the cases when = ∅ and {p}. When = ∅ is the empty set and W()= Q(ζ_{N}), we define
the functor EK : SCH_{/Q(ζ}_{N}_{)}→ SET S by

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

By the theory of Shimura-Deligne, E_{K} is represented by a quasi-projective scheme Sh_{K} over Q(ζ_{N}). We define
the functor EK : SCH/Q→ SET S by

E_{K}(S) =n

(A, ¯λ, ι, η) ∈ A^{()}_{K} (S) | η^{()}(L ⊗ZZ) = Hb _{1}(A_{s}, 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 E_{K}^{(p)}: SCH_{/W} → SET S by
E_{K}^{(p)}(S) =n

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

In [Kot92], Kottwitz shows E_{K}^{(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 K^{n}, 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^{∗}⊗ µ_{p}n,→ A[p^{n}]

as O-group schemes over S. We call j a level-p^{n} structure of A. Morphisms are
Hom_{A}(p)

K,n

((A, j), (A^{0}, j^{0})) =

φ ∈ Hom_{A}(p)
K(p)

(A, A^{0}) | φj = j^{0}

.

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,n}are relatively representable over E^{(p)}_{K} (cf. [SGA64, Prop. 3.12]), so it is represented by
a scheme over W, which we denote by I_{K,n}.

For n ≥ n^{0}> 0, the natural morphism πn,n^{0} : IK,n→ IK,n^{0} induced by the inclusion O^{∗}⊗ µ_{p}n0 ,→ O^{∗}⊗ µ_{p}n

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 I_{K,n} is smooth over Spec W. The image of π is the pre-image of ordinary abelian schemes in I_{K,n}⊗ ¯Fp.

2.4. Complex uniformization. We describe the complex points Sh_{K}(C). Put
X^{+}= {τ = (τσ)σ∈a∈ C^{a}| 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(A_{F ,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 = R^{a}. We can associate a AVRM to (τ, g) ∈ X^{+}×G(A_{F ,f}) as follows.

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

• The F+-orbit of polarization h , i_{can} onAg(τ ) is given by the Riemann form h , i ◦ p^{−1}_{τ} .

• The ι_{C}: O ,→ EndAg(τ ) ⊗_{Z}Q is induced from the pull back of the natural F -action on V via p_{τ}.

• 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 , i_{can}, 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 C^{a} 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 ⊗_{Z}C)dz = H^{0}(Ag(τ ), Ω_{A}_{g}_{(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 ∈ M_{k}(K_{1}^{n}, C) is a C-valued function f : X^{+}×G(A_{F ,f}) → C such that the function
f (−, gf) : X^{+} → C is holomorphic for each gf ∈ G(AF ,f) and

f (α(τ, g_{f})u) = J (α, τ )^{kΣ}f (τ, g_{f}) for all u ∈ K_{1}^{n} and α ∈ G(F )^{+}.
2.5.2. Fourier expansion. For every f ∈ M_{k}(K_{1}^{n}, C), we have the Fourier expansion

f (τ, gf) = X

β∈F+∪{0}

Wβ(f , gf)e^{2πiT}^{F /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

aβq^{β}, aβ∈ B.

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

(2.4) f |_{(a,b)}(q) = X

β∈(N^{−1}ab)_{≥0}

W_{β}(f ,b^{−1} 0

0 a

)q^{β}∈ CJ(N

−1ab)_{≥0}K.

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

M_{k}(K, B) =f ∈ Mk(K, C) | f |_{(a,b)}(q) ∈ BJ(N

−1ab)_{≥0}K at all cusps (a, b) .

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) = a^{∗}⊗Z Gm/q^{b} 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 ate_{a,b} induced by the isomorphism Lie(T atea,b(q)_{/Z((ab;}_{S ))}) = a^{∗}⊗ZLie(G^{m}) ' a^{∗}⊗ Z((ab;S )). Let
La,b=L ·b

a^{−1}

= be1⊕ a^{∗}e2. Then we have a level N -structure ηcan : N^{−1}La,b/La,b

→ T ate∼ a,b(q)[N ]
over Z[ζN]((N^{−1}ab;S )) induced by the fixed primitive N-th root of unity ζN. We write T ate_{a,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
η_{p,can}^{0} : O^{∗}⊗_{Z}µ_{p}n = a^{∗} ⊗_{Z} µ_{p}n ,→ T ate_{a,b}(q) be the canonical level p^{n}-structure induced by the natural
inclusion a^{∗}⊗Zµ_{p}n ,→ a^{∗}⊗ZGm.

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, G^{m}). 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
H^{0}(A, Ω_{A/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 (A^{0}, ω^{0}) ∈ C if (A, ω) ' (A^{0}, ω^{0}) over C,
(G2) For a B-algebra homomorphism ϕ : C → C^{0}, we have

f ((A, ω) ⊗CC^{0}) = ϕ(f (A, ω)),
(G3) f (A, aω) = κ(a^{−1})f (A, ω) for all a ∈ T (C) = (O ⊗_{Z}C)^{×},
(G4) f (T ate_{a,b}, ωcan) ∈ B[ζN]J(N

−1ab)≥0K at all cusps (a, b).

For a positive integer k, we regard k ∈ Hom(T, G^{m}) as the character t 7→ N_{F /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, g_{f}) = f (Ag(τ ), h , i_{can}, ι_{C}, η_{g}, 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 ate_{a,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 ate_{a,b}, ωcan) ∈ CJ(N

−1ab)_{≥0}K.

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

H^{0}(I_{K,n/B/p}mB, OI_{K,n}).

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

Let C be a B/p^{m}B-algebra. For each C-point [(A, j)] ∈ lim

−→^{m}lim

←−^{n}IK,n(C), the level p^{∞}-structure j induces
an isomorphism j_{∗}: O^{∗}⊗ZLie( bGm/C) = O^{∗}⊗ZC→ Lie(A). Let dt/t be the canonical invariant differential^{∼}
form of bGm. Then j^{∗}dt/t := dt/t ◦ j_{∗}is a generator of H^{0}(A, Ω_{A}) as a O ⊗_{Z}C-module. We thus have a natural
injection

(2.5)

M_{k}(K, B) ,→ V (K, B)

f 7→ bf (A, j) := f (A, j_{∗}dt/t)

which preserves the q-expansions in the sense that bf |_{(a,b)}(q) := bf (T ate_{a,b}, η^{0}_{p,can}) = f |_{(a,b)}(q). We will call bf
the p-adic avatar of f .

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 K_{0}(l) := {g ∈ K | e_{2}g ∈ O^{∗}e_{2}(mod lL )}. Define the Ul-operator on V (K_{0}(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(A_{F ,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, CD_{K/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 ⊂ F^{c}_{c}. 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) ϑ^{c}= −ϑ and Im σ(ϑ) > 0 for all σ ∈ Σ,
(d2) c(R) := D_{F}^{−1}(2ϑD_{K/F}^{−1} ) is prime to pD_{K/F}lCC^{c}.

Let ϑ^{Σ} := (σ(ϑ))_{σ∈Σ}∈ X^{+}. Let D = −ϑ^{2}∈ 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) =
q_{ϑ}(α) and q_{ϑ}(xα) = q_{ϑ}(x)ρ(α) for α, x ∈ K. Let C(Σ) be the K-module whose underlying space is C^{Σ}
with the K-action given α(xσ) = (σ(α)x_{σ}). Then we have a canonical isomorphism K ⊗_{Q}R = C(Σ), and
p_{ϑ}:= q_{ϑ}^{−1}: V ⊗_{Q}R→ K ⊗^{∼} _{Q}R = C(Σ) is the period map associated to ϑ^{Σ}.

3.2. A good level structure.

3.2.1. For each v|pFF^{c}, we decompose v = ww into two places w and w of K with w|FΣ_{p}. Here w|FΣ_{p}means
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 δ^{v}:= θv− θv be a fixed generator of the relative different
D_{K}_{w}_{/F}_{v}.

Fix a finite idele d_{F} = (d_{F}_{v}) ∈ A_{F ,f} such that il_{F}(d_{F}) = D_{F}. By (d2), we may choose d_{F}_{v} = 2ϑδ_{v}^{−1} if
v|D_{K/F}I (resp. d_{F}_{v} = −2ϑ_{w}if w|FΣ_{p}).

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 - plCC^{c}, we
choose {e1,v, e2,v} in R ⊗ Ov such that R ⊗O Ov = Ove1,v ⊕ O^{∗}_{v}e2,v. It is clear that {e1,v, e2,v} can be
taken to be {ϑ, 1} except for finitely many v. If v|pFF^{c}, let {e_{1,v}, e_{2,v}} = {e_{w}, d_{F}_{v}· e_{w}} with w|FΣ_{p}. If v
is inert or ramified, let {e1,v, e2,v} = {θv, dF_{v} · 1}. For every integer n ≥ 0, we let Rn = O + l^{n}R, and let
n

e^{(n)}_{1,l}, e^{(n)}_{2,l}o

:= {−1, −dF_{l}$_{l}^{n}θl} be a basis of Rn⊗OOl.

For v ∈ h, let ςv (resp. ς_{l}^{(n)}) be the element in GL2(Fv) such that eiς_{v}^{−1} = qϑ(ei,v) (resp. ei(ς_{l}^{(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)}×ς_{l}^{(n)} ∈
GL2(AF). Let ςf and ς_{f}^{(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} = q_{ϑ}(l^{−n}R_{n}⊗_{Z}Z).b

The matrix representation of ςv according to the basis {e1, e2} for v|plD_{K/F}CC^{c} is given as follows:

(3.1)

ςv=d_{F}_{v} −2^{−1}tv

0 d^{−1}_{F}

v

, tv= θv+ θv if v|D_{K/F}I,

ςv=

" _{d}

Fv

2 −^{1}_{2}

d_{Fv}

−2ϑw

−1 2ϑw

#

=−ϑw −^{1}_{2}
1 _{2ϑ}^{−1}

w

if v|pFF^{c} and w|FΣp,

ς_{l}^{(n)}=−b_{l} 1
al 0

d_{F}_{l}$_{l}^{n} 0

0 1

(θl= alϑ + bl, al∈ O^{×}_{l} , bl∈ Ol).

3.3. For every a ∈ A^{×}_{K,f}, we let

A_{n}(a)_{/C}:=Aρ(a)ς^{(n)}(ϑ^{Σ}) = (Aρ(g)ς^{(n)}(ϑ^{Σ}), h , i_{can}, ι_{can}, η(a)) ∈ Sh_{K}(C)

be the C-quadruple associated to (ϑ^{Σ}, ρ(a)ς_{f}^{(n)}) as in §2.4. Then A_{n}(a)_{/C} is an abelian variety with CM by
K. Let W be the p-adic completion of the maximal unramified extension of Z_{p} in C_{p}. By the general theory
of CM abelian varieties, the C-quadruple A_{n}(a)/C descends to a W -quadruple A_{n}(a). Moreover, since K is
p-ordinary, A_{n}(a) ⊗WF¯pis 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 x_{n} : A^{×}_{K,f} → lim←−^{m}I_{K,m}(W ) ⊂ I_{K,∞}(W ), which factors
through CK:= A^{×}_{K,f}/K^{×}the idele class group of K. The collection of points Cl^{∞}:= t^{∞}_{n=1}xn(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)^{−1}D_{F}^{−1} for the fractional ideal c(R) = D_{F}^{−1}(2ϑD_{K/F}^{−1} ). Then c(R) is
the polarization of CM points x_{0}(1). From the equation

D^{−1}_{F} det(ςf) = ∧^{2}L ς_{f}^{−1}= ∧^{2}R = c(R)^{−1}D^{−1}_{F} ,

we find that c(R) = (det(ςf)). Moreover, for a ∈ A^{×}_{K}, the polarization ideal of x0(a) is c(a) := c(R)N_{K/F}(a),
a= il_{K}(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 U_{n} = (C_{1})^{Σ}×(R_{n}⊗ bZ)^{×} in A^{×}_{K} = (C^{×})^{Σ}×A^{×}_{K,f}, where C_{1} is the unit circle
in C^{×}. Let Cl_{n} = K^{×}A^{×}_{F}\A^{×}_{K}/U_{n} and let [·]_{n} : A^{×}_{K} → Cl_{n} be the quotient map. Let Cl_{∞} = lim

←−^{n}Cl_{n}. For
a ∈ A^{×}_{K}, we let [a] := lim←−n[a]n ∈ Cl∞ be the holomorphic image in Cl∞. Henceforth, every ν ∈ X^{−}_{l} 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 ∈ U_{n}· A^{×}_{F},

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

(3.2)

Z

Cl∞

φdϕ_{E}:= al(E)^{−n}· X

[t]_{n}∈Cln

E(xn(t))χ(t)φ([t]b n).

3.5.2. Let ∆ be the torsion subgroup of Cl_{∞}. Let Cl^{alg} be the subgroup of Cl_{∞} generated by [a] for
a ∈ (A^{(l)}_{K})^{×} and ∆^{alg} = Cl^{alg}∩ ∆. 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^{−1}ρ(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) E^{R} =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 ∈ Cl_{F}
such that c = c(a) for some R-ideal a and for every u ∈ O prime to l, we can find β ≡ u mod l^{r(χ)}with
aβ(E^{R}, c) 6≡ 0 (mod mp),

where aβ(E^{R}, c) is the β-th Fourier coefficient of E^{R} at the cusp (O, c^{−1}). Then
Z

Cl∞

νdE 6≡ 0 (mod mp) for almost all ν ∈ X^{−}_{l} .

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|·|^{−}_{A}^{1}^{2}

K and χ+= χ|_{A}×
F.
Let K_{∞}^{0} :=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 K_{∞}^{0} -finite functions φ : G(A_{F}) → 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

E_{A}(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 W_{β}(φ_{v}, g_{v}) is defined by

Wβ(φv, gv) = Z

Fv

φv(w1 xv

0 1

gv)ψ(−βxv)dxv,
and the intertwining operator M_{w} is defined by

M_{w}φ_{v}(g_{v}) =
Z

Fv

φ_{v}(w1 xv

0 1

g_{v})dx_{v}.

By definition, M_{w}φ_{v}(g_{v}) 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|D_{F}|_{R} ·Y

v

Mwφv(gv) ; Wβ(EA, g) = 1

p|D_{F}|_{R} ·Y

v

Wβ(φv, 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 ∈ F_{v}^{×}.
4.3.1. We first introduce some notation and definitions. Let S^{◦} =v ∈ h | v - lCC^{c}D_{K/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
ψ_{L} := ψ ◦ T_{L/F}_{v}. Given a character µ : L^{×} → C, we recall that the epsilon factor (s, µ, ψL) in [Tat79] is
defined by

(s, µ, ψL) = |c|^{s}_{L}
Z

c^{−1}O^{×}_{L}

µ^{−1}(x)ψL(x)dLx (c = dL$_{L}^{a(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|^{s}_{L}d^{×}x (s ∈ C).

The local root number W (µ) is defined by

W (µ) := (1
2, µ, ψ_{L})

(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 = F_{v} (resp. E = K ⊗_{F}F_{v}) and let d_{F} = d_{F}_{v} be the fixed generator of
the absolute different DF in §3.2.1. Write χ (resp. χ+, χ^{∗}) for χv (resp. χ+,v, χ^{∗}_{v}). 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 I^{Y} 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 φ^{h}_{k,s,σ}∈ I_{v}(s, χ_{+}) of weight k by

(4.2) φ^{h}_{k,s,σ}(g) := J (g, i)^{−k}δ(g)^{s}.
The intertwining operator Mwφk,s,σ is given by

(4.3) Mwφ^{h}_{k,s,σ}(g) = i^{k}(2π)Γ(k + 2s − 1)

Γ(k + s)Γ(s) · J (g, i)^{k}det(g)^{−k}δ(g)^{1−s}.
For (x, y) ∈ R×R+ and β ∈ R^{×}, it is well known that

(4.4) W_{β}(φ^{h}_{k,s,σ},y x
0 1

)|_{s=0}= (2πi)^{k}

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

Define the section φ^{n.h.}_{k,κ}_{σ}_{,s,σ}∈ I(s, χ+) of weight k + 2κσ by

(4.5) φ^{n.h.}_{k,κ}_{σ}_{,s,σ}(g) := J (g, i)^{−k−κ}^{σ}J (g, i)^{κ}^{σ}δ(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.

Denote by V_{+}^{κσ} the operator (V^{+})^{κ}^{σ} acting on I_{v}(s, χ_{+}). By [JL70, Lemma 5.6 (iii)], we have
(4.6) V_{+}^{κ}^{σ}^{σ}φ^{h}_{k,s,σ}= 2^{κ}^{σ}Γ(k + κσ+ 2s)

Γ(k + 2s) φ^{n.h.}_{k,κ}

σ,s,σ.

4.3.3. v ∈ S^{◦}. In this case, χ is unramified. Define φ_{χ,s,v}(g) to be the spherical Godement section in I_{v}(s, χ_{+}).

To be precise, put

φχ,s,v(g) =fΦ_{v}(g) := |det g|^{s}
Z

F^{×}

Φv((0, t)g)χ+(t) |t|^{2s}d^{×}t, where
Φv = I^{O}v⊕O_{v}^{∗}.

It is well known that the local Whittaker integral is
(4.7) W_{β}(φ_{χ,s,v},cv

1

)|_{s=0}=1 − χ^{∗}($)^{v(βc}^{v}^{)+1}

1 − χ^{∗}($) · |D_{F}|^{−1}· IOv(βc_{v}),
and the intertwining operator is given by

(4.8) Mwφχ,s,v(c_{v}

1

) = Lv(2s − 1, χ+) |cv|^{1−s}χ^{−1}_{+} (cv).

4.3.4. v|FF^{c}. If v|FF^{c} is split in K, write v = ww with w|F and χ_{v} = (χ_{w}, χ_{w}). Then a(χ_{w}) ≥ a(χ_{w}). 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
ϕ_{µ}(x) = IO_{v}^{×}(x)µ(x) (x ∈ F ).

Define ϕw= ϕχ_{w} and

ϕw=
(ϕ_{χ}^{−1}

w if χwis ramified,
IOv if χ_{w}is unramified.

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

ϕb_{w}(y) =
Z

F

ϕ_{w}(x)ψ(yx)dx.

Define φχ,s,v∈ Iv(s, χ+) by

(4.9) φ_{χ,s,v}(g) = f_{Φ}_{v}(g) := |det g|^{s}
Z

F^{×}

Φ_{v}((0, t)g)χ_{+}(t) |t|^{2s}d^{×}t.

A straightforward calculation shows that the local Whittaker integral is

(4.10)

Wβ(φχ,s,v, 1) = Z

F^{×}

ϕw(x)ϕbw(−βx^{−1}) · χ+(x) |x|^{2s−1}d^{×}x

= Z

F^{×}

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

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

(4.11) M_{w}φ_{χ,s,v}(1) = 0.

4.3.5. v = l. Let φ_{χ,v,s}∈ I_{v}(s, χ_{+}) be the unique N (O_{v})-invariant section supported in the big cell B(F )wN (O_{v}^{∗})
and φχ,v,s(w) = 1. One checks easily that φ_{χ,s,v}|Ul given by

φ_{χ,s,v}|Ul(g) = X

u∈O^{∗}_{v}/lO^{∗}_{v}

φ_{χ,s,v}(g$ u
0 1

)

is also supported in the big cell and is invariant by N (O^{∗}_{v}). In particular, φ_{χ,s,v} is an U_{l}-eigenform, and the
eigenvalue is χ^{−1}_{+} ($_{l}).