Special Values of

Anticyclotomic Rankin-Selberg L-Functions

Ming-Lun Hsieh^{1}

Received: March 10, 2014 Revised: May 12, 2014

Communicated by Otmar Venjakob

Abstract. In this article, we construct a class of anticyclotomic p-adic Rankin-Selberg L-functions for Hilbert modular forms, generalizing the construction of Brakoc̆ević, Bertolini, Darmon and Prasanna in the elliptic case. Moreover, building on works of Hida, we give a necessary and sufficient criterion for the vanishing of the Iwasawa μinvariant of this p-adic L-function vanishes and prove a result on the non-vanishing modulo p of central Rankin-Selberg L-values with anticyclotomic twists. These results have future applications to Iwasawa main conjecture for Rankin-Selberg convolution and Iwasawa theory for Heegner cycles.

2010 Mathematics Subject Classification: 11F67 11G15 Keywords and Phrases: Iwasawa theory, p-adic L-functions, μinvariant.

Introduction

The purpose of this article is to (i) construct a class of anticyclotomic Rankin-Selberg p-adic L-functions for Hilbert modular forms and study the vanishing/non-vanishing of the associated Iwasawa μ-invariant, (ii) prove a result on the non-vanishing modulo p of central Rakin-Selberg L-values with anticyclotomic twists. Let F be a totally real algebraic number field of degree d over Q and K be a totally imaginary quadratic extension of F. Denote by z ↦ \overline{z} the non-trivial element in Gal(K/F). Let π be an irreducible cuspidal automorphic representation of GL_{2}(A_{F}) with unitary central character ω. Let π_{K} be a lifting of π to K constructed in [Jac72, Thm. 20.6]. Then π_{K} is an irreducible automorphic representation of GL_{2}(A_{K}), which is cuspidal if π is not obtained from the automorphic induction from K/F. Let λ : A_{K}^{×}/K^{×} → C^{×} be a unitary Hecke character of K^{×} such that

1The author is partly supported by National Science Council grant 101-2115-M-002-010-MY2

(0.1)  λ|_{A_{F}^{×}} = ω^{−1}.

The automorphic representation π_{K} ⊗ λ is therefore conjugate self-dual. For each place v of F, we can associate a local L-function L(s, π_{K_{v}} ⊗ λ_{v}) and a local epsilon factor ε(s, π_{K_{v}} ⊗ λ_{v}, ψ_{v}) (which depends on a choice of non-trivial character ψ_{v} : F_{v} → C^{×}) to the local constituent π_{K_{v}} ⊗ λ_{v} of π_{K} ⊗ λ ([JL70, Thm. 2.18 (iv)]). Denote by L(s, π_{K} ⊗ λ) the global L-function obtained by the meromorphic continuation of the Euler product of local L-functions at all finite places. In this paper, we study the p-adic variation of the central value L(\frac{1}{2}, π_{K} ⊗ λ) with anticyclotomic twists under certain hypotheses. To introduce our hypotheses precisely, we need some notation. Fix a CM-type Σ of K. Namely, Σ is a subset of Hom(K, C) such that

Σ ⊔ \overline{Σ} = Hom(K, C) ; Σ ∩ \overline{Σ} = ∅.

Then Σ induces an identification K ⊗_{Q} R ≃ C^{Σ}. We shall identify Σ with the set of archimedean places of F via the restriction. For each k = ∑_{σ∈Σ} k_{σ}σ ∈ Z[Σ], we write Γ_{Σ}(k) = ∏_{σ∈Σ} Γ(k_{σ}) (Γ is the usual Gamma function), and if x = (x_{σ}) ∈ (A^{×})^{Σ} for an algebra A, we let x^{k} := ∏_{σ∈Σ} x_{σ}^{k_{σ}}. For a Hecke character χ : A_{K}^{×}/K^{×} → C^{×}, we denote by χ_{∞} : (K ⊗_{Q} R)^{×} → C^{×} its archimedean component, and we say χ is of infinity type (k_{1}, k_{2}) for k_{1}, k_{2} ∈ 2^{−1}Z[Σ] such that k_{1} − k_{2} ∈ Z[Σ] if

χ_{∞}(z) = z^{k_{1}−k_{2}}(z\overline{z})^{k_{2}} for all z ∈ (K ⊗_{Q} R)^{×} ≃ (C^{×})^{Σ}.

For each ideal 𝔞 of F (resp. ideal 𝔄 of K), we have a unique factorization 𝔞 = 𝔞^{+}𝔞^{−} (resp. 𝔄 = 𝔄^{+}𝔄^{−}), where 𝔞^{+} (resp. 𝔄^{+}) is only divisible by primes split in K and 𝔞^{−} (resp. 𝔄^{−}) is divisible by primes inert or ramified in K. Let 𝔫 = 𝔫^{+}𝔫^{−} be the conductor of π. We define the normalized local root number attached to π_{K_{v}} and λ_{v} for each place v by

ε^{∗}(π_{K_{v}}, λ_{v}) := ε(\frac{1}{2}, π_{K_{v}} ⊗ λ_{v}, ψ_{v}) · ω_{v}(−1).

We remark that the value ε(\frac{1}{2}, π_{K_{v}} ⊗ λ_{v}, ψ_{v}) does not depend on the choice of ψ_{v}. We assume that π has infinity type k = ∑_{σ} k_{σ}σ ∈ Z_{>0}[Σ] and λ has infinity type (\frac{k}{2} , −\frac{k}{2} ). In other words, π_{σ} is a discrete series or limit of discrete series of weight k_{σ} with unitary central character at every archimedean place σ. In particular, this implies that {k_{σ}}_{σ∈Σ} have the same parity and the local root number ε^{∗}(π_{K_{v}}, λ_{v}) = +1 at every archimedean place. We further assume the following local root number hypothesis for (π, λ):

Hypothesis A. The local root number ε^{∗}(π_{K_{v}}, λ_{v}) = +1 for each v | 𝔫^{−}.

In particular, the above hypothesis holds if 𝔫^{−} = (1).

Special Values of Anticyclotomic L-Functions 711

We prepare some notation in Iwasawa theory. Let p be an odd rational prime. Fix an embedding ι_{∞} : \overline{Q} ↪ C and isomorphism ι : C →^{∼} C_{p} once and for all. Throughout this article, we make the following assumption

(ord) Σ is p-ordinary.

Let Σ_{p} be the set of p-adic places of K induced by Σ. Then the ordinary assumption (ord) means that Σ_{p} and its complex conjugation \overline{Σ}_{p} gives a full partition of the set of p-adic places of K. If L is a number field, we write G_{L} = Gal(\overline{Q}/L) for the absolute Galois group. Denote by rec_{K} : A_{K}^{×} → G_{K}^{ab} the geometrically normalized reciprocity law. Recall that we say a continuous character ϕ̂: G_{K}^{ab} → C_{p}^{×} is locally algebraic of weight (k_{1}, k_{2}) with k_{1}, k_{2} ∈ Z[Σ] if χ(rec_{K}(a)) = a^{k_{1}} \overline{a}^{k_{2}} for every a ∈ (K ⊗ Q_{p})^{×} close to 1 (See also [Ser68, Chapter III, §2]). Let K_{p^{∞}}^{−} be the maximal anticyclotomic Z_{p}^{[F:Q]}-extension of K. Let Γ^{−} = Gal(K_{p^{∞}}^{−}/K) and let Λ = \overline{Z}_{p}⟦Γ^{−}⟧ be the Iwasawa algebra of [F : Q]-variables. To each locally algebraic p-adic character ϕ̂ : Γ^{−} → C_{p}^{×} of weight (m, −m), we can associate a Hecke character ϕ : A_{K}^{×}/K^{×} → C^{×} of infinity type (m, −m) defined dy

ϕ(a) := ι^{−1}(ϕ̂(rec_{K}(a))a_{p}^{−m}\overline{a}_{p}^{m})a_{∞}^{m}\overline{a}_{∞}^{−m},

where a_{p} ∈ (K ⊗_{Q} Q_{p})^{×} and a_{∞} ∈ (K ⊗_{Q} R)^{×} are the p-component and the archimedean component of a respectively. We say ϕ̂ is the p-adic avatar of ϕ. We shall call 𝔛_{p}^{crit} the set of critical specializations, consisting of locally algebraic p-adic characters on Γ^{−} of weight (m, −m) with m ∈ Z_{≥0}[Σ] (See §5.4). Our first result is the construction of the anticyclotomic p-adic L-function attached to π_{K} ⊗ λ. We need more notation. Let D_{F} be the different of F and D_{K/F} be the relative different of K/F. Let 𝔑 be the prime-to-p conductor of π_{K} ⊗ λ. We have a unique decomposition 𝔑 = 𝔑^{+}𝔑^{−} and fix a decomposition 𝔑^{+} = 𝔉\overline{𝔉} with (𝔉, \overline{𝔉}) = 1 such that 𝔑^{−} is only divisible by prime inert or ramified in K/F and 𝔉 is only divisible by primes split in K/F. We choose δ ∈ K such that

• \overline{δ} = −δ,

• Im σ(δ) > 0 for all σ ∈ Σ,
• The polarization ideal 𝔠(O_{K}) := D_{F}^{−1}(2δD_{K/F}) is prime to p𝔑\overline{𝔑}.

Let (Ω_{∞}, Ω_{p}) be the complex and p-adic periods attached to (K, Σ) defined in [HT93, (4.4a), (4.4b)]. For each Hecke character χ of K^{×}, we define the p-adic multiplier e_{Σ_{p}}(π, χ) by

(0.2) e_{Σ_{p}}(π, χ) := ∏_{𝔓∈Σ_{p}, 𝔭=𝔓\overline{𝔓}} ε(\frac{1}{2}, π_{𝔭} ⊗ χ\overline{𝔓}, ψ_{𝔭})L(\frac{1}{2}, π_{𝔭} ⊗ χ\overline{𝔓})^{−2}χ−2 ; \overline{𝔓}(−2δ).

The shape of this modified p-Euler factor e_{Σ_{p}}(π, χ) has been suggested by J. Coates [Coa89].

Theorem A. In addition to (ord) and Hypothesis A, we further assume that

(sf)  𝔫^{−} is square-free.

Then there exists an element ℒ_{Σ_{p}}(π, λ) ∈ Λ such that for every ϕ̂ ∈ 𝔛_{p}^{crit} of weight (m, −m), we have

\frac{ϕ̂(ℒ_{Σ_{p}}(π, λ)^{2})}{Ω_{p}^{2(k+2m)}} =\frac{Γ_{Σ}(k + m)Γ_{Σ}(m + 1)}{(Im δ)^{k+2m}(4π)^{k+2m+1·Σ}} · e_{Σ_{p}}(π, λϕ) · \frac{L(\frac{1}{2}, π_{K} ⊗ λϕ)}{Ω_{K}^{2(k+2m)}}
× [O_{K}^{×} : O_{F}^{×}]^{2} · ϕ(𝔉^{−1}),

where Ω_{K} = (2πi)^{−1}Ω_{∞}, (Im δ) = (Im σ(δ))_{σ∈Σ} and (4π) means the diagonal element (4π)_{σ∈Σ} in (C^{×})^{Σ}.

If π is attached to a Hilbert new form f and χ is a Hecke character of A_{K}^{×}, then the L-function L(s, π_{K} ⊗ χ) can be identified with the Rankin-Selberg L-function L(f, χ, s) of f and the theta series associated to χ by

L(s, π_{K} ⊗ χ) = L(f, χ, s + \frac{k_{mx} − 1}{2}) (k_{mx} := max_{σ∈Σ} k_{σ}).

Therefore, ℒ_{Σ_{p}}(π, λ) is the p-adic L-function that interpolates the square root of Rankin-Selberg central L-values. We shall call L_{Σ_{p}}(π, λ) := ℒ_{Σ_{p}}(π, λ)^{2} the anticyclotomic p-adic L-function for π_{K} ⊗ λ with respect to the p-ordinary CM type Σ. If π is obtained from the automorphic induction of a Hecke character of K^{×}, one can see, by comparing the interpolation formulas on both sides, that L_{Σ_{p}}(π, λ) is a product of two p-adic Hecke L-functions for CM fields constructed by Katz [Kat78] and Hida-Tilouine [HT93] up to an explicit unit in Λ.

Remark. When F = Q, π arises from an elliptic new form f of weight k and level 𝔫, the construction of L_{Σ_{p}}(π, λ) can be recovered from [BDP13, Theorem 5.4] under some extra assumptions p ∤ 𝔫 and 𝔫^{−} is only divisible by ramified primes. In their notation, L(f, χ^{−1}, 0) = L(\frac{1−k}{2} , π_{K} ⊗ χ^{−1}) for χ ∈ Σ_{cc}^{(2)}(𝔑) defined in the page 1094 loc.cit.,and our set 𝔛_{p}^{crit} corresponds to Σ_{cc}^{(2)}(𝔑) by

ϕ̂ → λ^{−1}ϕ^{−1}|·|_{A_{K}}^{− \frac{k}{2}}, where |·|_{A_{K}} is the adelic absolute value of A_{K}^{×} ^{2}. Note that Hypothesis A on local root numbers is also imposed in the bottom of page 1903 loc.cit.. This kind of p-adic L-function with some extra Euler factors removed is also considered in [Bra11a] under different hypotheses.

Our second theorem is to prove the vanishing of the Iwasawa μ-invariant μ_{π,λ,Σ}^{−} of ℒ_{Σ_{p}}(π, λ) under certain hypothesis. Recall that the μ-invariant μ_{π,λ,Σ}^{−} is defined by

μ_{π,λ,Σ}^{−} = inf {r ∈ Q_{≥0} | p^{−r}ℒ_{Σ_{p}}(π, λ) ≢ 0 (mod 𝔪_{p}Λ)} ,

where 𝔪_{p} is the maximal ideal of \overline{Z}_{p}. To explain our hypothesis, we recall that thanks to the works of Deligne, Carayol, Blasius-Rogawski and Taylor et.al ([BR93], [Tay89], [Jar97]), there exists a finite extension L/Q_{p} and the p-adic Galois representation ρ_{p}(π) : G_{F} → GL_{2}(O_{L}) such that ρ_{p}(π) is unramified outside p𝔫; for each finite place v ∤ p𝔫,

2Our conventions for the infinity type are opposite to each other

ι^{−1}(L(s, ρ_{p}(π)|_{W_{Fv}})) = L(s + \frac{1 − k_{mx}}{2} , π_{v}^{∨}),

where W_{F_{v}} is the Weil group of F_{v}. Let 𝔠_{λ} be the conductor of λ. For each v | 𝔠_{λ}^{−}, let Δ_{λ,v} be the finite group λ(O_{K_{v}}^{×}).

Theorem B. With the assumptions in Theorem A, suppose further that (1) p is unramified in F,

(2) the residual Galois representation

\overline{ρ}_{p}(π_{K}) := ρ_{p}(π)|_{G_{K}} (mod 𝔪_{p}) is absolutely irreducible,

(3) p ∤ ∏_{v(𝔠_{λ}^{−})=1} ♯(Δ_{λ,v}).

Then μ_{π,λ,Σ}^{−} = 0.

Let ℓ ≠ p be a rational prime. We next consider the problem of the nonvanishing modulo p of L-values twisted by anticyclotomic characters of ℓ-power conductor. This problem has been studied in the literature in various settings (cf. [Vat03], [Hid04a], [Fin06], [Hsi12]). To state our result along this direction, we introduce some notation. Let 𝔩 be a prime of F above ℓ and let K_{𝔩^{∞}}^{−} be the anticyclotomic pro-ℓ extension in the ray class field over K of conductor 𝔩^{∞}. Let Γ_{𝔩}^{−} := Gal(K_{𝔩^{∞}}^{−}/K) and let 𝔛_{𝔩}^{0} be the set consisting of finite order characters ϕ : Γ_{𝔩}^{−} → μ_{ℓ^{∞}}. Let χ be a Hecke character of infinity type (\frac{k}{2} + m, −\frac{k}{2} − m) and of conductor 𝔠_{χ} with m ∈ Z_{≥0}[Σ] as before.

Theorem C. In addition to (ord), (sf) and Hypothesis A, we further assume that

(1) (p𝔩, 𝔫𝔠_{χ}D_{K/F}) = 1,

(2) the residual Galois representation \overline{ρ}_{p}(π_{K}) is absolutely irreducible,

(3) p ∤ ∏_{v(𝔠_{λ}^{−})=1} ♯(Δ_{χ,v}).

Then for almost all characters ϕ ∈ 𝔛_{𝔩}^{0}, we have

[O_{K}^{×} : O_{F}^{×}]^{2} · \frac{Γ_{Σ}(k + m)Γ_{Σ}(m + 1)}{(Im δ)^{k+2m}(4π)^{k+2m+1·Σ}} · \frac{L(\frac{1}{2}, π_{K} ⊗ χϕ)}{Ω_{K}^{2(k+2m)}} ≢ 0 (mod 𝔪_{p}).

Here almost all means "except for finitely many ϕ ∈ 𝔛_{𝔩}^{−}" if dim_{Q_{ℓ}} F_{𝔩} = 1 and "for ϕ in a Zariski dense subset of 𝔛_{𝔩}^{0}" if dim_{Q_{ℓ}} F_{𝔩} > 1 ([Hid04a, p.737]).

Note that Theorem C in particular implies a simultaneous non-vanishing of central L-values with anticyclotomic twists. This has application to the non-vanishing of Bessel models of theta lifts of GSp(4) in view of [PTB11, Thm. 3] and the existence of some explicit theta lifts [Nar13]. In addition, we would like to mention several future applications of these results in Iwasawa theory. I. Iwasawa main conjecture for Rankin-Selberg convolution. The congruences between Eisenstein series and cusp forms on unitary groups provide a general strategy to construct elements in Selmer groups in terms of L-values and has been used to prove one-sided divisibility in Greenberg-Iwasawa main conjectures for GL_{2} and CM fields ([SU14] [Hsi14a]). Usually the most difficult part in the method of Eisenstein congruence is to prove the non-vanishing modulo p of certain Eisenstein series, where the non-vanishing modulo p of L-values always play an important role. For example, Skinner and Urban use results of Finis and Vatsal to show the non-vanishing modulo p of certain Klingen-Eisenstein series on U(2, 2). In a recent work [Wan13b], Xin Wan applies the method of Eisenstein congruence on the unitary group U(3, 1) to obtain a one-sided divisibility result towards Greenberg-Iwasawa main conjecture for certain Rankin-Selberg convolution, and Theorem C is used to prove the non-vanishing modulo p of Fourier-Jacobi coefficients of certain Siegel-Eisenstein series on U(3, 1). His results along this direction lead to C. Skinner’s work on the converse of Gross-Zagier and Kolyvagin. Moreover, combining our Theorem B, he is able to deduce Perrin-Riou’s main conjecture for Heegner points [Wan13a] in some cases. II. Iwasawa theory for Heegner cycles. An immediate consequence of Theorem C is the non-vanishing of the p-adic L-function ℒ_{Σ_{p}}(π, λ). Combined with the work [BDP13] relating the p-adic logarithm of Heegner points and the values of ℒ_{Σ_{p}}(π, λ) outside the range of interpolation, this gives a new proof of Cornut-Vatsal theorem on Mazur conjecture for higher Heegner points when p is split in the imaginary quadratic field. In our forthcoming work [CH14] on the Perrin-Riou’s explicit reciprocity law for Heegner cycle Euler system with connection to ℒ_{Σ_{p}}(π, λ) when π is associated with an elliptic new form f ∈ S_{k}(Γ_{0}(N)) with Deligne’s p-adic Galois representation V_{f}, we also use this result to obtain the rank-zero case of Bloch-Kato conjecture for the Galois representation V_{f}(\frac{k}{2} ) ⊗ λ̂ as well as the analogue of Mazur conjecture for the image of higher Heegner cycles under the p-adic Abel-Jacobi map (the ℓ-adic case with ℓ ≠ p is proved by Howard [How06]).

The key ingredients in this article are the use of normalized toric cusp forms and the explicit calculations of their period integral formula. In representation theory, toric cusp forms are Gross-Prasad test vectors [GP91] in the space of cuspidal automorphic forms on GL_{2}(A_{F}). It seems they often serve the optimal choice in the application of toric period integrals to arithmetic. For example, Afalo and Nekovář [AN10] used Gross-Prasad test vectors in the setting of definite quaternion algebras to give an extension of the work of Cornut-Vatsal on Mazur conjecture. For the reader’s convenience, we recall the definition here. Let χ be a Hecke character of K^{×} such that χ|_{A^{×}} = ω^{−1} and let T ⊂ A_{K}^{×} be the subgroup consisting of ideles z = (z_{v}) ∈ ∏_{v} K_{v}^{×} with z_{v}/\overline{z_{v}} ∈ O_{K_{v}}^{×} for all primes v split in K. Fixing an embedding ι : K^{×} ↪ GL_{2}(F), we say an automorphic form φ : GL_{2}(F)\ GL_{2}(A_{F}) → C is a toric form of character χ if

φ(gι(t)) = χ^{−1}(t)φ(g) for all t ∈ T .

In addition, to obtain the optimal p-integrality, we will need to normalize toric forms so that their Fourier coefficients are not all congruent to zero modulo p. This is equivalent to choosing a normalized Gross-Prasad test vector in the local Whittaker model of π_{v} at each place v. The reader will find later that the normalization of toric forms is the most subtle and important part of this paper. We give a rough sketch of the proofs. We begin with an outline the construction of ℒ_{Σ_{p}}(π, λ) as follows.

(1) Construct a toric Hilbert modular form φ_{λϕ} of character λϕ for each ϕ̂ ∈ 𝔛_{p}^{crit} as above by a careful choice of toric local Whittaker functions in local Whittaker models of π (See Definition 3.1, Lemma 3.13).

(2) Make an explicit calculation of the Fourier expansion of φ_{λϕ}.

(3) Via the p-adic interpolation of the Fourier expansion, construct a p-adic distribution ℱ_{λ} on Γ^{−} valued in the space of p-adic modular forms, which interpolates toric forms φ_{λϕ} for ϕ̂ ∈ 𝔛_{p}^{crit}. The p-adic Lfunction ℒ_{Σ_{p}}(π, λ) is obtained by a weighted sum of the evaluation of this distribution ℱ_{λ} at a finite set of CM points.

The evaluation of φ_{χ} with χ = λϕ at CM points in the step (3) is essentially the toric period integral P_{χ}(φ_{χ}) given by

P_{χ}(φ_{χ}) = ∫_{A^{×}K^{×}\A_{K}^{×}} φ_{χ}(ι(t))χ(t)dt.

To prove the formula in Theorem A, we have to express the square P_{χ}(φ_{χ})^{2} in terms of the central L-value L(\frac{1}{2}, π_{K} ⊗ χ). This is usually referred to as an explicit Waldspurger formula. Such a formula has been exploited widely in the literature based on either explicit theta lifts ([Mur10], [Mur08], [Xue07], [Hid10a] and [BDP13]) or the technique of relative trace formula ([MW09]). In this paper we adopt a different approach, making use of a formula of Waldspurger which is indeed proved but not stated explicitly in [Wal85]. This formula decomposes the square P_{χ}(φ) of the global period toric integral into a product of local period integrals involving local Whittaker functions of φ. Explicit computation of these local integrals shows that P_{χ}(φ)^{2} is essentially equal to the central value of the L-function L(s, π_{K}⊗λ). We emphasize that this explicit formula does not depend on the choices of special Bruhat-Schwartz functions in the classical approach of theta lifting but on choices of local Whittaker functions which reflect the arithmetic of modular forms directly via the Fourier expansion. Now with the above construction of toric forms and explicit period integral formulas, the proofs of Theorem B and Theorem C when combined with fundamental works of Hida ([Hid10b] and [Hid04a]) are reduced to a study the vanishing/nonvanishing modulo p properties of the Fourier expansions of the toric cusp form φ_{λ} at cusps (𝔞, 𝔟) such that 𝔞𝔟^{−1} is the polarization of an abelian variety with CM by O_{K}. We give an explicit computation of Fourier coefficients of toric forms φ_{λ}, with which we can study the non-vanishing modulo p property of these Fourier coefficients. These calculations are elementary but quite tedious and lengthy. Finally, the connection between the Fourier coefficients of Hilbert modular forms and the trace of Frobenius of the associated Galois representations enables us to relate the non-vanishing modulo p of Fourier coefficients of φ_{λ} at these cusps and the irreducibility of the residual Galois representation \overline{ρ}_{p}(π)|_{G_{K}}. We end this introduction by making a few remarks on our assumptions. The restriction (sf) is merely due to the computational difficulty on the local period integrals and the local Fourier coefficients, and it is expected to be unnecessary. The global assumption on the irreducibility of residual Galois representation assures that the new form associated to π is not congruent to theta series arising from K. This assumption prevents the vanishing modulo p of ℒ_{Σ_{p}}(π, λ) from the possibility that ℒ_{Σ_{p}}(π, λ) is congruent to a product of two anticyclotomic Katz p-adic L-functions attached to self-dual Heck characters of the root number −1. The local assumption (3) is equivalent to saying that the local residual character λ_{v} (mod 𝔪_{p}) is ramified for all v|𝔠_{λ}^{−}. This hypothesis is used to avoid the vanishing of L-values due to sign change phenomenon. For example, if λ_{v} ≡ 1 (mod 𝔪_{p}) is unramified at some prime 𝔮|𝔠_{λ}^{−} with π_{𝔮} special, then one can construct λ′ ≡ λ (mod 𝔪_{p}) such that λ′ has the conductor 𝔮^{−1}𝔠_{λ}^{−} and the same infinity type with λ. This implies that π_{K} ⊗ λ′ has global root number −1, and hence the algebraic part of L(\frac{1}{2}, π_{K} ⊗ λ) is congruent to L(\frac{1}{2}, π_{K} ⊗ λ′) = 0. The above assumptions might be weaken with exceptional effort on the compuation of Fourier coefficients of toric forms. However, Hypothesis A is fundamental, the failure of which makes the period integral P_{χ}(φ_{χ}) vanish for all χ (and hence make the results null) by a well-known theorem of Saito-Tunnell ([Sai93], [Tun83]). This paper is organized as follows. After fixing notation and definitions in §1, we derive a key formula of Waldspurger on the decomposition of global toric period integrals into local toric period integrals (Proposition 2.1) in §2. The bulk of this article is §3, where we give the choices of local toric Whittaker functions W_{χ,v} and calculate explicitly these local period integrals attached to W_{χ,v}. The explicit Waldspurger formula is proved in Theorem 3.14, and a non-vanishing modulo p of these toric Whittaker functions is proved in Proposition 3.19. After reviewing briefly theory of complex and geometric Hilbert modular forms in §4, we prove Theorem A in §5. The key ingredient is Proposition 5.5, the construction of a p-adic measure ℱ_{λ,𝔠} on Γ^{−} with values in the space of p-adic modular forms, and the p-adic L-function ℒ_{Σ_{p}}(π, λ) is thus obtained by evaluating ℱ_{λ,𝔠} at suitable CM points. The precise evaluation formula of ℒ_{Σ_{p}}(π, λ)^{2} is established in Theorem 5.7. In §6, we study the μ-invariant of ℒ_{Σ_{p}}(π, λ) and prove Theorem B in Theorem 6.2. Finally, the non-vanishing of central L-values modulo p is considered in §7 and Theorem C is proved in Theorem 7.1.

Acknowledgments. The author thanks Prof. Atsushi Murase for sharing his preprint [Mur08], from which the author learnt many useful computations about local toric period integrals. He also thanks the referee for the suggestions and improvements.

1. Notation and definitions

1.1. Measures on local fields. We fix some general notation and conventions on local fields through this article. Let ψ_{Q} : A_{Q}/Q → C^{×} be the additive character such that ψ_{Q}(x_{∞}) = exp(2πix_{∞}) with x_{∞} ∈ R. Let q be a place of Q and let F be a finite extension of Q_{q}. Let ψ_{q} be the local component of ψ at q and let ψ_{F} := ψ_{q} ∘ T_{F/Q_{q}}, where T_{F/Q_{q}} is the trace from F to Q_{q}. Let dx be the Haar measure on F self-dual with respect to the pairing (x, x′) ↦ ψ_{F} (xx′). Let |·|_{F} be the absolute value of F normalized by d(ax) = |a|_{F} dx for a ∈ F^{×}. We often simply write |·| = |·|_{F} if it is clear from the context without possible confusion. We recall the definition of the local zeta function ζ_{F} (s). If F is non-archimedean, let ϖ_{F} be a uniformizer of F and let

ζ_{F} (s) = \frac{1}{1 − |ϖ_{F} |_{F}^{s}} .

If F is archimedean, then

ζ_{R}(s) = π^{−s/2}Γ(s/2); ζ_{C}(s) = 2(2π)^{−s}Γ(s).

The Haar measures d^{×}x on F^{×} is normalized by

d^{×}x = ζ_{F} (1) |x|_{F}^{−1} dx.

In particular, if F = R, then dx is the Lebesgue measure and d^{×}x = |x|_{R}^{−1} dx, and if F = C, then dx is twice the Lebesgue measure on C and d^{×}x = 2π^{−1}r^{−1}drdθ (x = re^{iθ}). Suppose that F is non-archimedean. Let O_{F} be the ring of integers of F and let D_{F} be the absolute different of F. Then D_{F}^{−1} is the Pontryagin dual of O_{F} with respect to ψ_{F}, and vol(O_{F} , dx) = |D_{F} |F^{\frac{1}{2}} . If μ : F^{×} → C^{×} is a character of F^{×}, define the local conductor a(μ) by

a(μ) = inf {n ∈ Z_{≥0} | μ(x) = 1 for all x ∈ (1 + ϖ_{F}^{n} O_{F}) ∩ O_{F}^{×}} .

1.2. If L is a number field, the ring of integers of L is denoted by O_{L}, A_{L} is the adele of L and A_{L,f} is the finite part of A_{L}. For a ∈ A_{L}^{×}, we put

𝔦𝔩_{L}(a) := a(O_{L} ⊗ Ẑ) ∩ L.

Denote by G_{L} the absolute Galois group and by rec_{L} : A_{L}^{×} → G_{L}^{ab} the geometrically normalized reciprocity law. We define ψ_{L} : A_{L}^{×}/L → C^{×} by ψ_{L}(x) = ψ_{Q} ∘ Tr_{L/Q}(x). Let v_{p} be the p-adic valuation on C_{p} normalized so that v_{p}(p) = 1. We regard L as a subfield in C (resp. C_{p}) via ι_{∞} : \overline{Q} ↪ C (resp. ι_{p} = ι^{−1} ∘ι_{∞} : \overline{Q} ↪ C_{p}) and Hom(L, \overline{Q}) = Hom(L, C_{p}).

Let \overline{Z} be the ring of algebraic integers of \overline{Q} and let \overline{Z}_{p} be the p-adic completion of \overline{Z} in C_{p}. Let \overline{Z} be the ring of algebraic integers of \overline{Q} and let \overline{Z}_{p} be the p-adic completion of \overline{Z} in C_{p} with the maximal ideal 𝔪_{p}.

1.3. Local L-functions. Let F be a non-archimedean local filed. Let μ, ν : F^{×} → C^{×} be two characters of F^{×}. Denote by I(μ, ν) the space consisting of smooth and GL_{2}(O_{F})-finite functions f : GL_{2}(F) → C such that

f((\begin{matrix}a & b \\ 0 & d \end{matrix}) g) = μ(a)ν(d) |\frac{a}{d}|^{\frac{1}{2}} f(g).

Then I(μ, ν) is an admissible representation of GL_{2}(F). Denote by π(μ, ν) the unique infinite dimensional subquotient of I(μ, ν). We call π(μ, ν) a principal series if μν^{−1} ≠ |·|^{±} and a special representation if μν^{−1} = |·|^{±}. Let E be a quadratic extension of F and let χ : E^{×} → C^{×} be a character. We recall the definition of local L-functions L(s, π_{K} ⊗ χ) ([Jac72, §20]) when π = π(μ, ν) is a subrepresentation of I(μ, ν). If E = F ⊕ F, then we write χ = (χ_{1}, χ_{2}) : F^{×} ⊕ F^{×} → C^{×} and put

L(s, π_{K} ⊗ χ) = \begin{cases}L(s, π ⊗ χ_{1})L(s, π ⊗ χ_{2}) & if μν^{−1} ≠ |·|^{±}, \\ L(s, μχ_{1})L(s, μχ_{2}) & if μν^{−1} = |·|. \end{cases}

If E is a field, then

L(s, π_{E} ⊗ χ) = \begin{cases}L(s, μ′χ)L(s, ν′χ) & if μν^{−1} ≠ |·|^{±}, \\ L(s, μ′χ) & if μν^{−1} = |·|. \end{cases}

Here μ′ = μ ∘ N_{E/F} , ν′ = ν ∘ N_{E/F} are characters of E^{×}.

1.4. Whittaker and Kirillov models. Let F be a local field. Let π be an irreducible admissible representation of GL_{2}(F) and let ψ : F → C^{×} be a non-trivial additive character. We let W(π, ψ) be the Whittaker model of π. Recall that W(π, ψ) is a subspace of smooth functions W : GL_{2}(F) → C such that

(1) W ((\begin{matrix}1 & x \\ 0 & 1 \end{matrix}) g) = ψ(x)W (g) for all x ∈ F.

(2) If v is archimedean, W ((\begin{matrix}a \\ & 1 \end{matrix})) = O(|a|^{N}) for some positive number

N.

(cf. [JL70, Thm. 6.3]). Let K(π, ψ) be the Kirillov model of π. If F is nonarchimedean, then K(π, ψ) is a subspace of smooth C-valued functions on F^{×}, containing all Bruhat-Schwartz functions on F^{×}. A function in K(π, ψ) shall be called a local Fourier coefficient of π. In addition, it is well known that we have the following GL_{2}(F)-equivariant isomorphism

(1.1)  \begin{matrix}W(π, ψ) & →^{∼} K(π, ψ) \\ W & ↦ ξ_{W} (a) := W ((\begin{matrix}a \\ & 1 \end{matrix})). \end{matrix}

2. Waldspurger formula

Let F be a number field and K be a quadratic field extension of F. Let A = A_{F}. Let G = GL_{2} _{/F}. Let π be an irreducible cuspidal automorphic representation of G(A) with unitary central character ω. Denote by A(π) the realization of π in the space A_{0}(G) of cusp forms on G(A). Let χ be a unitary Hecke character of K^{×} such that χ|_{A^{×}} = ω^{−1}. Let π_{K} be the quadratic base change of π to the quadratic extension K/F. The existence of π_{K} is established in [Jac72]. The goal of this section is to deduce from results in [Wal85] a formula (Proposition 2.1) which expresses the central value L(\frac{1}{2}, π_{K} ⊗ χ) in terms of a product of local toric period integrals of Whittaker functions. Let ψ := ψ_{F} : A/F → C^{×} be the standard non-trivial additive character. For a place v of F, we let G_{v} = G(F_{v}) and let χ_{v} : K_{v}^{×} → C^{×} (resp. ψ_{v} : F_{v} → C^{×}) denote the local constituent of χ (resp. ψ).

2.1. For x ∈ K, let T(x) := x + \overline{x} and N(x) = x\overline{x}. Let \{1, ϑ\} be a basis of K over F. We let ι : K → M_{2}(F) be the embedding attached to ϑ given by

(2.1)  ι(aϑ + b) = (\begin{matrix}aT(ϑ) + b & −aN(ϑ) \\ a & b \end{matrix}) (a, b ∈ F).

Put

J := (\begin{matrix}−1 & T(ϑ) \\ 0 & 1 \end{matrix}) .

Then M_{2}(F) = ι(K) ⊕ ι(K)J. It is clear that J^{2} = 1 and ι(t)J = Jι(\overline{t}) for all t ∈ K.

2.2. The local bilinear form and toric integral. For each place v of F, denote by π_{v} (resp. ψ_{v}) the local constituent of π (resp. ψ) at v. Define a C-bilinear form b_{v} : W(π_{v}, ψ_{v})×W(π_{v}, ψ_{v}) → C by

b_{v}(W_{1}, W_{2}) := ∑_{n=−∞}^{∞} ∫_{ϖ^{n}O_{F}^{×}} W_{1}((\begin{matrix}a \\ & 1 \end{matrix}))W_{2}((\begin{matrix}−a \\ & 1 \end{matrix}))ω^{−1}(a)d^{×}a

= ∫_{F_{v}^{×}} W_{1}((\begin{matrix}a \\ & 1 \end{matrix}))W_{2}((\begin{matrix}−a \\ & 1 \end{matrix}))ω^{−1}(a)d^{×}a.

It is known that this series converges absolutely as π_{v} is a local constituent of a unitary cuspidal automorphic representation. Moreover, the pairing b_{v} enjoys the property:

(2.2)  b_{v}(π(g)W_{1}, π(g)W_{2}) = ω(det g)b_{v}(W_{1}, W_{2}).

The pairing b_{v} thus gives rise to an isomorphism between the contragredient representation π^{∨} and π ⊗ ω^{−1}. The local toric period integral for W_{1}, W_{2} ∈ W(π_{v}, ψ_{v}) is given by

P (W_{1}, W_{2}, χ_{v}) := ∫_{K_{v}^{×}/F_{v}^{×}} b_{v}(π(ι(t))W_{1}, π(J)W_{2})χ_{v}(t)dt · \frac{L(1, τ_{K_{v}/F_{v}})}{ζ_{F_{v}}(1)}.

The above integral converges as χ_{v} is unitary ([Wal85, LEMME 7]).

2.3. A formula of Waldspurger. Let Λ(s, π_{K} ⊗ χ) be the completed Lfunction of π_{K} ⊗ χ given by

Λ(s, π_{K} ⊗ χ) := ∏_{v} L(s, π_{K_{v}} ⊗ χ_{v}) = L(s, π_{K} ⊗ χ) · ∏_{v|∞} L(s, π_{K_{v}} ⊗ χ_{v}).

It is well known that Λ(s, π_{K} ⊗ χ) converges absolutely for Re s ≫ 0 and has meromorphic continuation to all s ∈ C. Moreover, it satisfies the functional equation

Λ(s, π_{K} ⊗ χ) = ε(s, π_{K} ⊗ χ)Λ(1 − s, π_{K}^{∨} ⊗ χ^{−1}).

The global toric period integral for φ ∈ A(π) is defined by

P_{χ}(φ) := ∫_{K^{×}A^{×}\A_{K}^{×}} φ(ι(t))χ(t)dt.

The following proposition connects the global toric periods and central L-values of π_{K} ⊗ χ.

Proposition 2.1 (Waldspurger). Let φ_{1}, φ_{2} ∈ A(π) and let W_{φ_{1}}, W_{φ_{2}} be the associated global Whittaker functions. We suppose that W_{φ_{i}} = ∏_{v} W_{i,v}, where W_{i,v} ∈ W(π_{v}, ψ_{v}) such that W_{i,v}(1) = 1 for almost v (i = 1, 2). Then there exists a finite set S_{0} of places of F including all archimedean places such that for every finite set S ⊃ S_{0}, we have

P_{χ}(φ_{1})P_{χ}(φ_{2}) = Λ(\frac{1}{2}, π_{K} ⊗ χ) · ∏_{v∈S} \frac{1}{L(\frac{1}{2}, π_{K_{v}} ⊗ χ_{v})} · P (W_{1,v}, W_{2,v}, χ_{v}).

Proof. The proof is the combination of various formulae established in [Wal85]. We first recall some important local integrals. Let D = G×G. For each place v of F, let S_{v} = S(M_{2}(F_{v})) ⊗ S(F_{v}^{×}) and let D_{v} = G_{v}×G_{v}. Let r = r′×r′′ : G_{v}×D_{v} → End S_{v} be the Weil representation of G_{v}×D_{v} defined in [Wal85, §I.3 p.178] Let φ ∈ A(π) be an automorphic form in the automorphic realization of π. Recall that the global Whittaker function of φ is defined by

W_{φ}(g) = ∫_{F\A_{F}} φ((\begin{matrix}1 & x \\ 0 & 1 \end{matrix}) g)ψ(−x)dx.

Write W_{v} = W(π_{v}, ψ_{v}). We further assume that W_{φ} has the factorization W_{φ} = ∏_{v} W_{φ,v} ∈ ⊗′_{v}W_{v} such that W_{φ,v}(1) = 1 for almost v. For each v, let U : S_{v} → W_{v} ⊗ W_{v}, f_{v} → U_{f_{v}} be the G_{v}×G_{v}-equivaraint surjective morphism associated to W_{v} introduced in [Wal85, COROLLAIRE, p.187]. Define the following local integrals:

C(f_{v}) := ∫_{F_{v}^{×}} U_{f_{v}}((\begin{matrix}a \\ & 1 \end{matrix}) , (\begin{matrix}−a \\ & 1 \end{matrix})))ω^{−1}(a)d^{×}a,
B(f_{v}, 1) := ∫_{Z_{v}\G_{v}N_{v}} ∫_{F_{v}^{×}} W_{φ,v}(σ)r′(σ)f_{v}(x, x^{−2})dxdσ,
P (f_{v}, χ_{v}, \frac{1}{2}) := ∫_{F_{v}^{×}\K_{v}^{×}} B(r′′(ι(t), 1)f_{v}, 1)χ_{v}(t)dt.

The convergence and analytic properties of these local integrals are studied in [Wal85, LEMME 2, LEMME 3, LEMME 5]. Moreover, we have

B(f_{v}, 1) = C(f_{v}) · \frac{1}{ζ_{F_{v}}(1)}.

For each v, we take a special test function f_{v} ∈ S_{v} such that

(2.3)  U_{f_{v}} = W_{1,v} ⊗ π(J)W_{2,v}.

Note that f_{v} can be chosen to be the spherical test function f_{v}^{0} := 𝕀_{M_{2}(O_{Fv})} ⊗ 𝕀_{O_{Fv}^{×}} for all but finitely many v. With this particular choice of f_{v}, we have

(2.4)  \begin{aligned}P (f_{v}, χ_{v}, \frac{1}{2}) & = ∫_{F_{v}^{×}\K_{v}^{×}} C(r′′(ι(t), 1)f_{v})dt · \frac{1}{ζ_{F_{v}}(1)} \\ & = ∫_{F_{v}^{×}\K_{v}^{×}} b_{v}(π(ι(t))W_{1}, π(J)W_{2})χ_{v}(t)dt · \frac{1}{ζ_{F_{v}}(1)} \\ & = P (W_{1,v}, W_{2,v}, χ_{v}) · \frac{1}{L(1, τ_{K_{v}/F_{v}})}. \end{aligned}

Let S = ⊗S_{v} be the restricted product with respect to spherical test functions {f_{v}^{0}}_{v}. Define the theta kernel for f := ⊗f_{v} ∈ S by

θ_{f}(σ, g) := ∑_{(x,u)∈M_{2}(F)×F^{×}} r(σ, g)f(x, u) (σ ∈ G(A), g ∈ G(A)×G(A)),

and define the automorphic form θ(f, φ, g) on G(A)×G(A) by

θ(f, φ, g) = ∫_{G(F)\G(A)} φ(σ)θ_{f}(σ, g)dσ.

Note that according to (2.3), we have

θ(f, φ, g_{1}, g_{2}) = φ_{1}(g_{1})φ_{2}(g_{2}J).

We define the toric period integral P (f, χ) by

P (f, χ) := ∫_{[K^{×}A^{×}\A_{K}^{×}]^{2}} θ(f, φ, ι(t_{1}), ι(t_{2}))χ(t_{1})χ(\overline{t}_{2})dt_{1}dt_{2}.

By the relation Jι(t_{2})J = ι(\overline{t}_{2}) and the automorphy of φ_{2}, we find that

P (f, χ) = P_{χ}(φ_{1})P_{χ}(φ_{2}).

Let S_{0} be a finite set of places of F such that W_{φ,v}, W_{i,v} and f_{v} are spherical for all v ∉ S_{0}. From [Wal85, Prop. 4, p.196 and LEMME 7, p.219], we deduce the following formula for every finite set S ⊃ S_{0}:

P_{χ}(φ_{1})P_{χ}(φ_{2}) = Λ(\frac{1}{2}, π_{K} ⊗ χ) · ∏_{v∈S} P (f_{v}, χ_{v}, \frac{1}{2}) · \frac{L(1, τ_{K_{v}/F_{v}})}{L(\frac{1}{2}, π_{K_{v}} ⊗ χ)}.

We thus establish the desired formula in virtue of (2.4). □

3. Toric period integrals

3.1. Notation. Throughout we suppose that F is a totally real number field and K is a totally imaginary quadratic extension of F. We retain the notation in the introduction and §2.1. Let Σ be a fixed CM type of K. Let π be an irreducible automorphic cuspidal representation of GL_{2}(A). Let 𝔫 be the conductor of π. Suppose that π has infinity type k = ∑_{σ∈Σ} k_{σ}σ ∈ Z_{≥1}[Σ]. Let m = ∑_{σ} m_{σ}σ ∈ Z_{≥0}[Σ] and let χ be a Hecke character of infinity type (k/2 + m, −k/2 − m) such that χ|_{A^{×}} = ω^{−1}. Let h be the set of finite places of F. Recall that the set of infinite places of F is identified with the CM-type Σ. In this section, we will choose a special local Whittaker function at each place v of F in §3.6 and calculate their associated local toric period integrals in §3.7 and §3.8. Finally, we prove in §3.10 a non-vanishing modulo p result of these local Whittaker functions. This result plays an important role in the later application to the calculation of the μ-invariant. Let ℭ_{χ} (resp. 𝔠_{ω}) be the conductor of χ (resp. ω). Let 𝔠_{χ} = ℭ_{χ} ∩F. We further decompose 𝔫^{−} = 𝔫_{s}^{−}𝔫_{r}^{−}, where 𝔫_{s}^{−} is prime to 𝔠_{ω} and 𝔫_{r}^{−} is only divisible by prime factors of 𝔠_{ω}. Put

(3.1)  \begin{matrix}c_{v}(χ) = & inf \{n ∈ Z_{≥0} | χ = 1 on (1 + ϖ^{n}O_{E})^{×}\} , \\ m_{v}(χ, π) = & c_{v}(χ) − v(𝔫^{−}). \end{matrix}

It is clear that c_{v}(χ) = v(𝔠_{χ}). We put

A(χ) = {v ∈ h | K_{v} is a field, π_{v} is special and c_{v}(χ) = 0} .

Let p > 2 be a rational prime satisfying (ord). The assumption (ord) in particular implies that every prime factor of p in F splits in K. Let Σ_{p} be the p-adic places induced by Σ via ι_{p}. Thus Σ_{p} and its complex conjugation \overline{Σ}_{p} give a partition of the places of K above p. Let 𝔑 be the prime-to-p conductor of π_{K} ⊗ χ. We fix a decomposition 𝔑^{+} = 𝔉\overline{𝔉} such that (𝔉, \overline{𝔉}) = 1.

3.2. Galois representation attached to π. Let ρ_{p}(π) : G_{F} → GL_{2}(O_{L_{π}}) be the p-adic Galois representation associated to π as in the introduction. Let v ∤ p and let W_{F_{v}} be the local Weil group at v. Suppose that π_{v} = π(μ_{v}, ν_{v}) is a subquotient of the induced representations. By the local-global compatibility ([Car86], [Tay89] and [Jar97]), we have

(3.2)  ρ_{p}(π)|_{W_{Fv}} ≃ (\begin{matrix}μ_{v}^{−1}|·|^{\frac{1−kmx}{2}} & ∗ \\ 0 & ν_{v}^{−1}|·|^{\frac{1−kmx}{2}} \end{matrix} ) (k_{mx} = max_{σ} k_{σ}).

In particular, this implies that μ_{v}(ϖ_{F_{v}}) and ν_{v}(ϖ_{F_{v}}) are p-adic units in O_{L_{π}}^{×}.

3.3. Open compact subgroups. For each finite place v, we put

K_{v}^{0} = \{g = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ G_{v} | a, d ∈ O_{F_{v}}, b ∈ D_{F_{v}}^{−1}, c ∈ D_{F_{v}}, det g ∈ O_{F_{v}}^{×}\} ,

and for an integral ideal 𝔞 of F, we put

K_{v}^{0}(𝔞) = \{g = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ K_{v}^{0} | c ∈ 𝔞D_{F_{v}}, a − 1 ∈ 𝔞\} ,
U_{v}(𝔞) = {g ∈ GL_{2}(O_{F_{v}}) | g ≡ 1 (mod 𝔞)} .

Let K^{0} = ∏_{v∈h} K_{v}^{0} and U(𝔞) = ∏_{v∈h} U_{v}(𝔞) be open compact subgroups of GL_{2}(A_{f}).

3.4. The choices of ϑ and ς_{v}. We fix a finite idele d_{F} = (d_{F_{v}}) ∈ A_{F,f}^{×} such that d_{F_{v}} is a generator of the absolute different D_{F_{v}} at each finite place v and d_{F_{v}} = 1 for v ∤ D_{F}. Fixing an integral ideal 𝔯 ⊂ 𝔠_{χ}𝔫D_{K}^{2} of F, we choose ϑ ∈ K such that

(d1) Im σ(ϑ) > 0 for all σ ∈ Σ,

(d2) {1, d_{F_{v}}^{−1}ϑ} is an O_{F_{v}}-basis of O_{K_{v}} for all v | p𝔯, (d3) d_{F_{v}}^{−1}ϑ is a uniformizer of K_{v} for every v ramified in K.

The existence of such ϑ is guaranteed by strong approximation theorem. Then ϑ is a generator of K over F and determines an embedding K ↪ M_{2}(F) in (2.1). Let

δ = 2^{−1}(ϑ − \overline{ϑ}) ∈ K^{×}.

The condition (d2) allows us to choose d_{F_{v}} = 2δ at split v | p𝔯. For each finite place v, we also fix an O_{F_{v}}-basis {1, θ_{v}} of O_{K_{v}} such that θ_{v} = ϑ except for finitely many v and

θ_{v} = d_{F_{v}}^{−1}ϑ for v|p𝔯.

Write θ_{v} = a_{v}ϑ + b_{v} with a_{v}, b_{v} ∈ F_{v}. For every v split in K, we shall fix a place w of K above v throughout, and decompose K_{v} := K⊗_{F} F_{v} = F_{v}e\overline{w} ⊕F_{v}e_{w}, where e_{w} and e\overline{w} are the idempotents attached to w and \overline{w} respectively. If v|p𝔑^{+}, we further require that w|𝔉Σ_{p}, i.e. w|𝔉 or w ∈ Σ_{p}. We identify δ ∈ K_{w} = F_{v} and write ϑ_{v} = ϑ\overline{w}e\overline{w} + ϑ_{w}e_{w} for split v.

For each place v, we define ς_{v} ∈ GL_{2}(F_{v}) as follows:

(3.3)

ς_{v} = (\begin{matrix}Im σ(ϑ) & Re σ(ϑ) \\ 0 & 1 \end{matrix}) for v = σ ∈ Σ,
ς_{v} =(ϑ\overline{w} − ϑ_{w})^{−1} (\begin{matrix}d_{F_{v}}ϑ\overline{w} & ϑ_{w} \\ d_{F_{v}} & 1 \end{matrix}) for split v = w\overline{w},
ς_{v} = (\begin{matrix}a_{v} & 0 \\ −b_{v} & d_{F_{v}} \end{matrix}) for non-split finite v.

For t ∈ K_{v}, we put

ι_{ς_{v}}(t) := ς_{v}^{−1}ι(t)ς_{v}.

It is straightforward to verify that if v = σ ∈ Σ is archimedean and t = x+iy ∈ C^{×}, then

(3.4)  ι_{ς_{σ}}(t) = (\begin{matrix}x & −y \\ y & x \end{matrix}) ,

and if v = w\overline{w} is split and t = t_{1}e\overline{w} + t_{2}e_{w}, then

(3.5)  ι_{ς_{v}}(t) = (\begin{matrix}t_{1} \\ & t_{2} \end{matrix}) .

Moreover, for all finite places v

ι_{ς_{v}}(O_{K_{v}}^{×}) = ι_{ς_{v}}(K_{v}^{×}) ∩ K_{v}^{0}.

3.5. Running assumptions. In this section, we will assume Hypothesis A for (π, χ) and

(sf)  𝔫^{−} is square-free.

The assumption (sf) implies that π_{v} is an unramified special representation if v|𝔫_{s}^{−} and π_{v} is a ramified principal series if v|𝔫_{r}^{−}. In particular, for every place v inert or ramified in K, π_{v} is a sub-quotient of induced representations and the local L-function L(s, π_{v}) ≠ 1. We shall write π_{v} = π(μ_{v}, ν_{v}) such that L(s, π_{v}) = L(s, μ_{v}) for v|𝔫^{−}. By the local root number formulas [JL70, Prop. 3.5,Thm. 2.18], under the assumption (sf) Hypothesis A on the sign of local root numbers is equivalent to the following condition:

(R1)

Each v ∈ A(χ) is ramified in K and μ′_{v}χ_{v}(ϖ_{K_{v}}) = − |ϖ|^{\frac{1}{2}}
(μ′_{v} := μ_{v} ∘ N_{K_{v}/F_{v}}).

In what follows, we fix a place v of F. Let F = F_{v} and E = K_{v}. If v is finite, let O = O_{F} and let ϖ = ϖ_{F_{v}} and ϖ_{E} be uniformizers of O and O_{E} respectively. We shall suppress the subscript v and write π = π_{v}, χ = χ_{v}, ς = ς_{v} and ψ = ψ_{v}. For a ∈ F^{×}, we put

d(a) = (\begin{matrix}a \\ & 1 \end{matrix}) ∈ GL_{2}(F).

3.6. The choice of local toric Whittaker functions. If v is finite, we let W_{v}^{0} denote the new Whittaker function in W(π, ψ). In other words, W_{v}^{0} is the unique Whittaker function which is invariant by K_{v}^{0}(𝔫) and W_{v}^{0}(1) = 1. The existence and uniqueness of W_{v}^{0} are a consequence of the theory of new vectors [Cas73]. Now we introduce some special local Whittaker functions.

3.6.1. The archimedean case. Suppose that v = σ ∈ Σ is an archimedean place and F = R. Then π_{σ} = π(|·|^{\frac{kσ −1}{2}} , |·|^{\frac{1−kσ}{2}} sgn^{k_{σ}}) is the discrete series of minimal SO(2, R)-type k_{σ}. Let W_{k_{σ}} ∈ W(π_{v}, ψ_{v}) be the Whittaker function given by

(3.6)  W_{k_{σ}}(zd(a)κ_{θ}) = a^{\frac{kσ}{2}} e^{−2πa}𝕀_{R_{+}}(a) · e^{ik_{σ}θ} sgn(z)^{k_{σ}},

where z ∈ R^{×} and κ_{θ} = (\begin{matrix}cos θ & sin θ \\ − sin θ & cos θ \end{matrix}). Let V_{+} and V_{−} be the weight raising

and lowering differential operators in [JL70, p.165] given by

V_{±} = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) ⊗ 1 ± (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) ⊗ i ∈ Lie(GL_{2}(R)) ⊗_{R} C.

Define the normalized weight raising differential operator Ṽ_{+} by

(3.7)  Ṽ_{+} = \frac{1}{(−8π)} · V_{+}.

Then we have

(3.8)  Ṽ_{+}^{m_{σ}}W_{k_{σ}}(gκ_{θ}) = Ṽ_{+}^{m_{σ}}W_{k_{σ}}(g)e^{i(k_{σ}+2m_{σ})θ}.

3.6.2. The split case. Suppose that v = w\overline{w} is split with w|Σ_{p}𝔉 if v|p𝔑^{+}. We introduce some smooth functions a_{χ,v} on F^{×} in the Kirillov model K(π, ψ). Write χ = (χ_{w}, χ\overline{w}) : F^{×} ⊕F^{×} → C^{×}. If the local L-function L(s, π ⊗χ_{w}) = 1, we simply put

a_{χ,v}(a) = 𝕀_{O^{×}}(a)χ\overline{w}(a^{−1}).

Suppose that L(s, π ⊗ χ_{w}) ≠ 1. Then π = π(μ, ν) is a principal series or π = π(μ, ν) is special with μν^{−1} = |·| and μχ\overline{w} is unramified. If π ⊗ χ_{w} is unramified, we set

a_{χ,v}(a) = 𝕀_{O}(a) · χ−1 ; \overline{w}|·|^{\frac{1}{2}} (a) ∑_{i+j=v(a), i,j≥0} μχ\overline{w}(ϖ^{i})νχ\overline{w}(ϖ^{j}).

If μ_{i}χ_{w} is unramified and μ_{j}χ_{w} is ramified for {μ_{1}, μ_{2}} = {μ, ν}, we set

a_{χ,v}(a) = μ_{i}|·|^{\frac{1}{2}} (a)𝕀_{O}(a).

If π is special, we set

a_{χ,v}(a) = μ|·|^{\frac{1}{2}} (a)𝕀_{O}(a).

These functions a_{χ,v} indeed belong to the Kirillov model K(π, ψ) in virtue of the description of the Kirillov models [Jac72, Lemma 14.3]. For each ξ ∈ K(π, ψ), by the isomorphism (1.1) we denote by W_{ξ} ∈ W(π, ψ) the unique Whittaker function such that W_{ξ}(d(a)) = ξ(a). We put

W_{χ,v} := W_{a_{χ,v}}.

It follows from the choice of W_{χ,v} that

W_{χϕ,v} = W_{χ,v} if ϕ : E^{×} → C^{×} is unramified.

Recall that the zeta integral Ψ(s, W, χ\overline{w}) for W ∈ W(π, ψ) is defined by

Ψ(s, W, χ\overline{w}) := ∫_{F^{×}} W (d(a))χ\overline{w}(a) |a|^{s− \frac{1}{2}} d^{×}a.

Then the zeta integral for W_{χ,v} satisfies the following equation:

(3.9)  Ψ(s, W_{χ,v}, χ\overline{w}) = L(s, π ⊗ χ\overline{w}) |D_{F} |^{\frac{1}{2}} (vol(O_{F}^{×}, d^{×}a) = |D_{F} |^{\frac{1}{2}} ).

Suppose that v = w\overline{w} with w ∈ Σ_{p}. We define some p-modified Whittaker functions as follows. For each u ∈ O_{F}^{×}, we put

a_{u,v}(a) := 𝕀_{u(1+ϖO)}(a)χ\overline{w}(a^{−1}) and W_{χ,u,v} = W_{a_{u,v}}.

Let a_{χ,v}^{♭}(a) := 𝕀_{O^{×}}(a)χ_{w}(a^{−1}) and let W_{χ,v}^{♭} be the p-modified Whittaker function given by

(3.10)  W_{χ,v}^{♭} := W_{a_{χ,v}^{♭}} = ∑_{u∈U_{v}} W_{χ,u,v},

where U_{v} is the torsion subgroup of O^{×}. It is easy to verify that

(3.11)  Ψ(s, W_{χ,v}^{♭}, χ_{w}) = 1 ; π((\begin{matrix}a & b \\ 0 & d \end{matrix}))W_{χ,v}^{♭} =χ−1 ; \overline{w}(a)χ_{w}^{−1}(d)W_{χ,v}^{♭}

for a, d ∈ O^{×}, b ∈ D_{F}^{−1}.

3.6.3. The inert and ramified case. Suppose that v is an inert or ramified finite place. Then E is a non-archimedean local field. Define the operators R_{v} and P_{χ,ς} on W ∈ W(π, ψ) by

R_{v}W (g) :=W (g (\begin{matrix}1 \\ & ϖ \end{matrix})),
P_{χ,ς}W (g) :=v_{E}^{−1} · ∫_{E^{×}/F^{×}} π(ι_{ς}(t))W (g)χ(t)dt.
=v_{E}^{−1} ∫_{E^{×}/F^{×}} W (gς^{−1}ι(t)ς)χ(t)dt.

Note that

v_{E} = vol(E^{×}/F^{×}, dt) = e_{v} · |D_{E}|E^{\frac{1}{2}} |D_{F} |^{− \frac{1}{2}} , e_{v} = \begin{cases}1 & if v is inert, \\ 2 & if v is ramified. \end{cases}

We define the Whittaker function W_{χ,v} by

(3.12)  W_{χ,v} := P_{χ,ς}R_{v}^{m_{v}(χ,π)}W_{v}^{0}.

3.6.4. Define the subgroup T_{v} of E^{×} by

T_{v} = \begin{cases}O_{E}^{×}F^{×} & if v is split, \\ E^{×} & if v is non-split. \end{cases}

Then T_{v} = \{x ∈ E | x/\overline{x} ∈ O_{E}^{×}\} if v is finite.

Definition 3.1 (Toric Whittaker functions). We say that W ∈ W(π, ψ) is a toric Whittaker function of character χ if

π(ι_{ς}(t))W = χ^{−1}(t) · W for all t ∈ T_{v}.

Lemma 3.2. The Whittaker functions W_{χ,v} chosen as above are toric. To be precise, we have

(1) Ṽ_{+}^{m_{σ}}W_{k_{σ}} is a toric Whittaker function of the character χ_{σ} : C^{×} → C^{×}, z ↦ z^{k_{σ}+m_{σ}} \overline{z}^{−m_{σ}} |z\overline{z}|^{−k_{σ}/2}.

(2) If v is finite, then W_{χ,v} are toric Whittaker functions of character χ_{v}.

(3) If v|p, then W_{χ,v}^{♭} is toric, and for u ∈ O_{F}^{×}

π(ι_{ς}(t))W_{χ,u,v} = χ^{−1}(t)W_{χ,u.t^{1−c},v},

where u.t^{1−c} := ut_{w}t−1 ; \overline{w}, t = t\overline{w}e\overline{w} + t_{w}e_{w} ∈ O_{E}^{×} with w ∈ Σ_{p},

Proof. It follows immediately from the definitions of these Whittaker functions together with (3.8), (3.4) and (3.5). □

3.7. Local toric period integrals (I).

3.7.1. Define the local toric period integral for W ∈ W(π, ψ) by

P (W, χ) :=P (W, W, χ)
= ∫_{E^{×}/F^{×}} b_{v}(π(ι(t))W, π(J)W)χ(t)dt · \frac{L(1, τ_{E/F})}{ζ_{F} (1)}.

The main task of this section is to evaluate P (π(ς)W_{χ,v}, χ). We first treat the archimedean and split cases.

3.7.2. The archimedean case. Suppose v = σ ∈ Σ →^{∼} Hom(F, R) is an archimedean place.

Proposition 3.3. We have

P (π(ς)Ṽ_{+}^{m_{σ}}W_{k_{σ}}, χ) = 2^{3} · \frac{Γ(m_{σ} + 1)Γ(k_{σ} + m_{σ})}{(4π)^{k_{σ}+1+2m_{σ}}}.

Proof. Introduce the Hermitian inner product on W(π, ψ) defined by

⟨W_{1}, W_{2}⟩ := b_{v}(W_{1}, c(W_{2})), where c(W_{2})(g) := \overline{W ((\begin{matrix}−1 \\ & 1 \end{matrix}) g)}ω(det g).

Write k = k_{σ} and m = m_{σ}. It is clear that

⟨W_{k}, W_{k}⟩ = (4π)^{−k}Γ(k).

Documenta Mathematica 19 (2014) 709–767

Since c(V_{+}^{m}W_{k}) and π((\begin{matrix}−1 \\ & 1 \end{matrix}))V_{+}^{m}W_{k} are both nonzero Whittaker functions

of weight −k − 2m, there exists some constant γ such that

π((\begin{matrix}−1 \\ & 1 \end{matrix}))V_{+}^{m}W_{k} = γ · c(V_{+}^{m}W_{k}) ⟺ V_{+}^{m}W_{k}(d(a)) = γ · \overline{V_{+}^{m}W_{k}(d(a))}

for all a ∈ R_{+}. Let h_{m}(x) := V_{+}^{m}W_{k}(d(x)). Then h_{0}(x) = W_{k}(d(x)) is a real-valued function in view of the definition (3.6). A simple calculation shows that

h_{m+1} = 2x\frac{dh_{m}}{dx} + (k + 2m − 4πx)h_{m},

so by induction h_{m}(x) takes value in R (cf. [JL70, p.189]). This implies that γ = 1. We thus have

b_{v}(π(ς)V_{+}^{m}W_{k}, π(Jς)V_{+}^{m}W_{k}) = ⟨V_{+}^{m}W_{k}, V_{+}^{m}W_{k}⟩ (ς^{−1}Jς = (\begin{matrix}−1 \\ & 1 \end{matrix})).

To evaluate ⟨V_{+}^{m}W_{k}, V_{+}^{m}W_{k}⟩, note that by [JL70, p.166] we have

(3.13)  V_{−}^{m}V_{+}^{m}W_{k} = (−4)^{m} \frac{Γ(k + m)Γ(m + 1)}{Γ(k)} · W_{k},

and hence we find that

⟨V_{+}^{m}W_{k}, V_{+}^{m}W_{k}⟩ =(−1)^{m}⟨W_{k}, V_{−}^{m}V_{+}^{m}W_{k}⟩
=4^{m} \frac{Γ(k + m)}{Γ(k)}Γ(m + 1)⟨W_{k}, W_{k}⟩
=4^{m}(4π)^{−k} · Γ(k + m)Γ(m + 1).

Recall that dt = 2π^{−1}dθ witht = e^{iθ}, so vol(C^{×}/R^{×}, dt) = 2π^{−1} · π = 2. Combining these with Lemma 3.2 (1), we find that

P (π(ς)Ṽ_{+}^{m}W_{k}, χ) = 2 · (−8π)^{−2m} · b_{v}(π(ς)V_{+}^{m}W_{k}, π(Jς)V_{+}^{m}W_{k}) · \frac{ζ_{R}(2)}{ζ_{R}(1)}
= 2^{3}(4π)^{−2m−1}4^{−m} · ⟨V_{+}^{m}W_{k}, V_{+}^{m}W_{k}⟩
= 2^{3} · (4π)^{−k−2m−1}Γ(k + m)Γ(m + 1).  □

3.7.3. The split case. Suppose that v = w\overline{w} is a finite place split in E. Recall that we have assumed w|Σ_{p}𝔉 if v|p𝔑^{+}.

Lemma 3.4. We have

P (π(ς)W, χ)
=Ψ(\frac{1}{2}, W, χ\overline{w})^{2} · \frac{L(\frac{1}{2}, π ⊗ χ_{w})}{L(\frac{1}{2}, π ⊗ χ\overline{w})} · ε(\frac{1}{2}, π ⊗ χ\overline{w}, ψ) · ω^{−1}χ−2 ; \overline{w}(−d_{F})ω(det ς).

Proof. Let Ŵ(g) := W (g (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}))ω^{−1}(det g). By [JL70, Thm. 2.18 (iv)],

we have the local functional equation:

\frac{Ψ(1 − s, Ŵ, χ−1 ; \overline{w})}{L(1 − s, π^{∨} ⊗ χ−1 ; \overline{w})} = ε(s, π ⊗ χ\overline{w}, ψ) · \frac{Ψ(s, W, χ\overline{w})}{L(s, π ⊗ χ\overline{w})}.

We note that

ς_{v}^{−1}Jς_{v} = (\begin{matrix}0 & d_{F}^{−1} \\ d_{F} & 0 \end{matrix}) .

A straightforward computation shows that

P (π(ς)W, χ)
=ω(det ς) ∫_{F^{×}} ∫_{F^{×}} W (d(at_{1}))W (d(−a) (\begin{matrix}0 & d_{F}^{−1} \\ d_{F} & 0 \end{matrix}))χ\overline{w}(t_{1})ω^{−1}(a)d^{×}adt_{1}
=ω(− det ς)ω^{−1}χ−2 ; \overline{w}(d_{F}) · Ψ(\frac{1}{2}, W, χ\overline{w})Ψ(\frac{1}{2}, Ŵ, χ−1 ; \overline{w})
=ω(det ς)ω^{−1}χ−2 ; \overline{w}(−d_{F})Ψ(\frac{1}{2}, W, χ\overline{w})^{2} · ε(\frac{1}{2}, π ⊗ χ\overline{w}, ψ) · \frac{L(\frac{1}{2}, π^{∨} ⊗ χ−1 ; \overline{w})}{L(\frac{1}{2}, π ⊗ χ\overline{w})}.

The lemma thus follows. □

Proposition 3.5. We have

\frac{1}{L(\frac{1}{2}, π_{E} ⊗ χ)} · P (π(ς)W_{χ,v}, χ)

= |D_{F} | · \begin{cases}ε(\frac{1}{2}, π ⊗ χ\overline{w}, ψ)χ−2 ; \overline{w}(−d_{F}) & if v | 𝔑^{+}, \\ ω(det ς) & if v ∤ 𝔑^{+}. \end{cases}

If v = w\overline{w} with w ∈ Σ_{p}, then

\frac{1}{L(\frac{1}{2}, π_{E} ⊗ χ)} · P (π(ς)W_{χ,v}^{♭}, χ) =\frac{ε(\frac{1}{2}, π ⊗ χ\overline{w}, ψ)}{L(\frac{1}{2}, π ⊗ χ\overline{w})^{2}} · χ−2 ; \overline{w}(d_{F}) |D_{F} | .

Proof. The proposition follows immediately from Lemma 3.4. (3.9) and (3.11) combined with the equations det ς = d_{F} if v|p𝔑^{+} and

ε(\frac{1}{2}, π ⊗ χ\overline{w}, ψ) · ω^{−1}χ−2 ; \overline{w}(−d_{F}) = 1 if v ∤ 𝔑^{+}.  □

3.8. Local toric period integrals (II). In this subsection, we treat the case v is inert or ramified. A large part of the computation in this subsection is inspired by [Mur08]. Put

w = (\begin{matrix}0 & −d_{F}^{−1} \\ d_{F} & 0 \end{matrix});
K^{0}(ϖ) := \{g = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ K_{v}^{0} | a − 1 ∈ ϖO, c ∈ ϖD_{F} \} .

Let θ = θ_{v} ∈ O_{E} be the element chosen in §3.4 and write W^{0} for the new local Whittaker function W_{v}^{0} at v. Recall that {1, θ} is an O-basis of O_{E} and θ is a uniformizer if E/F is ramified.

3.8.1. We prepare some elementary lemmas.

Lemma 3.6. Suppose that v|𝔯. Let m be a non-negative integer and let

B^{1}(O) = \{(\begin{matrix}1 & x \\ 0 & d \end{matrix}) | x ∈ D_{F}^{−1}, d ∈ O^{×}\} ,
N(D_{F}^{−1}) = \{(\begin{matrix}1 & x \\ 0 & 1 \end{matrix}) | x ∈ D_{F}^{−1}\} .

If y ∈ ϖ^{m+1}O, then we have

d(ϖ^{m})ι_{ς}(1 + yθ)d(ϖ^{−m}) ∈ K^{0}(ϖ) (d(a) = (\begin{matrix}a \\ & 1 \end{matrix})).

If y ∈ ϖ^{r}O^{×} and 0 ≤ r ≤ m, then

d(ϖ^{m})ι_{ς}(1 + yθ)d(ϖ^{−m}) ∈ N(D_{F}^{−1}) (\begin{matrix}ϖ^{m−r} \\ & yϖ^{−m} \end{matrix}) wB^{1}(O).

If y ∈ ϖO, then

d(ϖ^{m})ι_{ς}(y + θ)d(ϖ^{−m}) ∈ N(D_{F}^{−1}) (\begin{matrix}ϖ^{m+e_{v}−1} \\ & ϖ^{−m} \end{matrix}) wB^{1}(O).

Proof. Recall that if v|𝔯, then θ = d_{F}^{−1}ϑ, ς = (\begin{matrix}d_{F} \\ & d_{F}^{−1} \end{matrix}), and hence

ι_{ς}(x + yθ) = (\begin{matrix}x + yT(θ) & yd_{F}^{−1}N(θ) \\ yd_{F} & x \end{matrix}) (x, y ∈ F).

Then the proof is a straightforward calculation, so we omit the details. □

Lemma 3.7. Suppose that χ|_{F^{×}} is trivial on 1 + ϖO. For each non-negative integer r, we set

X_{r} := ∫_{ϖ^{r}O} χ(1 + yθ)d′y,

where d′y is the Haar measure on O such that vol(O, d′y) =

L(1, τ_{E/F}) |D_{E}|E^{\frac{1}{2}} |D_{F} |^{− \frac{1}{2}} . Then X_{r} = 0 if c_{v}(χ) > 1 and 0 < r < c_{v}(χ)

and X_{r} = |ϖ^{r}| · L(1, τ_{E/F}) |D_{E}|E^{\frac{1}{2}} |D_{F} |^{− \frac{1}{2}} if r ≥ c_{v}(χ).

Proof. Let Q_{r} := 1 + ϖ^{r}O_{E}/1 + ϖ^{r}O. If 0 < r < c_{v}(χ), then χ is a non-trivial character on the group Q_{r}. Note that we have a bijection ϖ^{r}O →^{∼} Q_{r}, y ↦ 1 + yθ and the pull-back of the quotient measure dt on Q_{r} is d′y. Therefore, we have

X_{r} = ∫_{Q_{r}} χ(t)dt = \begin{cases}0 & if 0 < r < c_{v}(χ) \\ vol(ϖ^{r}O, d′y) & if r ≥ c_{v}(χ). \end{cases}

This finishes the proof. □

Define the matrix coefficient m^{0} : GL_{2}(F) → C by

m^{0}(g) :=b_{v}(π(g)W^{0}, π((\begin{matrix}−1 \\ & 1 \end{matrix}))W^{0})
= ∫_{F^{×}} W^{0}(d(a)g)W^{0}(d(a))ω^{−1}(a)d^{×}a.

Since W^{0} is invariant by K^{0}(ϖ), m^{0}(g) only depends on the double coset K^{0}(ϖ)gK^{0}(ϖ) by (2.2). Put

m = m_{v}(χ, π) = c_{v}(χ) − v(𝔫^{−}) ≥ −1.

We set (3.14)

P^{∗}(π(ς)R_{v}^{m}W^{0}, χ) := P (π(ς)R_{v}^{m}W^{0}, χ) · \frac{ζ_{F} (1)}{L(1, τ_{E/F})}ω(ϖ^{−m} det ς^{−1})

= ∫_{E^{×}/F^{×}} b_{v}(R_{v}^{−m}π(ι_{ς}(t))R_{v}^{m}W^{0}, π((\begin{matrix}−1 \\ & 1 \end{matrix}))W^{0})χ(t)dt

= ∫_{E^{×}/F^{×}} m^{0}(d(ϖ^{m})ι_{ς}(t)d(ϖ^{−m}))χ(t)dt.

Here we have used the fact that m + v(T(θ)) ≥ 0 in the second equality. It follows immediately from the definition of the projector P_{χ,ς} that

(3.15)  \begin{matrix}P (π(ς)W_{χ,v}, χ) = & P (π(ς)P_{χ,ς}R_{v}^{m}W^{0}, χ) \\ = & P^{∗}(π(ς)R_{v}^{m}W^{0}, χ) · \frac{ω(ϖ^{m} det ς)L(1, τ_{E/F})}{ζ_{F} (1)}. \end{matrix}

Using the decomposition

E^{×} = F^{×}(1 + Oθ) ⊔ F^{×}(ϖO + θ)

and Lemma 3.6, we find that (3.16)

P^{∗}(π(ς)R_{v}^{m}W^{0}, χ)
= ∫_{O} χ(1 + yθ)m^{0}(d(ϖ^{m})ι_{ς}(1 + yθ)d(ϖ^{−m}))d′y
+ ∫_{ϖO} χ(x + θ)m^{0}(d(ϖ^{m})ι_{ς}(y + θ)d(ϖ^{−m})) |y + θ|_{E}^{−1} d′y
=X_{m+1} · m^{0}(1) + ∑_{r=0}^{m} ∫_{ϖ^{r}O^{×}} χ(1 + yθ)ω(ϖ^{−m}y)d′y · m^{0}(d(ϖ^{2(m−r)})w)
+ Y_{0} · ω(ϖ^{−m})m^{0}(d(ϖ^{2m+e_{v}−1})w),

where

Y_{0} := ∫_{ϖO} χ(y + θ)d′y · |ϖ|^{1−e_{v}} .

In what follows, we use Lemma 3.7 and (3.16) to calculate P (π(ς)W_{χ,v}, χ).

3.8.2. The case v ∤ 𝔫_{r}^{−}. Suppose that v ∤ 𝔫_{r}^{−}, i.e. the central character ω is unramified. Then (sf) implies that π is either an unramified principal series or an unramified special representation.

Proposition 3.8. Suppose that π is an unramified principal series. Then

\frac{1}{L(\frac{1}{2}, π_{E} ⊗ χ)} · P (π(ς)W_{χ,v}, χ)

=ω(ϖ^{m}) |ϖ^{c_{v}(χ)}| |D_{E}|E^{\frac{1}{2}} · \begin{cases}ω(det ς) & if c_{v}(χ) = 0, \\ L(1, τ_{E/F})^{2} & if c_{v}(χ) > 0. \end{cases}

Proof. Since π is unramified, ω is unramified and m = c_{v}(χ). Write π = π(μ, ν) and let α = μ(ϖ) and β = ν(ϖ). The matrix coefficient m^{0} is a spherical function on GL_{2}(F) in the sense of [Car79, Definition 4.1, p.150], and m^{0}(g) only depends on the double coset K_{v}^{0}gK_{v}^{0}. By a standard computation (cf. [Wal85, LEMME 14, p.226)]), we obtain

m^{0}(1) =\frac{ζ_{F} (1)L(1, Ad π)}{ζ_{F} (2)} · |D_{F} |^{\frac{1}{2}} = \frac{(1 + |ϖ|)ζ_{F} (1)}{(1 − αβ^{−1} |ϖ|)(1 − α^{−1}β |ϖ|)} · |D_{F}|^{\frac{1}{2}} ; (3.17)

(3.18)  m^{0}(d(ϖ)) =\frac{|ϖ|^{\frac{1}{2}}}{1 + |ϖ|} · (α + β) · m^{0}(1);
(3.19)  m^{0}(d(ϖ^{2})) =\frac{|ϖ|}{1 + |ϖ|} · (α^{2} + β^{2} + (1 − |ϖ|)αβ) · m^{0}(1).

If v is inert and m = 0, then

ω(det ς^{−1})P (π(ς)W^{0}, χ) =m^{0}(1) · \frac{L(1, τ_{E/F})}{ζ_{F} (1)} · |D_{E}|E^{\frac{1}{2}} |D_{F} |^{− \frac{1}{2}}
=\frac{1}{(1 − αβ^{−1} |ϖ|)(1 − α^{−1}β |ϖ|)} · |D_{E}|E^{\frac{1}{2}}
=L(\frac{1}{2}, π_{E} ⊗ χ) · |D_{E}|E^{\frac{1}{2}} .

Suppose that either v is ramified or m > 0 (so v|𝔯 and det ς = 1). Then we deduce from (3.16) that

(3.20)  \begin{matrix}P^{∗}(π(ς)R_{v}^{m}W^{0}, χ) \\ = & X_{m} · m^{0}(1) + ∑_{r=0}^{m−1}(X_{r} − X_{r+1})ω(ϖ^{r−m}) · m^{0}(d(ϖ^{2(m−r)})) \\ + Y_{0} · ω(ϖ^{−m})m^{0}(d(ϖ^{2m+e_{v}−1})). \end{matrix}

Documenta Mathematica 19 (2014) 709–767

If v is ramified and m = 0, then X_{0} = |D_{E}|E^{\frac{1}{2}} |D_{F} |^{− \frac{1}{2}} and Y_{0} = χ(ϖ_{E}) |D_{E}|E^{\frac{1}{2}} |D_{F} |^{− \frac{1}{2}} . By (3.20), we find that

P (π(ς)W^{0}, χ)
=(m^{0}(1) + χ(ϖ_{E})m^{0}(d(ϖ)))\frac{L(1, τ_{E/F})}{ζ_{F} (1)} |D_{E}|E^{\frac{1}{2}} |D_{F} |^{− \frac{1}{2}}

=(1 + \frac{α + β}{1 + |ϖ|} · |ϖ|^{\frac{1}{2}} χ(ϖ_{E})) · m^{0}(1) · \frac{|D_{E}|E^{\frac{1}{2}} |D_{F} |^{− \frac{1}{2}}}{ζ_{F} (1)} (by (3.18))

=\frac{(1 + χ(ϖ_{E})α |ϖ|^{\frac{1}{2}} )(1 + χ(ϖ_{E})β |ϖ|^{\frac{1}{2}} )}{1 + |ϖ|} · m^{0}(1) · \frac{|D_{E}|E^{\frac{1}{2}} |D_{F} |^{− \frac{1}{2}}}{ζ_{F} (1)}
= |D_{E}|E^{\frac{1}{2}} · L(\frac{1}{2}, π_{E} ⊗ χ).

Suppose that m > 0. Note that since χ|_{O^{×}} = 1, Y_{0} = −X_{0} if v is inert and Y_{0} = X_{0} = 0 if v is ramified. Combining with Lemma 3.7, (3.19) and (3.20), we find that

P (π(ς)R_{v}^{m}W^{0}, χ) =X_{m} · (m^{0}(1) − ω(ϖ^{−1})m^{0}(d(ϖ^{2})) · ω(ϖ^{−m})\frac{L(1, τ_{E/F})}{ζ_{F} (1)}
=ω(ϖ^{m}) |ϖ^{m}| · \frac{(1 − αβ^{−1} |ϖ|)(1 − α^{−1}β |ϖ|)}{1 + |ϖ|} · m^{0}(1)

× \frac{L(1, τ_{E/F})^{2}}{ζ_{F} (1)} |D_{E}|E^{\frac{1}{2}} |D_{F} |^{− \frac{1}{2}}

=ω(ϖ^{m}) |ϖ^{m}| |D_{E}|E^{\frac{1}{2}} · L(1, τ_{E/F})^{2}.

The proposition follows immediately. □

Proposition 3.9. Suppose that π is an unramified special representation. Then

\frac{1}{L(\frac{1}{2}, π_{E} ⊗ χ)} · P (π(ς)W_{χ,v}, χ)

=ω(ϖ^{m}) |ϖ^{c_{v}(χ)}| |D_{E}|E^{\frac{1}{2}} · \begin{cases}L(1, τ_{E/F})^{2} & if c_{v}(χ) > 0, \\ 2 & if v is ramified and c_{v}(χ) = 0. \end{cases}

Proof. Suppose that v|𝔫_{s}^{−}. Then m = m_{v}(χ, π) = c_{v}(χ) − 1. Recall that π = π(μ, ν) is a special representation with a unramified character μ and μν^{−1} = |·|. We have

W^{0}(d(a)) =μ(a) |a|^{\frac{1}{2}} 𝕀_{O}(a),
W^{0}(d(a)w) = − μ(a) |a|^{\frac{1}{2}} |ϖ| 𝕀_{ϖ^{−1}O}(a)

(cf. [Sch02, Eq.(54)]). With the above formulas, we obtain by a direct computation that

m^{0}(1) =\frac{|D_{F}|^{\frac{1}{2}}}{1 − |ϖ|^{2}} ; m^{0}(w) = (− |ϖ|) · m^{0}(1) ;
m^{0}((\begin{matrix}1 \\ & ϖ \end{matrix}) w) =(−μ(ϖ) |ϖ|^{− \frac{1}{2}} ) · m^{0}(1).

If c_{v}(χ) > 0, then it follows from (3.16) and Lemma 3.7 that

P (π(ς)R_{v}^{m}W^{0}, χ) =X_{m+1} · (m^{0}(1) − m^{0}(w)) · ω(ϖ^{m})\frac{L(1, τ_{E/F})}{ζ_{F} (1)}
=ω(ϖ^{m}) |ϖ^{m+1}| · |D_{E}|E^{\frac{1}{2}} L(1, τ_{E/F})^{2}.

If c_{v}(χ) = 0 (m = −1), then v is ramified, X_{0} = |D_{E}|E^{\frac{1}{2}} |D_{F} |^{− \frac{1}{2}} ,, Y_{0} =

χ(ϖ_{E}) |D_{E}|E^{\frac{1}{2}} |D_{F} |^{− \frac{1}{2}} , and

P (π(ς)R_{v}^{m}W^{0}, χ)

= (X_{0} · m^{0}(1) + Y_{0} · m^{0}((\begin{matrix}1 \\ & ϖ \end{matrix}) w)) ω(ϖ^{−1}) · \frac{L(1, τ_{E/F})}{ζ_{F} (1)}
=(1 − μ(ϖ)χ(ϖ_{E}) |ϖ|^{− \frac{1}{2}} ) |D_{E}|E^{\frac{1}{2}} |D_{F} |^{− \frac{1}{2}} · m^{0}(1)(1 − |ϖ|)ω(ϖ^{−1})
=\frac{2 |D_{E}|E^{\frac{1}{2}} ω(ϖ^{−1})}{1 + |ϖ|} (by (R1))
=2 |D_{E}|E^{\frac{1}{2}} ω(ϖ^{−1}) · L(\frac{1}{2}, π_{E} ⊗ χ).  □

3.8.3. The case v|𝔫_{r}^{−}. We consider the case π is a ramified principal series. Recall that (sf) suggests that π = π(μ, ν), where μ is unramified and ν is ramified, and the conductor a(ν) = a(ω) = 1. Since χ|_{F^{×}} = ω^{−1}, we must have m = c_{v}(χ) − 1 ≥ 0. Let δ_{v} := θ − \overline{θ}. Let D_{E/F} be the discriminant of E/F. We begin with a lemma.

Lemma 3.10. Suppose that χ|_{O^{×}} ≠ 1 and χ|_{1+ϖO} = 1. Then

∫_{ϖ^{−m}O} χ(y + θ)d′y = χ(δ_{v}) |δ_{v}|E^{\frac{1}{2}} · \frac{ε(0, χ^{−1}, ψ_{E})}{ε(−1, ω, ψ)} · L(1, τ_{E/F}) |D_{E/F} |^{\frac{1}{2}} .

Proof. By [HKS96, Prop. 8.2], we have

∫_{F} χ(y + 2^{−1}δ_{v})dy := ∫_{F} χ(y + 2^{−1}δ_{v}) |y + 2^{−1}δ_{v}|_{E}^{−s− \frac{1}{2}} dy|_{s=− \frac{1}{2}}
=χ(δ_{v}) |δ_{v}|E^{\frac{1}{2}} · \frac{ε(0, χ^{−1}, ψ_{E})}{ε(−1, ω, ψ)}.

By the assumption, for all r ≥ m + 1 we have

∫_{ϖ^{−r}O^{×}} χ(y + θ)dy = χ(ϖ^{−r}) · ∫_{O^{×}} χ(y)dy = 0.

Thus

∫_{ϖ^{−m}O} χ(y + θ)dy = \underrightarrow{lim}_{r} ∫_{ϖ^{−r}O} χ(y + θ)dy = ∫_{F} χ(y + 2^{−1}δ_{v})dy.

The lemma follows from the fact that

d′y = L(1, τ_{E/F}) |D_{E/F} |^{\frac{1}{2}} · dy.  □

Proposition 3.11. We have

\frac{1}{L(\frac{1}{2}, π_{E} ⊗ χ)} · P (π(ς)W_{χ,v}, χ) = |ϖ^{c_{v}(χ)}| |D_{E}|E^{\frac{1}{2}} χ(δ_{v}d_{F}^{−1}) |δ_{v}|E^{\frac{1}{2}} · ε(0, χ, ψ_{E})
× L(1, τ_{E/F})^{2} · n_{v}^{2},

where n_{v} is given by

(3.21)  n_{v} := \frac{μ(ϖ) |ϖ|^{m/2} |D_{F} |^{\frac{1}{4}}}{ε(0, ω, ψ)} ∈ \overline{Z}_{(p)}^{×}.

Proof. We first recall that if ξ : F^{×} → C^{×} is a character of conductor a(ξ), then

ε(s, ξ, ψ) = ξ(c) |c|^{s− \frac{1}{2}} · ( ∑_{a∈O^{×}/(1+ϖ^{a(ξ)O})} ξ^{−1}(u)ψ(u/c)) (c = d_{F} ϖ^{a(ξ)}).

By the equation ε(s, ξ, ψ)ε(1 − s, ξ^{−1}, ψ) = ξ(−1) (cf. [Sch02, Eq.(7)]), we see that ε(0, ξ, ψ) belongs to \overline{Z}_{(p)}^{×} whenever v does not divide p and ξ takes values in \overline{Z}_{(p)}. This shows that n_{v} is a p-adic unit by the discussion in §3.2. We proceed to prove the toric integral. We have

W^{0}(d(a)) =ν|·|^{\frac{1}{2}} (a)𝕀_{O}(a),
W^{0}(d(a)w) =μ|·|^{\frac{1}{2}} (a)𝕀_{ϖ^{−1}O}(a) · \frac{ω(d_{F})μ(ϖ^{2})}{ε(0, ω, ψ)}

(cf. [Sch02, Eq.(50) and (51)]). A simple calculation shows that

m^{0}(1) = 0, m^{0}(w) = \frac{ω(d_{F})μ(ϖ^{2})}{ε(0, ω, ψ) · (1 − |ϖ|)} · |D_{F}|^{\frac{1}{2}} .

It is not difficult to show that if v is ramified, then

Y_{0} = ∫_{ϖO} χ(y + θ)d′y = 0,

and that if m = c_{v}(χ) − 1 > 0, then

∫_{ϖ^{r}O^{×}} χ(y^{−1} + θ)d′y = 0 for 0 < r < m and
∫_{O} χ(y + θ)d′y = 0.

From the above equations, we find that

P^{∗}(π(ς)R_{v}^{m}W^{0}, χ)
=X_{m+1} · m^{0}(1) + ∑_{r=0}^{m} ∫_{ϖ^{r}O^{×}} χ(y^{−1} + θ)d′y · ω(ϖ^{−m})m^{0}(d(ϖ^{2m−2r})w)
+ Y_{0} · ω(ϖ^{−m})m^{0}(d(ϖ^{2m+e_{v}−1})w)
= ∫_{ϖ^{−m}O} χ(y + θ)d′y · ω(ϖ^{−m}) |ϖ^{2m}| m^{0}(w).

By Lemma 3.10, we obtain

P (π(ς)R_{v}^{m}W^{0}, χ)

=P^{∗}(π(ς)R_{v}^{m}W^{0}, χ) · ω(ϖ^{m})\frac{L(1, τ_{E/F})}{ζ_{F} (1)}
= |ϖ^{2m}| χ(δ_{v}) |δ_{v}|E^{\frac{1}{2}} |D_{E}|E^{\frac{1}{2}} |D_{F} |^{− \frac{1}{2}} \frac{ε(0, χ^{−1}, ψ_{E})}{ε(−1, ω, ψ)} · \frac{μ(ϖ^{2})ω(d_{F})}{ε(0, ω, ψ)(1 − |ϖ|)}

× \frac{L(1, τ_{E/F})^{2}}{ζ_{F} (1)}

=\frac{L(1, τ_{E/F})^{2}μ(ϖ^{2}) |D_{F} |^{\frac{1}{2}} |ϖ^{2m+1}|}{ε(0, ω, ψ)^{2}} · |D_{E}|E^{\frac{1}{2}} χ(δ_{v}d_{F}^{−1}) |δ_{v}|E^{\frac{1}{2}} ε(0, χ^{−1}, ψ_{E}).

The last equality follows from

ε(−1, ω, ψ) = |ϖD_{F} |^{−1} ε(0, ω, ψ).

From the above computation and that L(s, π_{E} ⊗χ) = 1, the proposition follows. □

3.9. The global toric period integral. We return to the global situation. Let W_{χ,f}^{(p)} be the prime-to-p Whittaker function given by

W_{χ,f}^{(p)} = ∏_{v∈h, v∤p} W_{χ,v} ∈ ⨂_{v∈h,v∤p} W(π_{v}, ψ_{v}).

Definition 3.12. Let W_{χ,∞} := ∏_{σ∈Σ} W_{k_{σ}}. Define the p-modified toric Whittaker function W_{χ} by

(3.22)  W_{χ} =W_{χ,∞} · W_{χ,f}^{(p)} · ∏_{v|p} W_{χ,v}^{♭} ∈ W(π, ψ).

Let u = (u_{v}) ∈ (O_{F} ⊗_{Z} Z_{p})^{×} = ∏_{v} O_{F_{v}}^{×}. The u-component W_{χ,u} of W_{χ} is defined by

(3.23)  W_{χ,u} = W_{χ,∞} · W_{χ,f}^{(p)} · ∏_{v|p} W_{χ,u_{v},v}.

Recall that the automorphic form φ_{W} ∈ A(π) associated to W ∈ W(π, ψ) is defined by

(3.24)  φ_{W} (g) := ∑_{β∈F} W ((\begin{matrix}β \\ & 1 \end{matrix}) g).

Let φ_{χ} (resp. φ_{χ,u}) be the automorphic form associated to W_{χ} (resp. W_{χ,u}). Let U_{p} = ∏_{v|p} U_{v} be the torsion subgroup of (O_{F} ⊗_{Z} Z_{p})^{×}. It follows immediately from the definition (3.10) that

(3.25)  φ_{χ} = ∑_{u∈U_{p}} φ_{χ,u}.

Choose a sufficiently small prime-to-p integral ideal 𝔫_{1} such that W_{χ,v} is invariant by U_{v}(𝔫_{1}) for all v ∤ p. Let K = ∏_{v} K_{v} ⊂ GL_{2}(A_{f}) be an open compact subgroup such that

(3.26)  K_{v} = K_{v}^{0} if v | p; K_{v} ⊂ U_{v}(𝔫_{1}) if v ∤ p.

For each positive integer n, put

K_{1}^{n} := \{g ∈ K | g_{v} ≡ (\begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix}) (mod p^{n}) for all v|p\} .

One can verify that W_{χ} and W_{χ,u} (and hence φ_{χ} and φ_{χ,u}) are invariant by K_{1}^{n} for sufficiently large n. The following lemma immediately follows from Lemma 3.2,

Lemma 3.13. Let T = ∏′_{v} T_{v} ⊂ A_{K}^{×}. Then φ_{χ} is a toric automorphic form in the sense that for all t ∈ T , we have

π(ι_{ς}(t))Ṽ_{+}^{m}φ_{χ} =χ^{−1}(t)Ṽ_{+}^{m}φ_{χ}.

In addition, for all t ∈ T_{f} = ∏′_{v∈h} T_{v}, we have

π(ι_{ς}(t))φ_{χ,u} =χ^{−1}(t)φ_{χ,u·t^{1−c}},

where u · t^{1−c} := ut_{Σ_{p}}t_{\overline{Σ}_{p}}^{−1} ∈ (O_{F} ⊗_{Z} Z_{p})^{×}.

Decompose 𝔠_{χ}^{−} = 𝔠_{χ,1}^{−}𝔠_{χ,2}^{−} such that (𝔠_{χ,1}^{−}, 𝔫_{r}^{−}) = 1 and 𝔠_{χ,2}^{−} has the same support with 𝔫_{r}^{−}. Define a constant C′_{π}(χ) by

(3.27) \begin{matrix}C′_{π}(χ) := & 2^{♯(A(χ))+3[F:Q]} · N_{F/Q}(𝔠_{χ}^{−})^{−1}ω(𝔠_{χ,1}^{−})ω(𝔫_{s}^{−})^{−1} \\ × ∏_{v∤p𝔯} ω(det ς_{v}) · ∏_{w|𝔉, v=w\overline{w}} ε(\frac{1}{2}, π_{v} ⊗ χ\overline{w}, ψ_{v})χ−2 ; \overline{w}(−d_{F_{v}}) \\ × ∏_{v|𝔫_{r}^{−}} χ_{v}(−δ_{v}d_{F_{v}}^{−1}) |δ_{v}|K_{v}^{\frac{1}{2}} ε(0, χ_{v}^{−1}, ψ_{K_{v}}) \end{matrix}

Note that C′_{π}(χ) is actually a p-adic unit as p > 2 and (p, 𝔉𝔫^{−}) = 1. We introduce the normalization factor N(π, χ) given by

(3.28)  N(π, χ) := ∏_{v∈B(χ)} L(1, τ_{K_{v}/F_{v}})n_{v}.

We have the following central value formula of the toric integral P_{χ}(π(ς)Ṽ_{+}^{m}φ_{χ}).

Theorem 3.14. We have

P_{χ}(π(ς)Ṽ_{+}^{m}φ_{χ})^{2} =\frac{Γ_{Σ}(k + m)Γ_{Σ}(m + 1)}{(4π)^{k+2m+1·Σ}} · e_{Σ_{p}}(π, χ) · L(\frac{1}{2}, π_{K} ⊗ χ)
× |D_{K}|_{R}^{− \frac{1}{2}} · C′_{π}(χ)N(π, χ)^{2},

where e_{Σ_{p}}(π, χ) is the p-adic multiplier given by

e_{Σ_{p}}(π, χ) = ∏_{w∈Σ_{p},v=w\overline{w}} ε(\frac{1}{2}, π_{v} ⊗ χ\overline{w}, ψ_{v})L(\frac{1}{2}, π_{v} ⊗ χ\overline{w})^{−2}χ−2 ; \overline{w}(d_{F_{v}}).

Proof. Note that Ṽ_{+}^{m}φ_{χ} is the automorphic form associated to the Whittaker function

Ṽ_{+}^{m}W_{χ} = Ṽ_{+}^{m}W_{χ,∞} · W_{χ,f}^{(p)} · ∏_{v|p} W_{χ,v}^{♭}.

Hence, by Proposition 2.1 we find that

P_{χ}(π(ς)Ṽ_{+}^{m}φ_{χ})^{2}

= ∏_{σ∈Σ} P (π(ς_{σ})Ṽ_{+}^{m_{σ}}W_{k_{σ}}, χ_{σ}) ∏_{v|p} \frac{1}{L(\frac{1}{2}, π_{K_{v}} ⊗ χ_{v})} · P (π(ς_{v})W_{χ,v}^{♭}, χ_{v})

× ∏_{v∈h,v∤p} \frac{1}{L(\frac{1}{2}, π_{K_{v}} ⊗ χ_{v})} · P (π(ς_{v})W_{χ,v}, χ_{v}) · L(\frac{1}{2}, π_{K} ⊗ χ).

Combining the local calculations of toric integrals of our Whittaker functions (Proposition 3.3, Proposition 3.5, Proposition 3.8, Proposition 3.9 and Proposition 3.11) yields the central value formula. □

Remark 3.15. Let φ_{χ}^{0} be the automorphic form associated to the toric Whittaker function W_{χ}^{0} := W_{χ,∞} · ∏_{v∈h} W_{χ,v}. Then we obtain the following central value formula:

P_{χ}(π(ς)Ṽ_{+}^{m}φ_{χ}^{0})^{2} = |D_{K}|_{R}^{− \frac{1}{2}} \frac{Γ_{Σ}(k + m)Γ_{Σ}(m + 1)}{(4π)^{k+2m+1·Σ}}·L(\frac{1}{2}, π_{K}⊗χ)·C′_{π}(χ)N(π, χ)^{2}.

3.10. Non-vanishing of the local Fourier coefficients. In order to prove the non-vanishing of our toric form φ_{χ} later, we calculate its local Fourier coefficients in this subsection. Define a_{χ,v} : F^{×} → C the local Fourier coefficient associated to W_{χ,v} by

a_{χ,v}(a) = W_{χ,v}(d(a)).

To obtain the optimal p-integrality of φ_{χ}, we need the following normalization of the a_{χ,v}.

Definition 3.16 (Normalized local Fourier coefficients). Let

B(χ) = {v ∈ h | v is non-split with c_{v}(χ) > 0} .

For v ∈ B(χ), let n_{v} be defined as in (3.21) if v|𝔫_{r}^{−} and n_{v} = 1 if v ∤ 𝔫_{r}^{−}. Define the normalized local Fourier coefficient a_{χ,v}^{∗} by

a_{χ,v}^{∗} := a_{χ,v} · \begin{cases}n_{v}^{−1}L(1, τ_{K_{v}/F_{v}})^{−1} & if v ∈ B(χ), \\ 1 & if v ∈ A(χ), \\ e_{v} & otherwise. \end{cases}

Recall that e_{v} = 1 if v is unramified and e_{v} = 2 if v is ramified.

Let v ∤ p be a finite place. We shall show the normalized local Fourier coefficients a_{χ,v}^{∗} indeed take value in a finite extension of Z_{p} and is not identically zero modulo the maximal 𝔪_{p} of \overline{Z} induced by ι_{p} : \overline{Q} ↪ C_{p}. This is clear if v is split in view of the definition of a_{χ,v}^{∗} = a_{χ,v} in §3.6. The most difficult case is when v is inert and π is ramified at v. We begin with some formulae of a_{χ,v}.

Lemma 3.17. Suppose that c_{v}(χ) = 0. Then

a_{χ,v}^{∗}(a) = \begin{cases}W_{v}^{0}(d(a)) & if v ∤ 𝔫 is unramified, \\ W_{v}^{0}(d(a)) + W_{v}^{0}(d(aϖ))χ(ϖ_{E_{v}}) & if v ∤ 𝔫 is ramified. \end{cases}

If v | 𝔫, then v is ramified and

a_{χ,v}^{∗}(a) = μ|·|^{\frac{1}{2}} (a)𝕀_{ϖ^{−1}O}(a).

Proof. It is well-known that if π = π(μ, ν) is a unramified principal series, then

W_{v}^{0}(d(a)) = 𝕀_{O}(a) |a|^{\frac{1}{2}} · ∑_{i+j=v(a), i,j≥0} μ(ϖ^{i})ν(ϖ^{j})

(cf. [Bum97, Thm. 4.6.5]). It follows from the definition of W_{χ,v} that W_{χ,v} = W_{v}^{0} if v ∤ 𝔫 is unramified and

W_{χ,v}(g) = \frac{1}{2} · W_{v}^{0}(g) + \frac{1}{2} · W_{v}^{0}(gd(ϖ))χ(ϖ_{E}) if v ∤ 𝔫 is ramified .

If v|𝔫, then v ∈ A(χ). By (R1) v is ramified, and we find that

W_{χ,v}(g) = \frac{1}{2} · W_{v}^{0}(gd(ϖ)) + \frac{1}{2} · W_{v}^{0}(gw)ω(ϖ)χ(ϖ_{E}).

The assertion follows from the formulas of W_{v}^{0} in Proposition 3.9. □

To treat the case v is non-split with c_{v}(χ) > 0, i.e. v ∈ B(χ), we need to introduce certain partial Gauss sums. For a non-split place v, write π = π(μ, ν) with unramified μ and μν^{−1}(ϖ) ≠ |ϖ|^{−1} if π is unramified or special. Define a character Ψ_{π,χ,v} : E^{×} → C^{×} by

(3.29)  Ψ_{π,χ,v}(t) := μ(N(t)) · χ|·|K^{\frac{1}{2}}(t).

Recall that the partial Gauss sum Ã_{β}(Ψ_{π,χ,v}) in [Hsi12, (4.17)] is defined by

Ã_{β}(Ψ_{π,χ,v}) := lim_{n→∞} ∫_{ϖ^{−n}O} Ψ_{π,χ,v}^{−1}(x + θ)ψ(−d_{F}^{−1}βx)dx · |D_{F} |^{− \frac{1}{2}} (β ∈ F^{×}).

Lemma 3.18. Let v ∈ B(χ) be a non-split place with c_{v}(χ) > 0. Then we have

\frac{e_{v}}{L(1, τ_{E/F})} · a_{χ,v}(a) =Ã_{a}(Ψ_{π,χ,v}) · ν|·|^{\frac{1}{2}} (a)ν|·|^{\frac{1}{2}} (ϖ^{m})

× \begin{cases}1 & if v ∤ 𝔫^{−}, \\ −1 & if v | 𝔫_{s}^{−}, \\ |ϖ^{m}| χ(δ_{v}) |δ_{v}|E^{\frac{1}{2}} \frac{ε(0,χ^{−1},ψ_{E})}{ε(−1,ω,ψ)} |D_{F} |^{− \frac{1}{2}} & if v | 𝔫_{r}^{−}. \end{cases}

Proof. It seems very difficult to deduce the above formula of a_{χ,v}(a) by a straightforward computation, so we shall prove the formula by identifying the toric Whittaker function W_{χ,v} with the image of an explicit element in the induced representation corresponding to π_{v} via the Whittaker linear functional. Recall the Whittaker linear functional Λ : I(μ, ν) → C ([Bum97, (6.9), p.498]) is defined by

Λ(f) = ∫_{F} f((\begin{matrix}0 & −1 \\ 1 & x \end{matrix}))ψ(−x)dx := lim_{n→∞} ∫_{ϖ^{−n}O} f((\begin{matrix}0 & −1 \\ 1 & x \end{matrix}))ψ(−x)dx.

Let ς = ς_{v} = (\begin{matrix}d_{F} \\ & d_{F}^{−1} \end{matrix}) and m = m_{v}(χ, π). Define P_{χ,ς}R_{v}^{m} ∈ End_{C} I(μ, ν)

by

P_{χ,ς}R_{v}^{m}f(g) = vol(E^{×}/F^{×}, dt)^{−1} ∫_{E^{×}/F^{×}} f(g (\begin{matrix}1 \\ & ϖ^{m} \end{matrix}) ι_{ς}(t))dt.

By [Sch02, Lemma 2.2.1], there exists a local new vector section f^{0} ∈ I(μ, ν)^{K^{0}(ϖ)} such that

W_{v}^{0}(g) = Λ(π(g)f^{0}).

Put f_{χ}^{0} := P_{χ,ς}R_{v}^{m}f^{0}. Then

a_{χ,v}(a) = W_{χ,v}(d(a)) = Λ(π(d(a))f_{χ}^{0}).

We thus have

a_{χ,v}(a) =ν|·|^{\frac{1}{2}} (a) ∫_{F} f_{χ}^{0}((\begin{matrix}0 & −1 \\ 1 & x \end{matrix}))ψ(−ax)dx
=f_{χ}^{0}((\begin{matrix}d_{F} \\ & d_{F}^{−1} \end{matrix})) |D_{F} |^{−1} · ν|·|^{\frac{1}{2}} (a) ∫_{F} Ψ_{π,χ,v}^{−1}(x + θ)ψ(−d_{F}^{−1}ax)dx
=f_{χ}^{0}(ς)^{∗} · ν|·|^{\frac{1}{2}} (a)Ã_{a}(Ψ_{π,χ,v}) · e_{v}^{−1} |D_{E}|_{E}^{− \frac{1}{2}} ν|·|^{− \frac{1}{2}} (ϖ^{m}),

where f_{χ}^{0}(ς)^{∗} is the normalized value

f_{χ}^{0}(ς)^{∗} := ν^{−1}|·|^{\frac{1}{2}} (ϖ^{m})v_{E} · f_{χ}^{0}(ς) (v_{E} = e_{v} |D_{E}|E^{\frac{1}{2}} |D_{F} |^{− \frac{1}{2}} ).

To evaluate the value f_{χ}^{0}(ς)^{∗}, we use the computation in (3.16) and obtain

f_{χ}^{0}(ς)^{∗} = ∫_{O} χ(1 + yθ)f^{0}(ς · R_{v}^{−m}ι_{ς}(1 + yθ)R_{v}^{m})d′y
+ ∫_{ϖO} χ(y + θ)f^{0}(ς · R_{v}^{−m}ι_{ς}(y + θ)R_{v}^{m}) |D_{E}|_{E}^{−1} d′y
=X_{m+1} · f^{0}(ς) + ∑_{r=0}^{m} ∫_{ϖ^{r}O^{×}} χ(1 + yθ)ω(ϖ^{−m}y)d′y · f^{0}(ςd(ϖ^{2(m−r)}))
+ Y_{0} · ω(ϖ^{−m})f^{0}(ςd(ϖ^{2m+e_{v}−1})).

To proceed, we need to use explicit formulas for f^{0} ([Sch02, Prop. 2.1.2]). Suppose that π is a unramified principal series (v ∤ 𝔫^{−}) or special representation (v | 𝔫_{s}^{−}). Then

f_{χ}^{0}(ς)^{∗} =X_{m} · f^{0}(ς) + ∑_{r=0}^{m−1}(X_{r} − X_{r+1}) · ω(ϖ^{r−m}) · f^{0}(ςd(ϖ^{2(m−r)}))
+ Y_{0} · ω(ϖ^{−m})f^{0}(ςd(ϖ^{2m+e_{v}−1})).

Let f^{sph} be the unique K_{v}^{0}-invariant function in I(μ, ν) with f^{sph}(ς) = L(1, μν^{−1}) |D_{F}|^{\frac{1}{2}} . If π is an unramified principal series, then we can take f^{0} = f^{sph} ([Bum97, Prop. 4.6.8]), and following the computation of the case c_{v}(χ) > 0 in Proposition 3.8 we find that

f_{χ}^{0}(ς)^{∗} =X_{m} · (f^{0}(ς) − ω(ϖ^{−1})f^{0}(d(ϖ^{2})ς)) = X_{m} · (1 − μν^{−1}|·|(ϖ)) · f^{0}(ς)
= |ϖ^{m}| |D_{E}|E^{\frac{1}{2}} L(1, τ_{E/F}).

If π is special, then

f^{0} = f^{sph} − μ^{−1}|·|^{\frac{1}{2}} (ϖ)^{−1}π((\begin{matrix}1 \\ & ϖ \end{matrix}))f^{sph},

and following the computation of the case c_{v}(χ) > 0 in Proposition 3.9 we find that

f_{χ}^{0}(ς)^{∗} =X_{m+1} · (f^{0}(ς) − f^{0}(ς · w))
=X_{m+1} · (− |ϖ|^{−1} + |ϖ|)f^{sph}(ς)
=(−1) · |ϖ^{m}| |D_{E}|E^{\frac{1}{2}} · L(1, τ_{E/F}).

Finally, suppose that π is a ramified principal series with ramified ν (v | 𝔫_{r}^{−}). Let B(F) be the group of upper triangular matrices in GL_{2}(F). Let f^{0} ∈ I(μ, ν) be the function supported in B(F)wN(D_{F}^{−1}) such that

f^{0}(ςwn) = |D_{F} |^{\frac{1}{2}} for every n ∈ N(D_{F}^{−1}).

Then one checks easily that f^{0} does the job. Following the computation in Proposition 3.11, we find that

f_{χ}^{0}(ς)^{∗}

=ω(ϖ^{−m}) |ϖ^{2m}| · χ(δ_{v}) |δ_{v}|E^{\frac{1}{2}} \frac{ε(0, χ^{−1}, ψ_{E})}{ε(−1, ω, ψ)}L(1, τ_{E/F}) |D_{E}|E^{\frac{1}{2}} |D_{F} |^{−1} · f^{0}(ς · w)
= |ϖ^{m}| · L(1, τ_{E/F}) · ω^{−1}|·|(ϖ^{m}) · χ(δ_{v})\frac{ε(0, χ^{−1}, ψ_{E})}{ε(−1, ω, ψ)} |D_{E}|E^{\frac{1}{2}} |D_{F} |^{− \frac{1}{2}} .

This completes the proof in all cases. □

To investigate the p-integrality of a_{χ,v}^{∗}, we define the local invariant μ_{p}(Ψ_{π,χ,v}) by

(3.30)  μ_{p}(Ψ_{π,χ,v}) := inf_{x∈K_{v}^{×}} v_{p}(Ψ_{π,χ,v}(x) − 1).

By [Hsi12, (4.17)], Ã_{β}(Ψ_{π,χ,v}) is indeed an algebraic integer. Moreover, it is proved in [Hsi12, Lemma 6.4] that

μ_{p}(Ψ_{π,χ,v}) > 0 ⟺ Ã_{β}(Ψ_{π,χ,v}) ≡ 0 (mod 𝔪_{p}) for all β ∈ F^{×}.

Therefore, it follows from Lemma 3.18 that if v ∈ B(χ), then a_{χ,v}^{∗} takes values in \overline{Z}_{p} and

a_{χ,v}^{∗} ≡ 0 (mod 𝔪_{p}) ⟺ μ_{p}(Ψ_{π,χ,v}) > 0.

We summarize our discussion in the following proposition.

Proposition 3.19. Let 𝒪 be the finite extension of O_{L_{π}} generated by {a_{χ,v}^{∗}(1)}_{v∈B(χ)} and the values of χ̂. Then we have

(1) the normalized local Fourier coefficient a_{χ,v}^{∗} takes values in 𝒪 for every finite place v ∤ p,

(2) if either v ∉ B(χ) is unramified or v ∈ A(χ), then a_{χ,v}^{∗}(1) = 1,

(3) if v ∤ 𝔫 is ramified with c_{v}(χ) = 0, then a_{χ,v}^{∗}(ϖ^{−1}) = 1,

(4) if v ∈ B(χ), then μ_{p}(Ψ_{π,χ,v}) = 0 if and only if there exists η_{v} ∈ F^{×}

such that

a_{χ,v}^{∗}(η_{v}) ≢ 0 (mod 𝔪_{p}).

4. Review of Hilbert modular forms

In this section, we review some standard facts about Hilbert modular Shimura varieties and Hilbert modular forms.

4.1. Let V = Fe_{1} ⊕ Fe_{2} be a two dimensional F-vector space and ⟨ , ⟩ : V ×V → F be the F-bilinear alternating pairing defined by ⟨e_{1}, e_{2}⟩ = 1. Let L = O_{F}e_{1} ⊕ D_{F}^{−1}e_{2} be the standard O_{F}-lattice in V, which is self-dual with

respect to ⟨ , ⟩. For g = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ M_{2}(F), we define an involution g ↦ g′ :=

(\begin{matrix}d & −b \\ −c & a \end{matrix}). We identify vectors in V with row vectors according to the basis e_{1}, e_{2}, so G(F) = GL_{2}(F) has a natural right action on V. If g ∈ G(F), then g′ = g^{−1} det g. Define a left action of G on V by g ∗ x := x · g′, x ∈ V. Hereafter, we let K be an open compact subgroup of G(A_{f}) satisfying (3.26) and the following conditions: (neat) K is contained in U(N′) for some N′ ≥ 3 and det(K) ∩ O_{F,+}^{×} ⊂ (K ∩ O_{F}^{×})^{2}.

We also fix a prime-to-p positive integer N such that U(N) ⊂ K.

4.2. Kottwitz models. We recall Kottwitz models of Hilbert modular Shimura varieties following the exposition in [Hid04b].

Definition 4.1 (S-quadruples). Let □ be a finite set of rational primes not dividing N and let U be an open compact subgroup of K^{0} such that U(N) ⊂ U. Let W_{U} = Z_{(□)}[ζ_{N}] with ζ_{N} = exp(\frac{2πi}{N}). Define the fibered category A_{U}^{(□)} over the category SCH_{/W_{U}} of schemes over W_{U} as follows. Let S be a locally noetherian connected W_{U}-scheme and let \overline{s} be a geometric point of S. The objects are abelian varieties with real multiplication (AVRM) over S of level U, i.e. a S-quadruple (A, λ̄, ι, \overline{η}^{(□)})_{S} consisting of the following data:

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

(2) ι : O_{F} ↪ End_{S} A ⊗_{Z} Z_{(□)} .

(3) λ is a prime-to-□ polarization of A over S and λ̄ is the O_{F,(□),+}-orbit of λ. Namely

λ̄ = O_{F,(□),+}λ := {λ′ ∈ Hom(A, A^{t}) ⊗_{Z} Z_{(□)} | λ′ = λ ∘ a, a ∈ O_{F,(□),+}} .

(4) \overline{η}^{(□)} = η^{(□)}U^{(□)} is a π_{1}(S, \overline{s})-invariant U^{(□)}-orbit of the isomorphisms

of O_{F}-modules η^{(□)} : L⊗_{Z}A_{f}^{(□)} →^{∼} V^{(□)}(A\overline{s}) := H_{1}(A\overline{s}, Ẑ^{(□)})⊗_{Z}A_{f}^{(□)}.

Here we define η^{(□)}g for g ∈ G(A_{f}^{(□)}) by η^{(□)}g(x) = η^{(□)}(g ∗ x).

Furthermore, (A, λ̄, ι, \overline{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_{F}.

• Let e^{λ} be the Weil pairing induced by λ. Lifting the isomorphism Z/NZ ≃ Z/NZ(1) induced by ζ_{N} to an isomorphism ζ : Ẑ ≃ Ẑ(1), we can regard e^{λ} as an F-alternating form e^{λ} : V^{(□)}(A) × V^{(□)}(A) → D_{F}^{−1}⊗_{Z}A_{f}^{(□)}. Let e^{η} denote the F-alternating form on V^{(□)}(A) induced by e^{η}(x, x′) = ⟨xη, x′η⟩. Then

e^{λ} = u · e^{η} for some u ∈ A_{f}^{(□)}.

• As O_{F} ⊗_{Z} O_{S}-modules, we have an isomorphism Lie A ≃ O_{F} ⊗_{Z} O_{S}

locally under the Zariski topology of S. For two S-quadruples \underline{A} = (A, λ̄, ι, \overline{η}^{(□)})_{S} and \underline{A′} = (A′, \overline{λ′}, ι′, \overline{η′}^{(□)})_{S}, we define morphisms by

Hom_{A_{K}^{(□)}}(\underline{A}, \underline{A′}) = \{ϕ ∈ Hom_{O_{F}} (A, A′) | ϕ^{∗}\overline{λ′} = λ̄, ϕ ∘ \overline{η′}^{(□)} = \overline{η}^{(□)}\} .

We say \underline{A} ∼ \underline{A′} (resp. \underline{A} ≃ \underline{A′}) if there exists a prime-to-□ isogeny (resp. isomorphism) in Hom_{A_{K}^{(□)}}(\underline{A}, \underline{A′}).

We consider the cases when □ = ∅ and {p}. When □ = ∅ is the empty set and U is an open compact subgroup in G(A_{f}^{(□)}) = G(A_{f}), we define the functor E_{U} : SCH_{/W_{U}} → SET S by

E_{U}(S) = \{(A, λ̄, ι, \overline{η})_{S} ∈ A_{K}(S)\} / ∼ .

By the theory of Shimura-Deligne, E_{U} is represented by Sh_{U} which is a quasiprojective scheme over W_{U}. We define the functor 𝔈_{U} : SCH_{/W_{U}} → SET S by

𝔈_{U}(S) = \{(A, λ̄, ι, \overline{η}) ∈ A_{U}^{(□)}(S) | η^{(□)}(L ⊗_{Z} Ẑ) = H_{1}(A\overline{s}, Ẑ)\} / ≃ .

By the discussion in [Hid04b, p.136], we have 𝔈_{K} →^{∼} E_{K} under the hypothesis (neat). When □ = {p} and U = K, we let W = W_{K} = Z_{(p)}[ζ_{N}] and define functor E_{K}^{(p)} : SCH_{/W} → SET S by

E_{K}^{(p)}(S) = \{(A, λ̄, ι, \overline{η}^{(p)})_{S} ∈ A_{K^{(p)}}^{(p)}(S)\} / ∼ .

In [Kot92], Kottwitz shows E_{K}^{(p)} is representable by a quasi-projective scheme Sh_{K}^{(p)} over W if K is neat. Similarly we define the functor 𝔈_{K}^{(p)} : SCH_{/W} → SET S by

𝔈_{K}^{(p)}(S) = \{(A, λ̄, ι, \overline{η}^{(p)}) ∈ A_{K}^{(p)}(S) | η^{(p)}(L ⊗_{Z} Ẑ^{(p)}) = H_{1}(A\overline{s}, Ẑ^{(p)})\} / ≃ .

It is shown in [Hid04b, §4.2.1] that 𝔈_{K}^{(p)} →^{∼} E_{K}^{(p)}. Let 𝔠 be a prime-to-pN ideal of O_{F} and let c ∈ (A_{f}^{(pN)})^{×} such that 𝔠 = 𝔦𝔩_{F}(c). We say (A, λ, ι, \overline{η}^{(p)}) is 𝔠-polarized if λ ∈ λ̄ such that e^{λ} = ue^{η}, u ∈ c det(K). The isomorphism class [(A, λ, ι, \overline{η}^{(p)})] is independent of a choice of λ in \overline{λ} under the assumption (neat) (cf. [Hid04b, p.136]). We consider the functor

𝔈_{𝔠,K}^{(p)}(S) = \{𝔠-polarized S-quadruple [(A, λ, ι, \overline{η}^{(p)})_{S}] ∈ 𝔈_{K}^{(p)}(S)\} .

Then 𝔈_{𝔠,K}^{(p)} is represented by a geometrically irreducible scheme Sh_{K}^{(p)}(𝔠)_{/W}, and we have

(4.1)  Sh_{K}^{(p)}_{/W} = ⨆_{[𝔠]∈Cl_{F}^{+}(K)} Sh_{K}^{(p)}(𝔠)_{/W},

where Cl_{F}^{+}(K) is the narrow ray class group of F with level det(K).

4.3. Igusa schemes. Let n be a positive integer. Define the functor I_{K,n}^{(p)} : SCH_{/W} → SET S by

S ↦ I_{K,n}^{(p)}(S) = {(A, λ̄, ι, η^{(p)}, j)_{S}} / ∼,

where (A, λ̄, ι, η^{(p)})_{S} is a S-quadruple, j is a level p^{n}-structure, i.e. an O_{F}- group scheme morphism:

j : D_{F}^{−1} ⊗_{Z} μ_{p^{n}} ↪ A[p^{n}],

and ∼ means modulo prime-to-p isogeny. It is known that I_{K,n}^{(p)} is relatively representable over E_{K}^{(p)} (cf. [HLS06, Lemma (2.1.6.4)]) and thus is represented by a scheme I_{K,n}. Now we consider S-quintuples (A, λ, ι, η^{(p)}, j)_{S} such that [(A, λ, ι, η^{(p)})] ∈ 𝔈_{𝔠,K}^{(p)}(S). Define the functor I_{K,n}^{(p)}(𝔠) : SCH_{/W} → SET S by

S ↦ I_{K,n}^{(p)}(𝔠)(S) = {(A, λ, ι, η^{(p)}, j)_{S} as above} / ≃ .

Then I_{K,n}^{(p)}(𝔠) is represented by a scheme I_{K,n}(𝔠) over Sh_{K}^{(p)}(𝔠), and I_{K,n}(𝔠) can be identified with a geometrically irreducible subscheme of I_{K,n} ([DR80, Thm. (4.5)]). For n ≥ n′ > 0, the natural morphism π_{n,n′} : I_{K,n}(𝔠) → I_{K,n′}(𝔠) induced by the inclusion D_{F}^{−1} ⊗ μ_{p^{n′}} ↪ D_{F}^{−1} ⊗ μ_{p^{n}} is finite étale. The forgetful morphism π : I_{K,n}(𝔠) → Sh_{K}^{(p)}(𝔠) defined by π : (\underline{A}, j) ↦ \underline{A} is étale for all n > 0. Hence I_{K,n}(𝔠) is smooth over Spec W. We write I_{K}(𝔠) for \underleftarrow{lim}_{n} I_{K,n}(𝔠).

4.4. Complex uniformization. We describe the complex points Sh_{U}(C) for U ⊂ G(A_{f}). Put

X^{+} = {τ = (τ_{σ})_{σ∈Σ} ∈ C^{Σ} | Im τ_{σ} > 0 for all σ ∈ Σ} .

The action of g = (g_{σ})_{σ∈Σ} ∈ G(F ⊗_{Q} R) with g_{σ} = (\begin{matrix}a_{σ} & b_{σ} \\ c_{σ} & d_{σ} \end{matrix}) and det g_{σ} > 0

on X^{+} is given by τ = (τ_{σ}) ↦ gτ = (\frac{a_{σ}τ_{σ}+b_{σ}}{c_{σ}τ_{σ}+d_{σ}} ). 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 Shimura variety by

M(X^{+}, U) := G(F)^{+}\X^{+}×G(A_{f})/U.

It is well known that M(X^{+}, K) →^{∼} Sh_{U}(C) by the theory of abelian varieties over C (cf. [Hid04b, § 4.2]). Now we define this isomorphism explicitly. For τ = (τ_{σ})_{σ∈Σ} ∈ X^{+}, we let p_{τ} be the isomorphism V ⊗_{Q} R →^{∼} C^{Σ} defined by p_{τ} (ae_{1} + be_{2}) = aτ + b with a, b ∈ F ⊗_{Q} R = R^{Σ}. We can associate a AVRM to (τ, g) ∈ X^{+}×G(A_{f}) as follows.

• The complex abelian variety 𝒜_{g}(τ) = C^{Σ}/p_{τ} (g ∗ L).

• The F_{+}-orbit of polarization \overline{⟨ , ⟩}_{can} on 𝒜_{g}(τ) is given by the Riemann

form ⟨ , ⟩_{can} := ⟨ , ⟩ ∘ p_{τ}^{−1}.

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

• The level structure η_{g} : L ⊗_{Z} A_{f} →^{∼} (g ∗ L) ⊗_{Z} A_{f} = H_{1}(𝒜_{g}(τ), A_{f}) is

defined by η_{g}(v) = g ∗ v. Let \underline{𝒜_{g}(τ)} denote the C-quadruple (𝒜_{g}(τ), \overline{⟨ , ⟩}_{can}, ι_{C}, Kη_{g}). Then the map [(τ, g)] ↦ [\underline{𝒜_{g}(τ)}] gives rise to an isomorphism M(X^{+}, U) →^{∼} Sh_{U}(C). For a positive integer n, the exponential map gives the isomorphism exp(2πi−) : p^{−n}Z/Z ≃ μ_{p^{n}} and thus induces a level p^{n}-structure j(g_{p}):

j(g_{p}): D_{F}^{−1} ⊗_{Z} μ_{p^{n}} →^{∼} D_{F}^{−1}e_{2} ⊗_{Z} p^{−n}Z/Z ↪ L ⊗_{Z} p^{−n}Z/Z →^{∼}^{g∗} 𝒜_{g}(τ)[p^{n}].

Put

K_{1}^{n} := \{g ∈ K | g_{p} ≡ (\begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix}) (mod p^{n})\} .

We have a non-canonical isomorphism:

M(X^{+}, K_{1}^{n}) →^{∼} I_{K,n}(C)
[(τ, g)] ↦ [(𝒜_{g}(τ), \overline{⟨ , ⟩}_{can}, ι_{C}, \overline{η}_{g}^{(p)}, j(g_{p}))].

Let \underline{z} = \{z_{σ}\}_{σ∈Σ} be the standard complex coordinates of C^{Σ} and d\underline{z} = \{dz_{σ}\}_{σ∈Σ}. Then O_{F}-action on d\underline{z} is given by ι_{C}(α)^{∗}dz_{σ} = σ(α)dz_{σ}, σ ∈ Σ ≃ Hom(F, C). Let z = z_{id} be the coordinate corresponding to ι_{∞} : F ↪ \overline{Q} ↪ C. Then

(4.2)  (O_{F} ⊗_{Z} C)dz = H^{0}(𝒜_{g}(τ), Ω_{𝒜_{g}(τ)/C} ).

4.5. Hilbert modular forms. Let k = ∑_{σ} k_{σ}σ ∈ Z_{≥1}[Σ] such that

k_{σ_{1}} ≡ k_{σ_{2}} ≡ ⋯ ≡ k_{σ_{d}} (mod 2) for all σ_{1}, …, σ_{d} ∈ Σ.

For τ = (τ_{σ})_{σ∈Σ} ∈ X^{+} and g = ((\begin{matrix}a_{σ} & b_{σ} \\ c_{σ} & d_{σ} \end{matrix}))_{σ∈Σ} ∈ G(F ⊗_{Q} R), we put

\underline{J}(g, τ)^{k} = ∏_{σ∈Σ}(c_{σ}τ_{σ} + d_{σ})^{k_{σ}}.

Definition 4.2. Let k_{mx} = max_{σ∈Σ} k_{σ}. Denote by M_{k}(K_{1}^{n}, C) the space of holomorphic Hilbert modular forms of weight k and level K_{1}^{n}. Each f ∈ M_{k}(K_{1}^{n}, C) is a C-valued function f : X^{+}×G(A_{f}) → C such that the function f(−, g_{f}) : X^{+} → C is holomorphic for each g_{f} ∈ G(A_{f}), and for u ∈ K_{1}^{n} and α ∈ G(F)^{+},

f(α(τ, g_{f})u) = (det α)^{− \frac{kmxΣ+k}{2}} \underline{J}(α, τ)^{k} · f(τ, g_{f}).

Here det α is considered to be the element (σ(det α))_{σ∈Σ} in (C^{×})^{Σ}.

For every f ∈ M_{k}(K_{1}^{n}, C),we have the Fourier expansion

f(τ, g_{f}) = ∑_{β∈F_{+}∪{0}} W_{β}(f, g_{f})e^{2πi Tr_{F/Q}(βτ)}.

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

∑_{β∈L} a_{β}q^{β}, a_{β} ∈ B.

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

(4.3)  f|_{(𝔞,𝔟)}(q) = ∑_{β∈(N^{−1}𝔞𝔟)_{≥0}} W_{β}(f, (\begin{matrix}a^{−1} & 0 \\ 0 & b \end{matrix}))q^{β} ∈ C⟦(N^{−1}𝔞𝔟)_{≥0}⟧.

If B is a W-algebra in C, let M_{k}(𝔠, K_{1}^{n}, B) be the space consisting of functions f ∈ M_{k}(K_{1}^{n}, C) such that

f|_{(𝔞,𝔟)}(q) ∈ B⟦(N^{−1}𝔞𝔟)_{≥0}⟧ for all (𝔞, 𝔟) such that 𝔞𝔟^{−1} = 𝔠.

4.5.1. Tate objects. Let 𝒮 be a set of d linearly 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 L_{𝒮 ,n} = {x ∈ L | l(x) > −n for all l ∈ 𝒮 } and put B((L; 𝒮)) = lim_{n→∞} B⟦L_{𝒮 ,n} ⟧. To a pair (𝔞, 𝔟) of two prime-to-pN frac-

tional ideals, we can attach the Tate AVRM T ate_{𝔞,𝔟}(q) = 𝔞^{∗} ⊗_{Z} 𝔾_{m}/q^{𝔟} over Z((𝔞𝔟; 𝒮)) with O-action ι_{can} acting on 𝔞^{∗}, where 𝔞^{∗} := 𝔞^{−1}D_{F}^{−1}. As described in [Kat78], T ate_{𝔞,𝔟}(q) has a canonical 𝔞𝔟^{−1}-polarization λ_{can} and also carries a canonical O_{F} ⊗ Z((𝔞𝔟; 𝒮))-generator ω_{can} of Ω_{T ate_{𝔞,𝔟}} induced by the isomorphism Lie(T ate_{𝔞,𝔟}(q)_{/Z((𝔞𝔟;𝒮))} ) = 𝔞^{∗} ⊗_{Z} Lie(𝔾_{m}) ≃ 𝔞^{∗} ⊗ Z((𝔞𝔟; 𝒮)). Since 𝔞 is prime to p, the natural inclusion 𝔞^{∗} ⊗_{Z} μ_{p^{n}} ↪ 𝔞^{∗} ⊗_{Z} 𝔾_{m} induces a canonical level p^{n}-structure η_{p,can}: D_{F}^{−1} ⊗_{Z} μ_{p^{n}} = 𝔞^{∗} ⊗_{Z} μ_{p^{n}} ↪ T ate_{𝔞,𝔟}(q).

Let L_{𝔞,𝔟} = L · (\begin{matrix}𝔟 \\ & 𝔞^{−1} \end{matrix}) = 𝔟e_{1} ⊕ 𝔞^{∗}e_{2}. Then we have a level N-structure

η_{can}^{(p)} : N^{−1}L_{𝔞,𝔟}/L_{𝔞,𝔟} →^{∼} T ate_{𝔞,𝔟}(q)[N] over Z[ζ_{N}]((N^{−1}𝔞𝔟; 𝒮)) induced by

the fixed primitive N-th root of unity ζ_{N}. We write \underline{T ate}_{𝔞,𝔟} for the Tate Z((𝔞𝔟; 𝒮))-quadruple (T ate_{𝔞,𝔟}(q), λ_{can}, ι_{can}, \overline{η}_{can}^{(p)}, η_{p,can}) at (𝔞, 𝔟).

4.5.2. Geometric modular forms. We collect here definitions and basic facts of geometric modular forms. The whole theory can be found in [Kat78] and [Hid04b]. Let T be the algebraic torus over W defined by T (R) = (O_{F} ⊗_{Z} R)^{×} for every W-algebra R. Let k ∈ Hom(T, 𝔾_{m}_{/W}). Let B be a W-algebra. For a B-algebra C, we consider a triple (\underline{A}, j, ω) over C, consisting of [(\underline{A}, j)] = [(A, λ, ι, η^{(p)}, j)] ∈ I_{K,n}(𝔠)(C) (resp. [(\underline{A}, j)] = [(A, λ̄, ι, η^{(p)}, j)] ∈ I_{K,n}(C)) AVRM with level structures and an 1-form ω generating H^{0}(A, Ω_{A/C}) over O_{F} ⊗_{Z} C. A geometric modular form f of weight k on I_{K,n}(𝔠) (resp. I_{K,n}) over B is a rule of assigning to every triple (\underline{A}, j, ω) over C a value f(\underline{A}, j, ω) ∈ C satisfying the following axioms.

(G1) f(\underline{A}, j, ω) = f(\underline{A}′, j′, ω′) ∈ C if (\underline{A}, j, ω) ≃ (\underline{A}′, j′, ω′) over C, (G2) For a B-algebra homomorphism φ : C → C′, we have

f((\underline{A}, j, ω) ⊗_{C} C′) = φ(f(\underline{A}, j, ω)),

(G3) f((\underline{A}, j, aω) = k(a^{−1})f(\underline{A}, j, ω) for all a ∈ T (C) = (O_{F} ⊗_{Z} C)^{×}, (G4) f(\underline{T ate}_{𝔞,𝔟}, ω_{can}) ∈ B⟦(N^{−1}𝔞𝔟)_{≥0}⟧ at all cusps (𝔞, 𝔟) in I_{K,n}(𝔠) (resp.

I_{K,n}). For each k ∈ Z[Σ], we regard k ∈ Hom(T, 𝔾_{m}_{/W}) as the character x ↦ x^{k}, x ∈ (O_{F} ⊗_{Z} W)^{×}. We denote by M_{k}(𝔠, K_{1}^{n}, B) (resp. M_{k}(K_{1}^{n}, B)) the space of geometric modular forms over B of weight k on I_{K,n}(𝔠) (resp. I_{K,n}). For f ∈ M_{k}(K_{1}^{n}, B), we write f|_{𝔠} ∈ M_{k}(𝔠, K_{1}^{n}, B) for the restriction f|_{I_{K,n}(𝔠)}. For each f ∈ M_{k}(K_{1}^{n}, C), we regard f as a holomorphic Hilbert modular form of weight k and level K_{1}^{n} by

f(τ, g_{f}) = f(𝒜_{g}(τ), \overline{⟨ , ⟩}_{can}, ι_{C}, \overline{η}_{g}, 2πidz),

where dz is the differential form in (4.2). By GAGA this gives rise to an

isomorphism M_{k}(K_{1}^{n}, C) →^{∼} M_{k}(K_{1}^{n}, C) and M_{k}(𝔠, K_{1}^{n}, C) →^{∼} M_{k}(𝔠, K_{1}^{n}, C).

Moreover, as discussed in [Kat78, §1.7], we have the following important identity which bridges holomorphic modular forms and geometric modular forms

f|_{(𝔞,𝔟)}(q) = f(\underline{T ate}_{(𝔞,𝔟)}, ω_{can}) ∈ C⟦(N^{−1}𝔞𝔟)_{≥0}⟧.

By the q-expansion principle, if B is W-algebra in C and f ∈ M_{k}(𝔠, K_{1}^{n}, B) = M_{k}(𝔠, K_{1}^{n}, C), then f|_{𝔠} ∈ M_{k}(𝔠, K_{1}^{n}, B).

4.5.3. p-adic modular forms. Let B be a p-adic W-algebra in C_{p}. Let V (𝔠, K, B) be the space of Katz p-adic modular forms over B defined by

V (𝔠, K, B) := \underleftarrow{lim}_{m} \underrightarrow{lim}_{n} H^{0}(I_{K,n}(𝔠)_{/B/p^{m}B}, O_{I_{K,n}}).

In other words, Katz p-adic modular forms consist of formal functions on the Igusa tower. Let C be a B/p^{m}B-algebra. For each C-point [(\underline{A}, j)] = [(A, λ, ι, η^{(p)}, j] ∈ I_{K}(𝔠)(C) = \underleftarrow{lim}_{n} I_{K,n}(𝔠)(C), the p^{∞}-level structure j induces an isomorphism j_{∗} : D_{F}^{−1}⊗_{Z}C ≃ Lie A which in turn gives rise to a generator ω(j) of H^{0}(A, Ω_{A}) as a O_{F} ⊗_{Z} C-module. Then we have a natural injection

(4.4)  \begin{matrix}M_{k}(𝔠, K_{1}^{n}, B) & ↪ V (𝔠, K, B) \\ f & ↦ f̂(\underline{A}, j) := f(\underline{A}, j, ω(j)) \end{matrix}

which preserves the q-expansions in the sense that f̂|_{(𝔞,𝔟)}(q) := f̂(\underline{T ate}_{𝔞,𝔟}) = f|_{(𝔞,𝔟)}(q). We call f̂ the p-adic avatar of f.

4.6. CM points. Recall that we have fixed ϑ ∈ K in §3.4 satisfying (d1-3) and the associated embedding ι : K ↪ M_{2}(F) in (2.1). The map p_{ϑ} : V ⊗_{Q} R ≃ C^{Σ}, ae_{1} + be_{2} ↦ aϑ + b yields an isomorphism p_{ϑ} : V ≃ K satisfying

p_{ϑ}(x)α = p_{ϑ}(xι(α)) for x ∈ V, α ∈ K.

Let ς = ∏_{v} ς_{v} ∈ G(A), where ς_{v} ∈ G_{v} for each place v is defined in (3.3). Let ς_{f} ∈ G(A_{f}) be the finite part of ς. According to our choices of ς_{v}, we have

ς_{f} ∗ (L ⊗_{Z} Ẑ) = (L ⊗_{Z} Ẑ) · ς′_{f} = p_{ϑ}^{−1}(O_{K} ⊗_{Z} Ẑ).

Define x : A_{K}^{×} → X^{+}×G(A_{f}) by

a = (a_{∞}, a_{f}) ↦ x(a) := (ϑ_{Σ}, ι(a_{f})ς_{f}).

Let a ∈ (A_{K,f}^{(pN)})^{×} and let

(\underline{A}(a), j(a))_{/C} = (𝒜_{ι(a)ς_{f}} (ϑ_{Σ}), \overline{⟨ , ⟩}_{can}, ι_{can}, η^{(p)}(a), j(a))

be the C-quintuple associated to x(a) as in §4.4. The alternating pairing ⟨ , ⟩ : K×K :→ F defined by ⟨x, y⟩ = (\overline{x}y − x\overline{y})/(ϑ − \overline{ϑ}) induces an isomorphism O_{K}∧_{O_{F}} O_{K} = 𝔠(O_{K})^{−1}D_{F}^{−1} for the fractional ideal 𝔠(O_{K}) = D_{F}^{−1}((ϑ−\overline{ϑ})D_{K/F}^{−1}). The hypothesis (d2) on ϑ implies that

𝔠(O_{K}) is prime to p𝔠_{χ}𝔫D_{K/F}.

Note that 𝔠(O_{K}) descends to a fractional ideal of O_{F} and that 𝔠(O_{K}) is the polarization of x(1) = (\underline{A}(1), j(1)). In addition, x(a) = (\underline{A}(a), j(a))_{/C} is an abelian variety with CM by O_{K} with the polarization ideal of x(a) given by

𝔠(a) := 𝔠(O_{K})N(𝔞)^{−1} (𝔞 = 𝔦𝔩_{K}(a)).

It thus gives rise to a complex point [x(a)] in I_{K}(𝔠(a))(C). Let W_{p} be the p-adic completion of the maximal unramified extension of Z_{p} in C_{p}. The general theory of CM abelian varieties ([Shi98]) combined with the criterion of Serre and Tate ([ST68]) imply that [x(a)] indeed descends to a point in I_{K}(𝔠(a))(W_{p}) ↪ I_{K}(W_{p}), which is still denoted by x(a). The collection {[x(a)]}_{a∈(A_{K,f}^{(pN)})^{×}} ⊂ I_{K}(W_{p}) are called CM points in the Hilbert modular Shimura varieties.

5. Anticyclotomic Rankin-Selberg p-adic L-functions

5.1. Toric forms.

Definition 5.1 (Toric forms). We define the complex Hilbert modular form f_{χ} : X^{+}×G(A_{f}) → C associated to φ_{χ} by

(5.1) f_{χ}(τ, g_{f}) =φ_{χ}(g) · \underline{J}(g_{∞}, i)^{k}(det g_{∞})^{− \frac{kmxΣ+k}{2}} |det g|_{A}^{k_{mx}/2} ,

(i = (\sqrt{−1})_{σ∈Σ}, g = (g_{∞}, g_{f}), g_{∞}i = τ, det g_{∞} > 0).

Here det g_{∞} = (det g_{σ})_{σ∈Σ} ∈ (R^{×})^{Σ} and det g_{∞} > 0 means det g_{σ} > 0 for all σ ∈ Σ. Let f_{χ}^{∗} be the normalization of f_{χ} given by

f_{χ}^{∗} = N(π, χ)^{−1} |det ς_{f}|_{A_{f}}^{−k_{mx}/2} · f_{χ}.

Let δ_{k}^{m} be the Maass-Shimura differential operator (cf. [HT93, (1.21)]). Then the normalized differential operator Ṽ_{+}^{m} defined in (3.7) is the representation theoretic avatar of δ_{k}^{m} in the following sense:

δ_{k}^{m}f_{χ}(τ, g_{f}) = (Ṽ_{+}^{m}φ_{χ})(g_{∞}, g_{f})\underline{J}(g_{∞}, i)^{k+2m}(det g_{∞})^{− \frac{kmxΣ+k+2m}{2}} |det g|_{A}^{k_{mx}/2}

(cf. [Hsi12, §4.5]). We call δ_{k}^{m}f_{χ}^{∗} the normalized toric form of character χ associated with the Hilbert modular form f.

Similarly, for each u ∈ (O_{F} ⊗_{Z} Z_{p})^{×}, we let f_{χ,u}^{∗} be the normalized modular form associated to the u-component φ_{χ,u} (cf. W_{χ,u} in (3.23)). It is clear from (3.25) that

(5.2)  f_{χ}^{∗} = ∑_{u∈U_{p}} f_{χ,u}^{∗}.

Let K_{1}^{n} be the open compact subgroup defined in (3.26). Then f_{χ}^{∗} and {f_{χ,u}^{∗}}_{u∈U_{p}} belong to M_{k}(K_{1}^{n}, C) for sufficiently large n.

For a ∈ (A_{K,f}^{(p)})^{×}×(O_{K} ⊗ Z_{p})^{×}, we consider the Hecke action |[a] given by

|[a] : M_{k}(𝔠(a), K_{1}^{n}, C) → M_{k}(𝔠,_{a}K_{1}^{n}, C) (_{a}K_{1}^{n} := ι_{ς}(a)K_{1}^{n}ι_{ς}(a^{−1})),
f ↦ f|[a](τ, g_{f}) := f(τ, g_{f}ι_{ς}(a)).

The Hecke action |[a] can be extended to the spaces of p-integral modular forms (cf. [Hsi14b, §2.6]). It follows from Lemma 3.13 immediately that

(5.3) f_{χ,u}^{∗}|[a] = χ^{−1}|·|_{A_{K}}^{k_{mx}/2}(a) · f_{χ,u.a^{1−c}}^{∗} for all a ∈ T_{f} (u.a^{1−c} := ua_{Σ_{p}}a_{\overline{Σ}_{p}}^{−1}).

5.2. The toric period integral. Next we consider the toric period integral of f_{χ}^{∗}. Let U_{K} = (K ⊗_{Q} R)^{×}×(O_{K} ⊗_{Z} Ẑ)^{×} be a subgroup of A_{K}^{×} and let Cl_{−} = K^{×}A^{×}\A_{K}^{×}/U_{K}. Let R be the subgroup of A_{K}^{×} generated by K_{v}^{×} for all ramified places v and let Cl_{−}^{alg} be the subgroup of Cl_{−} generated by the image of R By Lemma 3.13 and the fact that T = A^{×}U_{K}R, we have

(5.4) P_{χ}(π(ς)Ṽ_{+}^{m}φ_{χ}) = vol(U_{K}, dt)♯(Cl_{−}^{alg}) · ∑_{[t]∈Cl_{−}/Cl_{−}^{alg}} Ṽ_{+}^{m}φ_{χ}(ι(t)ς)χ(t).

Let D_{1} be a set of representatives of Cl_{−}/Cl_{−}^{alg} in (A_{K,f}^{(pN)})^{×}. We define the χ-isotypic toric period by

P_{χ}(δ_{k}^{m}f_{χ}^{∗}) := ∑_{a∈D_{1}} δ_{k}^{m}f_{χ}^{∗}(x(a))χ|·|_{A_{K}}^{−k_{mx}/2}(a).

Proposition 5.2. Let D_{K/F} be the discriminant of K/F. We have

P_{χ}(δ_{k}^{m}f_{χ}^{∗})^{2} =\frac{Γ_{Σ}(k + m)Γ_{Σ}(m + 1)}{(Im ϑ)^{k+2m}(4π)^{2m+k+1}} · L(\frac{1}{2}, π_{K} ⊗ χ) · e_{Σ_{p}}(π, χ)

× [O_{K}^{×} : O_{F}^{×}]^{2} · C_{π}(χ), where

Documenta Mathematica 19 (2014) 709–767

C_{π}(χ) = C′_{π}(χ) · 4^{−[F:Q]} |N_{F/Q}(D_{K/F})|R^{\frac{1}{2}} (\frac{♯(Cl_{−})h_{F}}{♯(Cl_{−}^{alg})h_{K}} )^{2} ∈ \overline{Z}_{(p)}^{×}

and C′_{π}(χ) is defined in (3.27).

Proof. We first note that the ratio \frac{♯(Cl_{−})h_{F}}{♯(Cl_{−}^{alg})h_{K}} is a power of 2, so the constant

C_{π}(χ) is a p-adic unit. By definition, we have

f_{χ}(x(a)) = φ_{χ}(ι(a_{f})ς)(Im ϑ)^{−k/2} · |N(a) det ς_{f}|_{A}^{k_{mx}/2} .

By (5.4), we find that

vol(U_{E}, d^{×}t)♯(Cl_{−}^{alg}) · P_{χ}(δ_{k}^{m}f_{χ}^{∗}) = \frac{1}{N(π, χ) · (Im ϑ)^{k/2+m}} · P_{χ}(π(ς)Ṽ_{+}^{m}φ_{χ}).

From the well-known formula

2L(1, τ_{K/F}) = (2π)^{[F:Q]} · \frac{h_{K}/h_{F}}{|D_{K}|R^{\frac{1}{2}} |D_{F}|_{R}^{− \frac{1}{2}} · [O_{K}^{×} : O_{F}^{×}]},

we see that

vol(U_{K}, dt) = vol(K^{×}A^{×}\A_{K}^{×}, dt) · ♯(Cl_{−})^{−1}
=2π^{−[F:Q]}L(1, τ_{K/F}) · ♯(Cl_{−})^{−1} = \frac{2^{[F:Q]} |D_{F}|R^{\frac{1}{2}}}{|D_{K}|R^{\frac{1}{2}} [O_{K}^{×} : O_{F}^{×}]} · \frac{h_{K}}{h_{F}♯(Cl_{−})}.

The proposition follows form Theorem 3.14 immediately. □

5.3. The Fourier expansion of f_{χ,u}^{∗}. Let u = (u_{v}) ∈ U_{p}. We give an expression of the Fourier expansion of f_{χ,u}^{∗}. Let W_{χ,u,f} be the finite part of W_{χ,u}. By the definition of f_{χ,u}, we have (5.5)

f_{χ,u}(τ, g_{f}) = ∑_{β∈F} W_{χ,u,f}((\begin{matrix}β \\ & 1 \end{matrix}) g_{f})W_{χ,∞}((\begin{matrix}β \\ & 1 \end{matrix}) (\begin{matrix}y_{∞} & x_{∞} \\ 0 & 1 \end{matrix})) · y_{∞}^{−k/2}

= ∑_{β∈F_{+}} W_{χ,u,f}((\begin{matrix}β \\ & 1 \end{matrix}) g_{f})β^{k/2}e^{2πi Tr_{F/Q}(βτ)}.

(τ = x_{∞} + iy_{∞} = (x_{σ} + iy_{σ})_{σ∈Σ} ∈ X^{+})

The second equality follows from the choice of Whittaker functions at the archimedean places (3.6).

We define the global prime-to-p Fourier coefficient a_{χ}^{(p)} : (A_{f}^{(p)})^{×} → C by (5.6)

\begin{matrix}a_{χ}^{(p)}(a) := & N(π, χ)^{−1} · W_{χ,f}^{(p)}((\begin{matrix}a \\ & 1 \end{matrix})) (a = (a_{v}) ∈ A_{f}^{×}) \\ = & ∏_{v∈B(χ)} \frac{1}{n_{v}L(1, τ_{K_{v}/F_{v}})}W_{χ,v}((\begin{matrix}a_{v} \\ & 1 \end{matrix})) ∏_{v∉B(χ),v∤p} W_{χ,v}((\begin{matrix}a_{v} \\ & 1 \end{matrix})) \\ = & ∏_{v∈h,v∤p} a_{χ,v}^{∗}(a_{v}). \end{matrix}

Here a_{χ,v}^{∗} are the normalized local Fourier coefficients defined in Definition 3.16.

Proposition 5.3. Let 𝔠 be a prime-to-p ideal of F and let c ∈ (A_{K,f}^{(p)})^{×} such that 𝔦𝔩_{F}(c) = 𝔠. Then the Fourier expansion of f_{χ,u}^{∗} at the cusp (O_{F}, 𝔠) is given by

f_{χ,u}^{∗}|_{(O_{F} ,𝔠)}(q) = ∑_{β∈(N^{−1}𝔠)_{+}} a_{β}(f_{χ,u}^{∗}, 𝔠)q^{β},

where

a_{β}(f_{χ,u}^{∗}, 𝔠) = β^{k/2}a_{χ}^{(p)}(βc^{−1}) ∏_{w∈Σ_{p},v|w} χ\overline{w}(β^{−1})𝕀_{u_{v}(1+ϖ_{v}O_{Fv})}(β).

In particular, f_{χ,u}^{∗} ∈ M_{k}(K_{1}^{n}, 𝒪) by Proposition 3.19, and the Fourier expansion of f_{χ}^{∗} at the cusp (O_{F}, 𝔠) is given by

f_{χ}^{∗}|_{(O_{F} ,𝔠)}(q) = ∑_{β∈(N^{−1}𝔠)_{+}} a_{β}(f_{χ}^{∗}, 𝔠)q^{β},

where

a_{β}(f_{χ}^{∗}, 𝔠) = β^{k/2}a_{χ}^{(p)}(βc^{−1}) ∏_{w∈Σ_{p}} χ\overline{w}(β^{−1}) · 𝕀_{O_{F,(p)}^{×}}(β).

Proof. It follows from the definition of W_{χ,u} that

W_{χ,u,f}((\begin{matrix}βc^{−1} \\ & 1 \end{matrix})) =W_{χ,f}^{(p)}((\begin{matrix}βc^{−1} \\ & 1 \end{matrix})) · ∏_{v|p} W_{χ,u_{v},v}((\begin{matrix}β \\ & 1 \end{matrix}))

=W_{χ,f}^{(p)}((\begin{matrix}βc^{−1} \\ & 1 \end{matrix})) · ∏_{w∈Σ_{p},v|w} χ−1 ; \overline{w}(a_{v})𝕀_{u_{v}(1+ϖ_{v}O_{Fv})}(β).

The proposition follows from (5.5) immediately. The Fourier expansion of f_{χ}^{∗} follows from (5.2). □

5.4. p-adic L-functions. Now we resume the setting in the introduction. Let K_{p^{∞}}^{−} be the maximal anticyclotomic Z_{p}^{[F:Q]}-extension of K and let Γ^{−} = Gal(K_{p^{∞}}^{−}/K). Let C(Γ^{−}) = C(Γ^{−}, \overline{Z}_{p}) be the space of continuous functions φ : Γ^{−} → \overline{Z}_{p}. The reciprocity law rec_{K} at Σ_{p} induces a morphism

rec_{Σ_{p}}: (F ⊗_{Q} Q_{p})^{×} ≃ ∏_{w∈Σ_{p}} K_{w}^{×} ⟶^{rec_{K}} Γ^{−}.

Let 𝔛_{p}^{crit} be the set of critical specializations, consisting of p-adic characters ϕ̂: Γ^{−} → C_{p}^{×} such that for some m ∈ Z_{≥0}[Σ],

ϕ̂(rec_{Σ_{p}}(x)) = x^{m} for all x ∈ (O_{F} ⊗ Z_{p})^{×} sufficiently close to 1.

Let ϕ be an anticyclotomic Hecke character of p-power conductor and of infinity type (m, −m) with m ∈ Z_{≥0}[Σ]. Then ϕ is unramified outside p and ϕ|_{A^{×}} = 1. The p-adic avatar ϕ̂ of ϕ belongs to 𝔛_{p}^{crit}. To be precise, let ϕ_{Σ_{p}} := ∏_{w∈Σ_{p}} ϕ_{w}. Then we have

(5.7)  ϕ̂(rec_{Σ_{p}}(x)) = ϕ_{Σ_{p}}(x)x^{m} for every x ∈ (F ⊗_{Q} Q_{p})^{×}.

Hereafter, we let λ be a Hecke character of K^{×} and assume that Hypothesis A and (sf) hold for (π, λ). Note that Hypothesis A and (sf) also hold for (π, λϕ). We will apply our calculations in §3 to the pair (π, χ) = (π, λϕ).

Lemma 5.4. Let ϕ be as above. Then

(1) a_{λϕ}^{(p)} = a_{λ}^{(p)}.
(2) C′_{π}(λϕ) = C′_{π}(λ)ϕ(𝔉).

Proof. If v ∤ p is split, we have remarked that W_{χ_{v}ϕ_{v}^{−1},v} = W_{χ_{v},v}. If v is inert or ramified, then ϕ_{v} = 1 as ϕ_{v} is unramified and p > 2. Therefore, we have W_{χϕ,f}^{(p)} = W_{χ,f}^{(p)}. Part (1) follows from the definition of a_{λϕ}^{(p)} (5.6) immediately. Next, recall that we have defined C′_{π}(χ) for a Hecke character χ in (3.27). Since ϕ is anticyclotomic and unramified outside p, part (2) follows from the fact [Sch02, Eq.(11)]

ε(\frac{1}{2}, π_{v} ⊗ λ\overline{w}ϕ\overline{w}, ψ_{v}) = ε(\frac{1}{2}, π_{v} ⊗ λ\overline{w}, ψ)ϕ\overline{w}(D_{F}^{2}𝔉) (v = w\overline{w}, w|𝔉). □

Let O_{p} := O_{F} ⊗_{Z} Z_{p} and let Γ′ := rec_{Σ_{p}}(1 + pO_{p}) be an open subgroup of Γ^{−}. Let {θ(σ)}_{σ∈Σ} be the Dwork-Katz p-adic differential operators ([Kat78, Cor. (2.6.25)]) and let θ^{m} := ∏_{σ∈Σ} θ(σ)^{m_{σ}}.

Proposition 5.5. There exists a unique p-adic distribution ℱ_{λ,𝔠} : C(Γ^{−}) → V (𝔠, K, \overline{Z}_{p}) such that

(i) ℱ_{λ,𝔠} is supported in Γ′,

(ii) for every ϕ̂ ∈ 𝔛_{p}^{crit} of weight (m, −m), we have
ℱ_{λ,𝔠}(ϕ̂) = θ^{m}f̂_{λϕ,𝔠}^{∗}.

Proof. We denote by ℱ_{λ,𝔠}(q) the p-adic measure with values in the space of formal q-expansions such that for every φ ∈ C(Γ^{−}),

ℱ_{λ,𝔠}(q)(φ) = ∑_{β∈(N^{−1}𝔠)_{+}} a_{β}(f_{λ}^{∗}, 𝔠)φ(rec_{Σ_{p}}(β))q^{β}.

Note that a_{β}(f_{λ}^{∗}, 𝔠) = 0 unless β ∈ O_{F,(p)}^{×}, and thus ℱ_{λ,𝔠} has support in Γ′ by definition.

754 M.-L. Hsieh Let ϕ̂ be the p-adic avatar of a Hecke character ϕ of infinity type (m, −m). By [Kat78, (2.6.27)] (cf. [HT93, §1.7 p.205]), the q-expansion of θ^{m}f̂_{λϕ}^{∗} is given by

θ^{m}f̂_{λϕ}^{∗}|_{(O_{F} ,𝔠)}(q) = ∑_{β∈(N^{−1}𝔠)_{+}} a_{β}(f_{λϕ}^{∗}, 𝔠)ϕ_{Σ_{p}}(β)β^{m}q^{β}.

Therefore, by Lemma 5.4 and (5.7) we find that

(5.8)  ℱ_{λ,𝔠}(q)(ϕ̂) = θ^{m}f̂_{λϕ,𝔠}^{∗}(q).

By the q-expansion principle, this measure descends to the p-adic measure ℱ_{λ,𝔠} with values in the space of p-adic modular forms V (𝔠, K, \overline{Z}_{p}). □ We are ready to define the p-adic L-function. Definition 5.6. For a ∈ A_{K}^{×}, define |[a] ∈ End(C(Γ^{−})) by φ ↦ φ|[a](σ) := φ(σ rec_{K}(a)|_{Γ^{−}}). Fix a square root \sqrt{C_{π}(λ)} ∈ \overline{Z}_{p}^{×} of the constant C_{π}(λ) and let λ̃ = λ · |·|_{A_{K}}^{−k_{mx}/2}. We define the p-adic integral distribution ℒ_{Σ_{p}}(π, λ) : C(Γ^{−}) → \overline{Z}_{p} by

(5.9)  ℒ_{Σ_{p}}(π, λ)(φ) = \frac{1}{\sqrt{C_{π}(λ)}} · ∑_{a∈D_{1}} λ̃(a) · (ℱ_{λ,𝔠(a)}(φ|[a])) (x(a)).

We will still denote by ℒ_{Σ_{p}}(π, λ) ∈ \overline{Z}_{p}⟦Γ^{−}⟧ the corresponding power series. We give the evaluation formula of ℒ_{Σ_{p}}(π, λ) at critical specializations. Let (Ω_{∞}, Ω_{p}) ∈ (C^{×})^{Σ}×(\overline{Z}_{p}^{×})^{Σ} be the complex and p-adic CM periods of (K, Σ) introduced in [HT93, (4.4 a,b) p.211] (cf. (Ω, c) in [Kat78, (5.1.46), (5.1.48)]) and let Ω_{K} = (2πi)^{−1}Ω_{∞}. For each Hecke character χ of infinity type (m, −m), we define the algebraic L-value by

(5.10) L^{alg}(\frac{1}{2}, π_{K} ⊗ χ) := \frac{Γ_{Σ}(m)Γ_{Σ}(k + m)}{(Im ϑ)^{k+2m}(4π)^{k+2m+1·Σ}} · \frac{L(\frac{1}{2}, π_{K} ⊗ χ)}{Ω_{K}^{2(k+2m)}} ∈ \overline{Q}.

The algebraicity of this L-value is due to Shimura [Shi78]. Theorem 5.7. Suppose that Hypothesis A and (sf) hold. Then for each p-adic character ϕ̂ ∈ 𝔛_{p}^{crit} of weight (m, −m), we have the evaluation formula

(\frac{ℒ_{Σ_{p}}(π, λ)(ϕ̂)}{Ω_{p}^{k+2m}} )^{2} =[O_{K}^{×} : O_{F}^{×}]^{2} · e_{Σ_{p}}(π, λϕ)L^{alg}(\frac{1}{2}, π_{K} ⊗ λϕ) · ϕ^{−1}(𝔉).

Proof. It follows from [Kat78, (2.4.6), (2.6.8), (2.6.33)] that

\frac{1}{Ω_{p}^{k+2m}} θ^{m}f̂_{λϕ}^{∗}(x(a)) = \frac{1}{Ω_{K}^{k+2m}} δ_{k}^{m}f_{λϕ}^{∗}(x(a)).

We thus have

\frac{1}{Ω_{p}^{k+2m}} · ℒ_{Σ_{p}}(π, λ)(ϕ̂) = \sqrt{C_{π}(λ)}^{−1} ∑_{a∈D_{1}} λ̃ϕ(a) · \frac{1}{Ω_{p}^{k+2m}} θ^{m}f̂_{λϕ}^{∗}(x(a))

=\frac{1}{Ω_{K}^{k+2m} · \sqrt{C_{π}(λ)}} · P_{λϕ}(δ_{k}^{m}f_{λϕ}^{∗}).

Combined with Proposition 5.2 and Lemma 5.4 (2), the above equation yields the proposition. □

6. The μ-invariant of p-adic L-functions

In this section, we use the explicit computation of Fourier coefficients of {f_{λ,u}^{∗}}_{u∈U_{p}} to study the μ-invariant of the p-adic L-function ℒ_{Σ_{p}}(π, λ) by the

approach of Hida [Hid10b].

6.1. The t-expansion of p-adic modular forms. We begin with a brief review of the t-expansion of p-adic modular forms. A functorial point in I_{K}(𝔠) can be written as [(\underline{A}, j)] = [(A, λ, ι, \overline{η}^{(p)}, j)]. Enlarging W_{p} if necessary, we let W_{p} be the p-adic ring generated by the values of λ on finite ideles over the Witt ring W (𝔽̄_{p}). Let 𝔪_{W_{p}} be the maximal ideal of W_{p} and fix an isomorphism

W_{p}/𝔪_{W_{p}} →^{∼} 𝔽̄_{p}. Let T := O_{F}^{∗} ⊗_{Z} μ_{p^{∞}} and let T̂ = O_{F}^{∗} ⊗_{Z} 𝔾̂_{m}_{/W_{p}} be the formal

completion. Let {ξ_{1}, …, ξ_{d}} be a basis of O_{F} over Z and let t be the character 1 ∈ O_{F} = X^{∗}(O_{F}^{∗} ⊗_{Z} 𝔾_{m}) = Hom(O_{F}^{∗} ⊗_{Z} 𝔾_{m}, 𝔾_{m}). Then

OT̂ →^{∼} W_{p}⟦S_{1}, …, S_{d}⟧ (S_{i} = t^{ξ_{i}} − 1).

Let x := x(1)_{/W_{p}} ∈ I_{K}(𝔠)(W_{p}) be the CM point introduced in §4.6and let x_{0} = x ⊗_{W_{p}} 𝔽̄_{p} = (\underline{A}_{0}, j_{0}) ∈ I_{K}(𝔠)(𝔽̄_{p}) be the reduction. The theory of Serre-Tate coordinates ([Kat81]) tells us the deformation space Ŝ_{x_{0}} of x_{0} is isomorphic to the formal torus T̂, and the p^{∞}-level structure j_{0} of A_{0} induces a canonical isomorphism φ_{x_{0}} : T̂_{/W} →^{∼} Ŝ_{x_{0}} = Spf Ô_{I_{K}(𝔠),x_{0}} (cf. [Hid10b, (3.15)]). We will regard the character t on T̂ as a function on Ŝ_{x_{0}} via φ_{x_{0}}. Then x is the canonical lifting of x_{0}, i.e. t(x) = 1. For f ∈ V (𝔠, K, W_{p}), we define

f(t) := φ_{x_{0}}^{∗}(f) ∈ OT̂ = W_{p}⟦S_{1}, …, S_{d}⟧.

The formal power series f(t) is called the t-expansion around x_{0} of f.

6.2. The vanishing of the μ-invariant. Let π_{−} : (A_{K,f}^{(pN)})^{×} → Γ^{−} be the natural map induced by the reciprocity law. Let Z′ = π_{−}^{−1}(Γ′) be a subgroup of (A_{K,f}^{(pN)})^{×} and let Cl′_{−} ⊃ Cl_{−}^{alg} be the image of Z′ in Cl_{−}. Let D′_{1} (resp. D′′_{1}) be a set of representative of Cl′_{−}/Cl_{−}^{alg} (resp. Cl_{−}/Cl′_{−}) in (A_{K,f}^{(pN)})^{×}. Let D_{1} := D′_{1}D′′_{1} be a set of representative of Cl_{−}/Cl_{−}^{alg}. Recall that U_{p} is the torsion subgroup of O_{p}^{×}. Let U be the torsion subgroup of K^{×} and let U^{alg} = (K^{×})^{1−c} ∩ O_{K}^{×} be a subgroup of U. We regard U^{alg} as a subgroup of O_{p} by the imbedding induced by Σ_{p}. Let D_{0} be a set of representatives of U_{p}/U^{alg} in U_{p}. Let 𝔠 := 𝔠(O_{K}) be the polarization ideal of the chosen CM point x(1). The following theorem reduces the calculation of the μ-invariant μ_{π,λ,Σ}^{−} to the determination of p-adic valuation of the q-expansion of f_{λ,u}^{∗}.

Theorem 6.1. Suppose that p is unramified in F. Then

μ_{π,λ,Σ}^{−} = inf_{\begin{matrix}(a,u)∈D_{1}×D_{0} \\ β∈F_{+} \end{matrix}} v_{p}(a_{β}(f_{λ,u}^{∗}, 𝔠(a))).

Proof. For every pair (u, a) ∈ U_{p}×D_{1}, we let f_{u,a}^{∗} := f_{λ,u}^{∗}|_{𝔠(a)} ∈ M_{k}(𝔠(a), K, 𝒪). Let ℱ_{u,a} be the p-adic avatar of f_{u,a}^{∗}. Fix a sufficient large finite extension L over Q_{p} so that χ and f_{u,a}^{∗}|[a] are defined over O_{L} for all (u, a), and hence ℱ_{u,a}|[a] ∈ V (𝔠, O_{L}). For each z ∈ Z′, let ⟨z⟩ be the unique element in 1 + pO_{p} such that rec_{Σ_{p}}(⟨z⟩) = π_{−}(z) ∈ Γ^{−}. For (a, b) ∈ D_{1}×D′′_{1}, we define

ℱ̃_{a}(t) = ∑_{u∈U_{p}} ℱ_{u,a}(t^{u^{−1}}),
ℱ^{b}(t) = ∑_{a∈bD_{1}} λ̃(ab^{−1})ℱ̃_{a}|[a](t^{⟨^{ab^{−1}}⟩}).

Let ℒ_{Σ}^{b}(π, λ) be the p-adic measure on 1+pO_{p} ≃ Γ′ obtained by the restriction of ℒ_{Σ_{p}}(π, λ) to π_{−}(b)Γ′. In other words, for each continuous function φ : Γ′ → \overline{Z}_{p}, we have

ℒ_{Σ}^{b}(π, λ)(φ) := ℒ_{Σ_{p}}(π, λ)(𝕀_{b.Γ′} · φ|[b^{−1}])
= ∑_{a∈bD′_{1}} λ̃(a)ℱ_{λ,𝔠(a)}(φ|[ab^{−1}])(x(a)).

Here the second equality follows from the fact that ℱ_{λ,𝔠(a)} has support in Γ′ (Proposition 5.5 (i)). The argument of [Hsi14b, Prop. 5.2] shows that ℱ^{b}(t) is the power series expansion of the measure ℒ_{Σ}^{b}(π, λ) regarded as a p-adic measure on O_{p} and that

μ_{π,λ,Σ}^{−} = inf_{b∈D′′_{1}} μ(ℱ^{b}), where
μ(ℱ^{b}) := inf {r ∈ Q_{≥0} | p^{−r}ℱ^{b} ≢ 0 (mod 𝔪_{p})} .

By (5.3) we find that

ℱ̃_{a}(t) = ♯(U^{alg}) · ∑_{u∈D_{0}} ℱ_{u,a}(t^{u^{−1}}),

and hence

ℱ^{b}(t) = ♯(U^{alg}) · ∑_{(u,a)∈D_{0}×bD′_{1}} λ̃(ab^{−1})ℱ_{u,a}|[a](t^{⟨ab^{−1}⟩u^{−1}}).

Proceeding along the same lines in [Hsi14b, Thm. 5.5], we deduce the theorem from the above equation by the linear independence of p-adic modular forms modulo p acted by the automorphisms in D_{0}×D′_{1} ([Hid10b, Thm. 3.20, Cor. 3.21]) and the q-expansion principle for p-adic modular forms. □

Theorem 6.2. In addition to Hypothesis A and (sf), we suppose that p is unramified in F and

(aiK) the residual Galois representation \overline{ρ}_{p}(π_{K}) is absolutely irreducible.

Then μ_{π,λ,Σ}^{−} = 0 if and only if

∑_{v|𝔠_{λ}^{−}} μ_{p}(Ψ_{π,λ,v}) = 0,

where μ_{p}(Ψ_{π,λ,v}) are the local invariants defined as in (3.30).

Proof. It is not difficult to deduce from the formula of a_{β}(f_{λ,u}^{∗}, 𝔠(a)) in Proposition 5.3 and Proposition 3.19 that

μ_{p}(Ψ_{π,λ,v}) > 0 for some v|𝔠_{λ}^{−} ⇒ a_{β}(f_{λ,u}^{∗}, 𝔠(a)) ≡ 0 (mod 𝔪_{p}) for all a ∈ A_{f}^{×},

and hence μ_{π,λ,Σ}^{−} > 0 by Theorem 6.1. Conversely, we suppose that μ_{p}(Ψ_{π,λ,v}) = 0 for all v|𝔠_{λ}^{−}. We are going to show μ_{π,λ,Σ}^{−} = 0 by contradiction. Assume that μ_{π,λ,Σ}^{−} > 0. By Proposition 5.3 Theorem 6.1, for each a ∈ A_{K,f}^{(pN)} we find that

a_{β}(f_{λ,u}^{∗}, 𝔠(a)) ≡ 0 (mod 𝔪_{p}) for all u ∈ U_{p} and β ∈ F_{+}
⟺ a_{λ}^{(p)}(βc^{−1}N(a^{−1})) ≡ 0 (mod 𝔪_{p}) for all β ∈ O_{F,(p)}^{×}.

Therefore, as a function on (A_{f}^{(p)})^{×}, we have

(6.1)

a_{λ}^{(p)}(a) ≡ 0 (mod 𝔪_{p}) for all
a ∈ O_{F,(p)}^{×}c^{−1} det(U(N))N((A_{K,f}^{(pN)})^{×}) = F^{×}c^{−1}N((A_{K,f}^{(p)})^{×}).

By Proposition 3.19, there exists η = (η_{v}) ∈ ∏_{v|𝔠_{χ}^{−}} F_{v}^{×} such that a_{λ,v}^{∗}(η_{v}) ≢ 0 (mod 𝔪_{p}) for each v|𝔠_{λ}^{−}. We extend η to be the idele in A_{f}^{×} such that η_{v} = 1 at v ∤ 𝔠_{χ}. Therefore, (6.1) together with the factorization formula of a_{λ}^{(p)} (5.6) imply that for each uniformizer ϖ_{v} at v ∤ p𝔯, we have (6.2)

a_{λ}^{(p)}(ηϖ_{v}) ≡ 0 (mod 𝔪_{p}) ⟺ W_{v}^{0}((\begin{matrix}ϖ_{v} \\ & 1 \end{matrix})) ≡ 0 (mod 𝔪_{p}) whenever
ϖ_{v} ∈ [η^{−1}c^{−1}] := F^{×}η^{−1}c^{−1}N((A_{K,f}^{(p)})^{×}).

On the other hand, by (3.2), we find that

Tr ρ_{p}(π)(Frob_{v}) = ω(ϖ_{v})^{−1} |ϖ_{v}|^{−k_{mx}/2} W_{v}^{0}((\begin{matrix}ϖ_{v} \\ & 1 \end{matrix})) for all v ∤ p𝔫.

Let rec_{K/F} : A_{K}^{×} → Gal(K/F) be the surjection induced by the reciprocity law. Combined with (6.2), the above equation yields that

Tr ρ_{p}(π)(Frob_{v}) ≡ 0 (mod 𝔪_{p}) whenever
Frob_{v}|_{K} = rec_{K/F}(ϖ_{v}) = rec_{K/F}(η^{−1}c^{−1}).

This in particular implies that rec_{K/F}(η^{−1}c^{−1}) must be the complex conjugation c, and hence we arrive at a contradiction to (aiK) by the following Lemma 6.3. □

Lemma 6.3. Let p > 2 be a prime. Let G be a finite group and H ⊂ G be a index two subgroup. Let ρ : G ↪ GL_{2}(𝔽̄_{p}) be a faithful irreducible representation of G. Let T = Tr ρ : G → 𝔽̄_{p} be the trace function. Assume that

(1) There exists an order two element c ∈ G − H,

(2) T(hc) = 0 for all h ∈ H. Then ρ|_{H} is reducible.

Proof. The assumption (2) implies that T(c) = 0, and hence det ρ(c) = −1.

We may assume that ρ(c) = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}). Suppose that p ∤ ♯(G). By the usual

representation theory of finite groups, we have

1 = ⟨T, T⟩ := \frac{1}{♯(G)} ∑_{g∈G} T(g)T(g^{−1}) = \frac{1}{2♯(H)} ∑_{h∈H} T(h)T(h^{−1}) = \frac{1}{2} ·⟨T|_{H}, T|_{H}⟩.

Since ⟨T|_{H}, T|_{H}⟩ = 2, we conclude that ρ|_{H} is not irreducible. Now we assume that p | ♯(H). For each b ∈ M_{2}(𝔽̄_{p}) with b^{2} = 0, define the p-subgroup P_{b} of ρ(H) by

P_{b} = {h ∈ ρ(H) | h = 1 + xb for some x ∈ 𝔽̄_{p}} .

Let h ∈ H be an element of p-power order. It is well known that (ρ(h)−1)^{2} = 0, and hence T(h) = 2 and det ρ(h) = 1. Combined with T(hc) = 0, these equations imply that

ρ(h) ∈ P_{b_{1}} or P_{b_{2}} with b_{1} = (\begin{matrix}1 & 1 \\ −1 & −1 \end{matrix}) , b_{2} = (\begin{matrix}1 & −1 \\ 1 & −1 \end{matrix}) .

Note that either P_{b_{1}} or P_{b_{2}} is trivial. Indeed, if h_{1} ≠ 1 ∈ P_{b_{1}} and h_{2} ≠ 1 ∈ P_{b_{2}}. Then h_{1}h_{2} ∈ H and Tr(h_{1}h_{2}c) ≠ 0, which is a contradiction. In particular, we conclude that elements of p-power order in H are commutative with each other and that there is only one p-Sylow subgroup of H, which we denote by P. It is clear that H normalizes P. Since P ≠ {1}, there is a unique line fixed by ρ(P), which is an invariant subspace of ρ(H). We find that ρ|_{H} is reducible if p | ♯(H). □

Remark 6.4. The assumption (3) in Theorem B in the introduction implies the vanishing of μ_{p}(Ψ_{π,λ,v}) for all v|𝔠_{λ}^{−}.

7. Non-vanishing of central L-values with anticyclotomic twists

In this section, we consider the problem of non-vanishing of L-values modulo p with anticyclotomic twists. Let ℓ ≠ p be a rational prime and let 𝔩 be a prime of F above ℓ. Let Γ_{𝔩}^{−} := Gal(K_{𝔩^{∞}}^{−}/K) be the Galois group of the maximal anticyclotomic pro-ℓ extension K_{𝔩^{∞}}^{−} in the ray class field of K of conductor 𝔩^{∞}. Let 𝔛_{𝔩}^{0} be the set consisting of finite order characters ϕ : Γ_{𝔩}^{−} → μ_{ℓ^{∞}}. Fix a Hecke character χ of infinity type (k/2 + m, −k/2 − m). We assume

(p𝔩, 𝔫D_{K/F}) = 1.

When p ∤ D_{K}, we know the algebraic L-value L^{alg}(\frac{1}{2}, π_{K} ⊗χϕ) ∈ \overline{Z}_{p} in (5.10) in view of Theorem 5.7. Recall that 𝔪_{p} is the maximal ideal of \overline{Z}_{p}. This section is devoted to proving the following result:

Theorem 7.1. With the same assumptions in Theorem 6.2, we further assume that

(1) (p𝔩, 𝔫𝔠_{χ}D_{K/F}) = 1.
(2) μ_{p}(Ψ_{π,χ,v}) = 0 for all v|𝔠_{χ}^{−}.

Then for almost all ϕ ∈ 𝔛_{𝔩}^{0} we have

L^{alg}(\frac{1}{2}, π_{K} ⊗ χϕ) ≢ 0 (mod 𝔪_{p}).

Here almost all means "except for finitely many ϕ ∈ 𝔛_{𝔩}^{0}" if dim_{Q_{ℓ}} F_{𝔩} = 1 and "for ϕ in a Zariski dense subset of 𝔛_{𝔩}^{0}" if dim_{Q_{ℓ}} F_{𝔩} > 1 (See [Hid04a, p.737]).

When F = Q, an imprimitive version of the above result under different assumptions is treated in [Bra11b].

7.1. After introducing some notation, we outline the approach of Hida [Hid04a] to study this problem. We shall take 𝔯 = 𝔠_{χ}𝔫D_{K/F}𝔩 to be the fixed ideal in §3.4. For every n ∈ Z_{≥0}, let R_{𝔩^{n}} := O_{F} + 𝔩^{n}O_{K} be the order in K of conductor 𝔩^{n}. Let U_{𝔩^{n}} = (K ⊗_{Q} R)^{×}(R_{n} ⊗_{Z} Ẑ)^{×} and let Cl_{𝔩^{n}}^{−} := K^{×}A^{×}\A_{K}^{×}/U_{𝔩^{n}} be the anticyclotomic ideal class group of conductor 𝔩^{n}. Denote by [·]_{n} : A_{K}^{×} → Cl_{𝔩^{n}}^{−} the quotient map. Let Cl_{𝔩^{∞}}^{−} = \underleftarrow{lim}_{n} Cl_{𝔩^{n}}^{−}. Let I_{𝔩} be the 𝔩-adic Iwahori subgroup of K_{𝔩}^{0} given by

I_{𝔩} = \{g = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ K_{𝔩}^{0} | c ∈ ϖ_{𝔩}D_{F_{𝔩}}\} .

Let K_{0}(𝔩) := K^{𝔩}I_{𝔩} = {g ∈ K | g_{𝔩} ∈ I_{𝔩}} be an open compact subgroup of GL_{2}(A_{f}). Recall that the U_{𝔩}-operator on M_{k}(K_{0}(𝔩), C) is given by

F | U_{𝔩}(τ, g_{f}) = ∑_{u∈O_{F} /𝔩O_{F}} F (τ, g_{f} (\begin{matrix}ϖ_{𝔩} & ud_{F_{𝔩}}^{−1} \\ 0 & 1 \end{matrix})).

We briefly outline the approach of Hida to prove Theorem 7.1 as follows:

(1) Construct a suitable p-adic modular form f̂_{χ}^{†} which is an eigenfunction of U_{𝔩}-operator with p-adic unit eigenvalue.

(2) Consider Hida’s measure φ_{χ}^{†} on Cl_{𝔩^{∞}}^{−} attached to f̂_{χ}^{†} (7.1) and show the evaluation formula of this measure is related to central values L^{alg}(\frac{1}{2}, π_{K} ⊗ χϕ) (Proposition 7.3).

(3) The Zariski density of CM points in Hilbert modular varieties modulo p reduces the proof of Theorem 7.1 to the non-vanishing of certain Fourier coefficients of f̂_{χ}^{†} at some cusp ([Hid04a, Thm. 3.2 and Thm. 3.3]).

We remind, as the reader will note, that the proof is very close to Theorem 5.7 and Theorem 6.2. The essential new inputs in this section are the choice of U_{𝔩}-eigenforms and the computation of local period integral at 𝔩.

7.2. CM points of conductor 𝔩^{n}. Let n ∈ Z_{≥0}. We choose ς_{𝔩}^{(n)} ∈ G_{𝔩} as follows. If 𝔩 = 𝔏\overline{𝔏} splits in K, writing ϑ = ϑ\overline{𝔏}e\overline{𝔏} + ϑ_{𝔏}e_{𝔏} (so d_{F_{𝔩}} = ϑ_{𝔏} − ϑ\overline{𝔏} is a generator of D_{F_{𝔩}}), we put

ς_{𝔩}^{(n)} = (\begin{matrix}ϑ_{𝔏} & −1 \\ 1 & 0 \end{matrix}) (\begin{matrix}ϖ_{𝔩}^{n} \\ & 1 \end{matrix}) .

If 𝔩 is inert, then we put

ς_{𝔩}^{(n)} = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) (\begin{matrix}ϖ_{𝔩}^{n} \\ & 1 \end{matrix}) .

Let ς^{(n)} := ς_{𝔩}^{(n)} ∏_{v≠𝔩} ς_{v}. According to this choice of ς_{𝔩}^{(n)}, we have

ς_{f}^{(n)} ∗ (ℒ ⊗_{Z} Ẑ) = q_{ϑ}(R_{𝔩^{n}}).

Define x_{n} : A_{K}^{×} → X^{+}×G(A_{f}) by

x_{n}(a) := (ϑ_{Σ}, a_{f}ς_{f}^{(n)}).

This collection {x_{n}(a)}_{a∈A_{K}^{×}} of points is called CM points of conductor 𝔩^{n}. As discussed in §4.6, {x_{n}(a))}_{a∈(A_{K,f}^{(p)} )^{×}} descend to CM points in I_{K}(W_{p}).

7.3. The measures associated to U_{𝔩}-eigenforms. We construct the U_{𝔩}- eigenform f_{χ}^{†} as follows. Write π_{𝔩} = π(μ_{𝔩}, ν_{𝔩}). Define the local Whittaker function W_{𝔩}^{†} ∈ W(π_{𝔩}, ψ_{𝔩}) by

W_{𝔩}^{†}(g) = W_{𝔩}^{0}(g) − μ_{𝔩}|·|^{\frac{1}{2}} (𝔩)W_{𝔩}^{0}(g (\begin{matrix}ϖ_{𝔩}^{−1} \\ & 1 \end{matrix})).

It is not difficult to verify that

• W_{𝔩}^{†} is invariant by I_{𝔩},

• W_{𝔩}^{†}((\begin{matrix}a \\ & 1 \end{matrix})) = ν_{𝔩}|·|^{\frac{1}{2}} (a)𝕀_{O_{F_{𝔩}}} (a),

• W_{𝔩}^{†} is an U_{𝔩}-eigenfunction with the eigenvalue ν_{𝔩}(ϖ_{𝔩}) |ϖ_{𝔩}|^{− \frac{1}{2}} .

Define the normalized global Whittaker function W_{χ}^{†} by

W_{χ}^{†} := N(π, χ)^{−1} |det ς_{f}|_{A_{f}}^{−k_{mx}/2} · ∏_{σ∈Σ} W_{k_{σ}} · ∏_{v∈h,v≠𝔩} W_{χ,v} · W_{𝔩}^{†},

where N(π, χ) is the normalization factor in (3.28). Let φ_{χ}^{†} be the automorphic form associated to W_{χ}^{†} as in (3.24) and let f_{χ}^{†} be the associated Hilbert modular form as in Definition 5.1. The following lemma follows from the choice of our Whittaker function W_{χ}^{†} and the construction of f_{χ}^{†}.

Lemma 7.2. Recall that R is the group generated by K_{v}^{×} for all ramified places v in A_{K}^{×}. We have

(1) f_{χ}^{†} is toric of character χ outside 𝔩, and

f_{χ}^{†}(x_{n}(ta)) = f_{χ}^{†}(x_{n}(t))χ^{−1}|·|_{A_{K}}^{k_{mx}/2}(a) for all a ∈ R · (R_{𝔩^{n}} ⊗_{Z} Ẑ)^{×}.

(2) f_{χϕ}^{†} = f_{χ}^{†} for every ϕ ∈ 𝔛_{𝔩}^{0}.

Proof. Part (1) follows immediately from the fact that W_{χ}^{†} is a toric Whittaker function outside 𝔩 in view of Lemma 3.2. In addition, for every ϕ ∈ 𝔛_{𝔩}^{0}, ϕ is anticyclotomic and unramified outside 𝔩. We thus have W_{χ}^{†} = W_{χϕ}^{†}, which verifies part (2) (cf. Lemma 5.4). □

Following [Hid04a, (3.9)], we define a p-adic \overline{Z}_{p}-valued measure φ_{χ}^{†} on Cl_{𝔩^{∞}}^{−} attached to the p-adic avatar f̂_{χ}^{†} of f_{χ}^{†} as follows. For a locally constant function ϕ : Cl_{𝔩^{∞}}^{−} → \overline{Z}_{p} factoring through Cl_{𝔩^{n}}^{−}, we define

(7.1)  ∫_{Cl_{𝔩^{∞}}^{−}} ϕdφ_{χ}^{†} = α_{𝔩}^{−n} ∑_{[a]_{n}∈Cl_{𝔩^{n}}^{−}} θ^{m}f̂_{χ}^{†}(x_{n}(a))χ̂(a)ϕ([a]_{n}),

where α_{𝔩} = ν_{𝔩}(ϖ_{𝔩}) |ϖ_{𝔩}|^{\frac{−kmx}{2}} and χ̂ is the p-adic avatar of χ|·|_{A_{K}}^{−k_{mx}/2}. One checks that the right hand side does not depend on the choice of n since f_{χ}^{†} is an U_{𝔩}-eigenform with the eigenvalue α_{𝔩}. Let ϕ ∈ 𝔛_{𝔩}^{0} be a character of conductor 𝔩^{n}. We view ϕ as a character on Cl_{𝔩^{n}}^{−} by the reciprocity law. Following the arguments in Proposition 5.2 and Theorem 5.7, we can write the measure as a toric period integral of Ṽ_{+}^{m}φ_{χϕ}^{†}:

(7.2)

\frac{Ω_{K}^{k+2m}}{Ω_{p}^{k+2m}} · ∫_{Cl_{𝔩^{∞}}^{−}} ϕdφ_{χ}^{†}

=α_{𝔩}^{−n} vol(U_{𝔩^{n}}, dt)^{−1} \frac{1}{(Im ϑ)^{k/2+m}} · P_{χϕ}(π(ς^{(n)})Ṽ_{+}^{m}φ_{χ}^{†})
= vol(U_{K}, dt)^{−1} \frac{1}{(Im ϑ)^{k/2+m}} · \frac{α_{𝔩}^{−n}}{L(1, τ_{K_{𝔩}/F_{𝔩}}) |ϖ_{𝔩}|^{n}} · P_{χϕ}(π(ς^{(n)})Ṽ_{+}^{m}φ_{χϕ}^{†}).

Here we used the fact that

vol(U_{𝔩^{n}}, dt) = vol(U_{K}, dt) · L(1, τ_{K_{𝔩}/F_{𝔩}}) |ϖ_{𝔩}|^{n} .

Documenta Mathematica 19 (2014) 709–767

We have the following evaluation formula.

Proposition 7.3. Suppose that (𝔩, 𝔠_{χ}𝔫D_{K/F}) = 1. For ϕ ∈ 𝔛_{𝔩}^{0} of conductor 𝔩^{n} with n ≥ 1, we have

(\frac{1}{Ω_{p}^{k+2m}} · ∫_{Cl_{𝔩^{∞}}^{−}} ϕdφ_{χ}^{†})^{2} = \frac{|ϖ_{𝔩}|^{−n}}{α_{𝔩}^{2n}} · [O_{K}^{×} : O_{F}^{×}]^{2} · L^{alg}(\frac{1}{2}, π_{K} ⊗ χϕ) · C_{π}(χ)ϕ(𝔉).

Proof. In view of (7.2), it remains to compute P_{χϕ}(π(ς^{(n)})Ṽ_{+}^{m}φ_{χϕ}^{†})^{2}, which can be written as a product of local toric period integrals as in the proof of Theorem 3.14. We have computed these local period integrals in §3.7 and §3.8 except for the local integral at 𝔩, which will be carried out in the following Lemma 7.4. The desired formula is obtained by combining these calculations.

□

Lemma 7.4. Suppose that χ_{𝔩} is unramified and ϕ ∈ 𝔛_{𝔩}^{0} has conductor 𝔩^{n}, n ≥ 1. Then

P (π(ς_{𝔩}^{(n)})W_{𝔩}^{†}, χϕ) = |D_{K_{𝔩}}|K_{𝔩}^{\frac{1}{2}} · ω_{𝔩}(ϖ_{𝔩}^{n}) |ϖ_{𝔩}^{n}| L(1, τ_{K_{𝔩}/F_{𝔩}})^{2}.

Proof. Write F = F_{𝔩} (resp. E = K_{𝔩}) and ϖ = ϖ_{𝔩}. For t ∈ E, we put

ι_{ς}^{(n)}(t) := (ς_{𝔩}^{(n)})^{−1}ι(t)ς_{𝔩}^{(n)}.

First we suppose 𝔩 is split. A direct computation shows that

ι_{ς}^{(n)}(t) = (\begin{matrix}1 & ϖ^{−n}d_{F}^{−1} \\ 0 & 1 \end{matrix}) \binom{t_{𝔏}}{0 t\overline{𝔏}} (\begin{matrix}1 & −ϖ^{−n}d_{F}^{−1} \\ 0 & 1 \end{matrix});
(ς_{𝔩}^{(n)})^{−1}Jς_{𝔩}^{(n)} = (\begin{matrix}1 & 0 \\ ϖ^{n}d_{F} & −1 \end{matrix}) .

By the definition of W_{𝔩}^{†}, we find that

ω^{−1}(det ς_{𝔩}^{(n)})P (π(ς_{𝔩}^{(n)})W_{𝔩}^{†}, χϕ)
= ∫_{O_{F}^{×}} ψ(−d_{F}^{−1}ϖ^{−n}x)ϕ_{𝔏}(x)d^{×}x · ∫_{O_{F}} ψ(d_{F}^{−1}ϖ^{−n}a)ϕ_{𝔏}^{−1}(a)d^{×}a
=ε(1, ϕ_{𝔏}^{−1}, ψ)ϕ_{𝔏}(−1)ε(1, ϕ_{𝔏}, ψ) · ζ_{F} (1)^{2}
= |ϖ^{n}D_{F} | L(1, τ_{E/F})^{2}.

This proves the formula in the split case. Now we suppose that 𝔩 is inert. We shall retain the notation in §3.8. Define m^{†} : G_{𝔩} → C by

m^{†}(g) := b_{𝔩}(π(g)W_{𝔩}^{†}, W_{𝔩}^{†}).

Then m^{†}(g) only depends on the double coset I_{𝔩}gI_{𝔩}. Put

P^{∗} := ∫_{E^{×}/F^{×}} m^{†}(ι_{ς}^{(n)}(t))χϕ(t)dt.

It is clear that

(7.3)  P (π(ς_{𝔩}^{(n)})W_{𝔩}^{†}, χϕ) = P^{∗} · \frac{L(1, τ_{E/F})ω(det ς_{𝔩}^{(n)})}{ζ_{F} (1)}.

For y ∈ ϖ^{r}O_{F}^{×}, it is easy to verify that ι_{ς}^{(n)}(1 + yθ) ∈ I_{𝔩} if r ≥ n and

ι_{ς}^{(n)}(1 + yθ) ∈ I_{𝔩}w (\begin{matrix}ϖ^{n−r} \\ & ϖ^{r−n} \end{matrix}) I_{𝔩} if 0 ≤ r < n (w = (\begin{matrix}0 & −d_{F}^{−1} \\ d_{F} & 0 \end{matrix})).

If x ∈ ϖO_{F}, then

ι_{ς}^{(n)}(x + θ) ∈ I_{𝔩}w (\begin{matrix}ϖ^{n} \\ & ϖ^{−n} \end{matrix}) I_{𝔩}.

Note that n = c_{𝔩}(ϕ) = c_{𝔩}(χϕ) as in (3.1). Combined with the above observations and Lemma 3.7, a direct computation shows that

P^{∗} = X_{n} · m^{†}(1) + (−X_{n}) · m^{†}(w (\begin{matrix}ϖ \\ & ϖ^{−1} \end{matrix}))

= b_{𝔩}(W_{𝔩}^{0} − π((\begin{matrix}ϖ^{−1} \\ & ϖ \end{matrix}))W_{𝔩}^{0}, W_{𝔩}^{†}) · X_{n}

= (\frac{μ_{𝔩}(ϖ)}{1 − |ϖ|} − \frac{ν_{𝔩}(ϖ)}{1 − μ_{𝔩}^{−1}ν_{𝔩}|·|(ϖ)}) · \frac{1 − μ_{𝔩}^{−1}ν_{𝔩}|·|(ϖ)}{μ_{𝔩}(ϖ) − ν_{𝔩}(ϖ)} · |D_{F} |^{\frac{1}{2}} X_{n}

= \frac{1}{1 − |ϖ|} · L(1, τ_{E/F}) |ϖ^{n}| |D_{E}|E^{\frac{1}{2}} .

The formula in the inert case follows from (7.3) immediately. □

7.4. Proof of Theorem 7.1. We prove Theorem 7.1 in this subsection. By the evaluation formula Proposition 7.3, it boils down to proving that

(7.4)  ∫_{Cl_{∞}^{−}} ϕdφ_{χ}^{†} ≢ 0 (mod 𝔪_{p}) for almost all ϕ ∈ 𝔛_{𝔩}^{0}.

By [Hid04a, Thm. 3.2, 3.3] together with the toric property of f_{χ}^{†} Lemma 7.2 (cf. [Hsi12, Lemma 6.1 and Remark 6.2]), the validity of (7.4) is reduced to verifying the following condition: (H′) For every u ∈ O_{F_{𝔩}} and a positive integer r, there exist β ∈ O_{F,(p)}^{×} and

a ∈ (A_{K,f}^{(pN)})^{×} such that β ≡ u (mod 𝔩^{r}) and
a_{β}(f_{χ}^{†}, 𝔠(a)) ≢ 0 (mod 𝔪_{p}).

The verification of (H′) under the assumptions (aiK) and μ_{p}(Ψ_{π,χ,v}) = 0 for all v|𝔠_{χ}^{−} follows from the same argument in Theorem 6.2. Note that for a polarization ideal 𝔠(a) (𝔠 = 𝔠(O_{K}), a ∈ (A_{K,f}^{(p𝔩)})^{×}) and a totally positive β ∈ O_{F,(p)}^{×} ∩ O_{F_{𝔩}}, we have (𝔠(a), p𝔩) = 1 and

a_{β}(f_{χ}^{†}, 𝔠(a)) = β^{k/2} ∏_{v∤p𝔩} a_{χ,v}^{∗}(βc_{v}^{−1}N(a_{v}^{−1})) · ν_{𝔩}|·|F_{𝔩}^{\frac{1}{2}}(β) (𝔦𝔩_{F}(c) = 𝔠).

Let u ∈ O_{F_{𝔩}} and let η^{u} = (η_{v}^{u}) be the idele in A^{×} such that a_{λ,v}^{∗}(η_{v}^{u}) ≢ 0 (mod 𝔪_{p}) for all v|𝔠_{χ}^{−}, η_{𝔩}^{u} = u and η_{v}^{u} = 1 for all v ∤ 𝔩𝔠_{χ}^{−}. To verify (H′), we simply proceed the Galois argument in Theorem 6.2, replacing η in by η^{u} therein. This completes the proof of Theorem 7.1.

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Ming-Lun Hsieh Department of Mathematics National Taiwan University No. 1, Sec. 4, Roosevelt R. Taipei 10617 Taiwan mlhsieh@math.ntu.edu.tw 768

Documenta Mathematica 19 (2014)