Waldspurger's period integral for newforms
Yueke Hu Jie Shu Hongbo Yin
math.NT [2010]{Primary 11F70}

Waldspurger's period integral for newforms]Waldspurger's period integral for newforms
Yueke Hu, Jie Shu and Hongbo Yin]Yueke Hu, Jie Shu and Hongbo Yin



In this paper we discuss Waldspurger's local period integral for newforms in new cases. The main ingredient is the work <cit.> on Waldspurger's period integral using the minimal vectors, and the explicit relation between the newforms and the minimal vectors. We use a representation theoretical trick to simplify computations for newforms. As an example, we compute the local integral coming from a special arithmetic setting which was used to study 3-part full BSD conjecture in <cit.>.

Department of Mathematics, ETH, Zurich, Switzerland
huyueke2012@gmail.com

School of Mathematical Sciences, Tongji University, Shanghai 200092
shujie@tongji.edu.cn
School of Mathematics, Shandong University, Jinan 250100,
P.R.China
yhb2004@mail.sdu.edu.cn

[2010]Primary 11F70

Yueke Hu is supported by SNF-169247; Jie Shu is supported by NSFC-11701092; Hongbo Yin is supported by NSFC-11701548 and The Fundamental Research Funds of Shandong University.

[
    [
    December 30, 2023
=====================







§ INTRODUCTION

In this paper we study Waldspurger's local period integral for newforms in new cases. Most of the paper is purely local, so we shall skip subscript v when there is no confusion.

Suppose that p≠ 2. Let π be a smooth irreducible self-dual representation of _2 over a p-adic field  with central character w_π, and χ be any character over a quadratic field extension / such that  w_πχ|_^×=1. 
For any test vector φ∈π, denote the local Waldspurger's period integral against a character χ on ^× by

    ={φ,φ} where {φ,φ'}=∫_t∈^×\^×Φ_φ,φ'(t)χ(t)dt.

Here Φ_φ,φ'(t)=(π(t)φ,φ') is the matrix coefficient associated to φ, φ'. 
We are particularly concerned about the case when π is a supercuspidal representation and φ is a proper translate of the newform for a fixed embedding of  (or equivalently φ is a fixed newform and the embedding of  can vary).

The explicit knowledges of the local period integral and the particular choice of test vectors have been useful for analytic number theory as well as arithmetic geometry problems. For example, they were used to study the moments and the subconvexity bound of L-functions in <cit.> and <cit.>.
In <cit.>, they were used to give general explicit Gross-Zagier and Waldspurger formulae, which is important to attack the refined BSD conjecture. 
In the most recent example, we used the explicit Gross-Zagier formula in <cit.> to study the 3-part full BSD conjecture for the elliptic curves related to the Sylvester conjecture. The byproduct of this paper  is a special example of  the local period integral which is not covered in the previous literatures, and is used in <cit.> to establish the explicit Gross-Zagier formula there.


 §.§ A short history of local test vector

The study of test vectors when there are ramifications was initiated in <cit.>. It assumes disjoint ramifications, and describes a test vector in terms of invariance by proper compact subgroups. 
The work in <cit.>   gives test vectors in more general situations on _2 side. In particular it solves the case when  is split. When  is a quadratic field extension, 
its method can deal with the range c(π_χ)>c(π), (here π_χ is the representation of _2 associated to χ via the Theta correspondence or Langlands correspondence,) and the test vectors used are diagonal translates of newforms.

The recent work <cit.> under mild assumptions provided test vectors for the complementary range c(π_χ)≤ c(π) when π is supercuspidal, using a new type of test vectors called minimal vectors. A particular minimal vector is given in the Kirillov model in Lemma <ref> below, and any single translate of this element is still considered to be a minimal vector.  Such test vectors arise naturally from compact induction theory for supercuspidal representations and have better properties than the standard newforms in terms of their matrix coefficients and Whittaker functionals.   For some applications however (in particular with <cit.> in mind), it's still necessary to understand the local period integral for the classical newforms. The purpose of this paper is thus to make use of <cit.> to predict the local period integral for the newforms.


 §.§ Main results

The goal of this paper is mainly to present the method, rather than exhausting all the cases. So we restrict ourselves to a special setting to avoid lengthy discussions. We shall assume that π is associated to a character θ over a ramified extension  via compact induction theory with c(θ)=2n, θ|_^×=w_π=1, 
 ≃ is also ramified,   and c(π_χ)≤ c(π). We also assume ϵ(π_×χ)=1 so that _^×(π,χ^-1)≠ 0.

For simplicity we shall fix the embedding of  into M_2 by

    x+y√(D')↦xyyD'x,

where v(D')=1. Any other embedding differs by a conjugation, which is effectively equivalent to a single translate for the test vector.

Using the explicit Kirillov model for a particular minimal vector φ_0 in Lemma <ref>, we can write the newform as a sum of minimal vectors in Corollary <ref>. Correspondingly we can write the period integral for the newform as a sum of period integrals for the minimal vectors in Corollary <ref>, which specialises in our setting to the following

    {φ_new,φ_new}=1/(q-1)q^⌈n/2 ⌉-1∑_x,x'∈ (_/ϖ^⌈n/2⌉_)^×{φ_x,φ_x'}.

Here φ_new=πϖ^n001φ_new is a diagonal  translate of the standard newform, and φ_x=πx001φ_0. The work in <cit.> computed the diagonal terms {φ_x,φ_x} which is either 0 or some constant only depending on the associated conductors.

In the case when there is a single diagonal term {φ_x,φ_x} in the family which is nonvanishing, one can use the representation theoretical trick in Lemma <ref> to easily get the following

When φ_x,φ_x≠ 0 for a single x∈ (_/ϖ^⌈n/2⌉_)^×, all off-diagonal terms vanish and

    φ_new,φ_new=1/(q-1)q^⌈n/2⌉-1φ_x,φ_x.


Note here that φ_x,φ_x is already computed in <cit.>.


The more challenging case is when there are several  non-vanishing diagonal terms. The corresponding off-diagonal terms will also be non-vanishing and have the same absolute value as the diagonal terms by Lemma <ref>. The main innovation of this paper is that we devised a way to detect the phase factor for the off-diagonal terms with relatively simple computations. 
Roughly speaking, the sizes of the non-vanishing off-diagonal terms are the same as those for the non-vanishing diagonal terms. The support of the integral (see Definition <ref>) can also be computed directly. By comparing the support of the integral with the size of the integral, we shall see that the integrand must be constant on the support of the integral. This constant is the phase factor and can be easily detected by taking special values.

In particular we obtain the following result.

Suppose that ≃ are ramified, c(θ)=2n, w_π=1, ϵ(π_×χ)=1, and 0<c(θχ)=2l≤ 2n with n-l even. Then

    I(φ_new,χ)=1/(q-1)q^⌈n/2 ⌉-11/q^⌊ l/2⌋ (1+θχ(√(D)))^2    .

In particular it is either 0 or asymptotically 1/C(π×π_χ)^1/4, where C(π×π_χ) is the conductor for the associated Rankin-Selberg L-function.




  * One feature used in our particular setting in Proposition <ref> is that there are exactly two non-vanishing diagonal terms in the expansion (<ref>). According to <cit.>, this is true in most of the other cases.

  * Unlike the range c(π_χ)>c(π), we see here the additional obstruction for using diagonal translate of newforms. 

  * Hopefully the same strategy works for more general cases, at least for those mentioned in (1).





We would like to point out that both Proposition <ref> and Proposition <ref> are new cases not known before.
As a motivation for these discussions, we give the local period integral in Proposition <ref> coming from a special arithmetic setting, 
where Proposition <ref> alone turns out to be enough. Such result is one of the key ingredients used in <cit.> to discuss the 3-part full BSD conjecture. 




§ NOTATIONS AND PRELIMINARY RESULTS



 §.§ Notations and basics









For a real number a, let ⌊ a⌋≤ a be the largest possible integer, and ⌈ a⌉≥ a be the smallest possible integer.

Let  be a p-adic field with residue field of order q, uniformizer ϖ=ϖ_, ring of integers _ and p-adic valuation v_. Let ψ be an additive character of . 
Assume that 2∤ q. For n≥ 1, let U_(n)=1+ϖ_^n _.
Let π be a supercuspidal representation over  with central character w_π=1. 

Let Ł  be a quadratic field extension over . Let e_Ł=e(Ł/) be the ramification index and v_Ł be the valuation on Ł. Let ϖ_Ł be a uniformizer for Ł. When Ł is unramified we shall identify ϖ_Ł with ϖ_. Otherwise we suppose that ϖ_Ł^2=ϖ_. Let x↦x to be the unique nontrivial involution of Ł/.
 Let ψ_Ł=ψ∘_Ł/. One can make similar definitions for a possibly different quadratic field extension . Note that we shall assume that ϖ_^2=ξϖ_ for ξ∈_^×-(_^×)^2 if  and Ł are both ramified and distinct.

For χ a multiplicative character on ^×, let c(χ) be  the smallest integer such that χ is trivial on 1+ϖ_^c(χ)_. Similarly c(ψ) is the smallest integer such that  ψ is trivial on ϖ_^c(ψ)_. We choose ψ  to be unramified, or equivalently,   c(ψ)=0. Then c(ψ_Ł)=-e_Ł+1.
Let c(π) be the power of the conductor of π.

When χ is a character over a quadratic extension, denote χ(x)=χ(x).

Suppose that p is large enough.
For a multiplicative character ν over  with c(ν)≥ 2, there exists α_ν∈^× with v_(α_ν)=-c(ν)+c(ψ) such that for u∈ϖ__

    ν(1+u)=ψ(α_νlog(1+u))

where log(1+u) is the standard Taylor expansion for logarithm 

    log(1+u)=u-u^2/2+u^3/3+⋯.

In particular when p>2 we have that 

    ν(1+u)=ψ(α_ν (u-u^2/2))

for any u∈ϖ_^⌊ c(ν)/2⌋_.



For a multiplicative character ν over  with c(ν)≥ 2, there exists α_ν∈^× with v_(α_ν)=-c(ν)+c(ψ) such that

    ν(1+u)=ψ(α_ν u)

for any u∈ϖ_^⌈ c(ν)/2⌉_. α_ν is determined ϖ_^-⌈ c(ν)/2⌉+c(ψ)_.

One can easily check this lemma by using that ν(1+u) becomes an additive character in u for u∈ϖ_^⌈ c(ν)/2⌉_.


We shall also need some basic results for compact induction theory and matrix coefficient.
In general, let G be a unimodular locally profinite group with center Z. Let H⊂ G be an open and closed subgroup containing Z with H/Z compact. Let ρ be an irreducible smooth representation of H with unitary central character and 

    π=c-_H^G(ρ)={f:G→ρ| f(hg)=ρ(h)f(g) ∀ h∈ H,  f is compactly supported}.

  By the assumption on H/Z, ρ is automatically unitarisable, and we shall denote the unitary pairing on ρ by ·,·_ρ. Then one can define the unitary pairing on π by

    ϕ,ψ=∑_x∈ H\ Gϕ(x),ψ(x)_ρ.

If we let y∈ H\ G and {v_i} be a basis for ρ, the elements

    f_y,v_i(g)=ρ(h)v_i,    if  g=hy∈ Hy;
    
    0,    otherwise.
 
form a basis for π.

For y,z∈ H\ G,

    π(g)f_y,v_i,f_z,v_j=ρ(h)v_i,v_j_ρ,     if g=z^-1hy∈ z^-1Hy;
    
    0,    otherwise.



We also recall the multiplicity one result and Tunnell-Saito's epsilon value test.

Suppose that w_π=χ|_^×. Let π^ be the image of π under the Jacquet-Langlands correspondence.
The space _^×(π^⊗χ^-1,) is at most one-dimensional. It is nonzero if and only if 

    ϵ(π_×χ^-1)=χ(-1)ϵ().

Here π_ is the base change of π to . ϵ()=1 if it is a matrix algebra, and -1 if it's a division algebra.




 §.§ Compact induction theory for supercuspidal representations and minimal vectors

We shall review the compact induction theory for supercuspidal representations on _2. For more details, see <cit.>.

We shall fix the embeddings and work out everything explicitly. For v_(D')=0 or 1, we shall refer to the following embedding of a quadratic field extension =(√(D')) as a standard embedding:

    x+y√(D')↦xyyD'x.


The supercuspidal representations are parametrized via compact induction by characters θ over some quadratic field extension Ł. More specifically we have the following quick guide. 

  Case 1.c(π)=2n+1 corresponds to e_=2 and c(θ)=2n .

  Case 2. c(π)=4n corresponds to e_=1 and c(θ)=2n.

  Case 3. c(π)=4n+2 corresponds to e_=1 and c(θ)=2n+1 .



For e_=1,2, let 
    _e_=
    M_2(_),  if e_=1,
    __ϖ__, otherwise.

Its Jacobson radical is given by

    _e_=ϖ M_2(_),  if e_=1,
    ϖ__ϖ_ϖ_, otherwise.



Define the filtrations of compact open subgroups as follows:

    K__e_(n)=1+_e_^n,    U_(n)=1+ϖ_^n_.

Note that each K__e_(n) is normalised by ^× which is embedded as in (<ref>).

Denote 
J=^× K__e_(⌊ c(θ)/2⌋), J^1=U_(1)K__e_(⌊ c(θ)/2⌋), H^1=U_(1)K__e_(⌈ c(θ)/2⌉). Then θ on ^× can be extended to be a character θ̃ on H^1 by

    θ̃(l (1+x))=θ(l)ψ∘ (α_θ x),

where l∈^×, 1+x∈ K__e_(⌈ c(θ)/2⌉) and α_θ∈^×⊂ M_2() is associated to θ by Lemma <ref> under the fixed embedding.

When c(θ) is even, H^1=J^1 and θ̃ can be further extended to J by the same formula. In this case denote Λ=θ̃  and π=c-_J^GΛ.

When c(θ) is odd, J^1/H^1 is a two dimensional vector space over the residue field. This case only occurs when c(π)=4n+2 as listed above.
Then there exists a q-dimensional representation Λ of J such that
Λ|_H^1 is a multiple of θ̃, and
Λ|_^×=⊕θν where ν is over Ł^×, c(ν)=1 and ν|_^×=1. More specifically, let B^1 be any intermediate group between J^1 and H^1 such that B^1/H^1 gives a polarisation of J^1/H^1 under the  pairing given by ψ_α_θ.

    (1+x,1+y)↦ψ∘ (α_θ [x, y]).

 Then θ̃ can be extended to B^1 by the same formula (<ref>) and  Λ|_J^1=_B^1^J^1θ̃.
 Again π=c-_J^GΛ in this case.

 In the case J^1=H^1, we take B^1=J for uniformity. In either cases, we have w_π=θ|_^×.


There exists a unique element φ_0∈π such that B^1 acts on it by θ̃. (Type 1 minimal vector in the notation of <cit.>.)
We also call any single translate π(g)φ_0 a minimal vector,

Note that the conjugated group gB^1g^-1 acts on π(g)φ_0 by the conjugated character θ̃^g and They form an orthonormal basis of π.

Let Φ_φ_0 be the matrix coefficient associated to a minimal vector φ_0 as above. Then Φ_φ_0 is supported on J, and

    Φ_φ_0(bx)=Φ_φ_0(xb)=θ̃(b)Φ_φ_0(x)

for any b∈ B^1. Furthermore when Λ≠ 1, Φ_φ_0|_J^1 is supported only on B^1.

Note that φ_0 is basically f_1,v_i as in (<ref>) for the coset representative 1∈ J\ G. The corollary follows immediately from the definition of φ_0 and Lemma <ref>.



 §.§ Local Langlands correspondence and compact induction

Here we describe the relation between the compact induction parametrisation and the local Langlands correspondence. See <cit.> Section 34 for more details.

For a field extension Ł/ and an additive character ψ over , let λ_Ł/(ψ) be the Langlands λ-function in <cit.>. When Ł/ is a quadratic field extension, let η_Ł/ be the associated quadratic character. By <cit.>, we have for ψ_β(x)=ψ(β x),

    λ_Ł/(ψ_β)=η_Ł/(β)λ_Ł/(ψ).





  * If  is inert, define Δ_θ to be the unique unramified character of ^× of order 2.


  * If  is ramified and θ is a character over  with c(θ)>0 even, associate α_θ to θ as in Lemma <ref>.




Then define Δ_θ to be the unique level 1 character of ^× such that

    Δ_θ|_^×=η_/, Δ_θ(ϖ_)=η_/(ϖ_^c(θ)-1α_θ)λ_/^c(θ)-1(ψ).







Note that in  <cit.>   ψ is chosen to be of level 1. We have adapted the formula there to our choice of ψ using (<ref>).
The definition is also independent of the choice of ϖ_.    

If π is associated by compact induction to a character θ over a quadratic extension , then its associated Deligne-Weil representation by local Langlands correspondence is σ=_^ (Θ), where Θ=θΔ_θ^-1, or equivalently θ=ΘΔ_Θ.

Note here that Θ and θ always differ by a  level ≤ 1 character, so α_Θ can be chosen to be the same as α_θ in Lemma <ref> and Δ_Θ=Δ_θ.





 §.§ Using minimal vectors for Waldspurger's period integral 

Now we review the local Waldspurger's period integral for minimal vectors, in the setting e_Ł=2 and ≃Ł. For more details and proofs, see <cit.>, and in particular its appendix for explicit treatment.

For simplicity we pick 
    D'=1/α_θ^2ϖ_Ł^2c(θ),

identify 1/α_θϖ_Ł^c(θ) with √(D') and Ł with (√(D')), and use the standard embedding (<ref>).





By this choice, we have v_(D')=0 if e_Ł=1 and v_(D')=1 if e_Ł=2.


    α_θ= 1/ϖ_Ł^c(θ)1/√(D')↦1/ϖ^c(θ)/e_Ł01/D'10.

Such choice is not essential. A different choice will result in slightly different formulation, for example, in (<ref>), but the final results are similar.

Note that when e_Ł=2, c(θ) must be even when θ|_^*=1. 
 Assume that =(√(D)) for v_(D)=0,1 is also embedded as

    x+y√(D)↦xyyDx.


In <cit.>, test vectors of form π(g)φ_0 were used to study  for general combinations of π,  and χ. 
For the purpose of this paper we shall only review the case when ≃ are ramified,  w_π=1, c(π)=2n+1 is odd and c(π_χ)≤ c(π). 

We shall normalize the Haar measure on ^× so that ( _^×\_^×)=1. Then (^×\^×)=2.

In this case we use test vectors of the form  
    φ=π1u01v001φ_0

for some u∈_,v∈_^×. Recall that when e_=2, 

    K__e_(n)=1+ϖ^⌈ n/2⌉_ϖ^⌊ n/2⌋_ϖ^⌊ n/2⌋+1_ϖ^⌈ n/2⌉_,

and J=^× K__e_(n) acts on φ_0 by a character, so we can assume that v∈ (_/ ϖ^⌈ n/2⌉_)^×.

There are two situations depending on min{c(θχ), c(θχ)
}. Note that for the embedding of  fixed above, we have

    -1001xyyDx-1001=x-y-yDx

and thus

    I(φ,χ)=Iπ-1001φ, χ.

So we shall always assume that c(θχ)≤ c(θχ).


The first situation is when c(θχ)=0, then the Tunnell-Saito's test requires θχ to be trivial for  to be nonzero. In that case we can take u=0 and then there is a unique vϖ^⌈ n/2⌉ such that ≠ 0, and 
for this v we have 
    =(^×\^×)=2.


The second situation is when 0<c(θχ)=2l≤ 2n. In this case, α_θχ can be associated to θχ by Lemma <ref> and φ would be a test vector if v, u are solutions of the following quadratic equation:

    D/D'v^2-2ϖ^nα_θχ√(D)-2√(D/D')v+(1-Du^2)≡ 0ϖ^n-⌊l/2⌋.


This implies that for fixed u, the discriminant of the equation 

    Δ(u)=4ϖ^nα_θχ√(D)ϖ^nα_θχ√(D)-2√(D/D')+4D/D'Du^2

has to be a square ϖ^n-⌊l/2⌋.
When n-l is even, we can pick u=0 directly. Whether Δ(0) is a square is consistent with Tunnell-Saito's test.
When Δ(0) is indeed a square, we get two solutions of vϖ^⌈ n/2⌉. 
For each of these two solutions we have

    =1/q^⌊ l/2⌋.


Now if n-l is odd, v_(Δ(0))=n-l is odd, thus Δ(0) can never be a square. We need to pick u such that v_(u)=n-l-1/2 and Δ(0)+4D/D'Du^2 can be of higher evaluation and a square. Whether this is possible is again consistent with Tunnell-Saito's test.
 In this case it's possible to get more or less solutions of vϖ^⌈ n/2⌉. For each solution we have

    =1/q^⌊ l/2⌋.



 §.§ Kirillov model for minimal vectors

Here we describe the minimal vectors explicitly in the Kirillov model. For this purpose, we choose the special intermediate subgroup B^1=U_(1)K__2(2n+1) in the case e_=1 and c(π)=4n+2. Recall we choose D' such that  

    α_θ= 1/ϖ_^c(θ)1/√(D')↦1/ϖ^c(θ)/e_01/D'10.

We define the intertwining operator from π to its Whittaker model by

    φ↦ W_φ(g)=∫_Φ_φϖ^⌊ c(π)/2⌋0011n01gψ(-n)dn.

A particular minimal vector was given in the Kirillov model in <cit.> under this operator.
 

Up to a constant multiple, a minimal vector φ_0 is given in the Kirillov model by the following: 

  * When c(π)=4n, φ_0=(ϖ^-2nU_(n)).

  * When c(π)=2n+1, φ_0=(ϖ^-nU_(⌈ n/2 ⌉)).

  * When c(π)=4n+2, φ_0=(ϖ^-2n-1U_(n+1)).















The newform φ_new can be related to φ_0 by the following formula

    φ_new=1/√((q-1)q^⌈c(θ)/2 e_⌉-1)∑_x∈ (_/ϖ^⌈c(θ)/2 e_⌉_)^×πϖ^-c(θ)/e_x001φ_0.

Here φ_0 and φ_new are both L^2-normalized.


By the previous lemma one can uniformly write

    φ_0=√((q-1)q^⌈c(θ)/2 e_⌉-1)(ϖ^-c(θ)/e_ U_(⌈c(θ)/2 e_⌉)).

The coefficient comes from the L^2-normalization of φ_0, as 
    (U_(⌈c(θ)/2 e_⌉))=1/(q-1)q^⌈c(θ)/2 e_⌉-1.

Then one just has to use that φ_new=(_^×) in the Kirillov model. 






§ WALDSPURGER'S PERIOD INTEGRAL USING NEWFORMS

In the last section we reviewed the local Waldspurger's period integral for minimal vectors. In this section we show how to work out the local Waldspurger's period integral for newforms. Using the relation between the newform and the minimal vectors in Corollary <ref>, and the bilinearity of Waldspurger's period integral, we can write the integral for the newform as a sum of integrals for the minimal vectors. In the case when there is a single non-vanishing term, we get the integral very easily in Section <ref>, which turns out to be enough for the special example in Section <ref>. We further illustrate how to evaluate the off-diagonal terms in Section <ref> in more general cases for possible future applications.


 




As in the introduction we denote

    φ_1,φ_2=∫_t∈^×\^×(π(t)φ_1,φ_2)χ(t)dt

for the embedding of  as in (<ref>). Using its bilinearity and Corollary <ref>, we immediately have the following:


Let φ_new=πϖ^c(θ)/e_001φ_new, φ_x=πx001φ_0 Then

    φ_new,φ_new=1/(q-1)q^⌈c(θ)/2 e_⌉-1∑_x,x'∈ (_/ϖ^⌈c(θ)/2e_⌉_)^×φ_x,φ_x'.






  * Suppose that φ_x,φ_x=0, then  φ_x',φ_x=φ_x,φ_x'=0 for any x'.

  * Suppose that |φ_x,φ_x|=|φ_x',φ_x'|. Then |φ_x,φ_x'|= |φ_x,φ_x|.




For any nontrivial functional ℱ∈_^×(π⊗χ,), we have φ_1,φ_2=C ℱ(φ_1)ℱ(φ_2) for some non-zero constant C independent of the test vectors, as _^×(π⊗χ,)≤ 1.
Then

    |φ_x,φ_x|=|Cℱ(φ_x)^2|,


    |φ_x',φ_x'|=|Cℱ(φ_x')^2|,


    |φ_x,φ_x'|=|Cℱ(φ_x)ℱ(φ_x')|.

Now the results are clear.



Note that the diagonal terms where x=x' are already known by Section <ref>. 










 §.§ The special case


When φ_x,φ_x≠ 0 for a single x∈ (_/ϖ^⌈c(θ)/2e_⌉_)^×, all off-diagonal terms vanish and

    φ_new,φ_new=1/(q-1)q^⌈c(θ)/2 e_⌉-1φ_x,φ_x.



It follows from Lemma <ref> that all the off-diagonal terms vanish, and only a single diagonal terms is non-vanishing. The proposition now follows directly from Corollary <ref>.


This case happens mostly when c(θχ) or c(θχ)≤ 1. But there are other possibilities, as we shall see in the special example in Section <ref>.






 §.§ The general cases

Now we explain how to work out the off-diagonal terms in Corollary <ref> in more general cases. 

By the support of the local Waldspurger's period integral {φ,φ'}, we mean the set ^×∩Φ_φ,φ'.


The main idea is to get the size of the off-diagonal terms by Lemma <ref>, and to get the support of the integral in Lemma <ref>. The volume of the support of the integral is exactly the size of the integral, while the integrand is absolutely bounded by 1. This forces the integrand to be constant (with absolute value 1) on the support of the integral. Then one can easily detect this constant by looking at the value of the integrand at any point in the support of the integral.

For simplicity, however, we stay in the setting where ≃ are ramified, 0<c(θχ)=2l≤ 2n. We further assume that n-l is even. By Section <ref>, we can pick u=0, and there exists 2 solutions v,v' ϖ^⌈ n/2⌉ to (<ref>), while the diagonal terms are always 1/q^⌊ l/2⌋ for these 2 solutions.


According to Lemma <ref>, we can write that φ_v,φ_v'= γφ_v,φ_v for some phase factor γ with |γ|=1. Then

    I(φ_new,χ)   =1/(q-1)q^⌈c(θ)/2 e_⌉-1(φ_v,φ_v+φ_v,φ_v'+φ_v',φ_v+φ_v',φ_v')
       =1/(q-1)q^⌈c(θ)/2 e_⌉-11/q^⌊ l/2⌋(1+γ)(1+γ).


To study γ,  we first study the support of the integral. Without loss of generality, we can assume that v, v' satisfy

    D/D'v^2-2ϖ^nα_θχ√(D)-2√(D/D')v+1=0,

compared to (<ref>) while taking u=0.

Let k=v001 and k'=v'001.
Then for t=abbDa,

    k'^-1tk=v v'^-1a v'^-1bvbDa,

and

    φ_v,φ_v'=∫_^×\^×Φ_φ_0(k'^-1tk)χ(t)dt.


For the setting as above, we have vv'D=D', and v_(v/v'-1)= n-l/2.
In particular the support of the integral 
is v_(b)=0, v_(a)≥⌈l+1/2⌉.


According to (<ref>), v and v' satisfy 

    v v'=D'/D,   v+v'=2(ϖ^nα_θχ√(D')-1)√(D'/D)

so the first result is direct.
For the second result, we note that

    v/v'-1^2=(v+v')^2-4vv'/v'^2=D'/D4ϖ^nα_θχ√(D')(ϖ^nα_θχ√(D')-2)/v'^2.

Thus v_((v/v'-1)^2)=n-l and v_(v/v'-1)= n-l/2. 
Now for k'^-1tk∈ J=^× K__e_(n), there are two parts to consider: either  v_(b)=0, v_(a)>0 or v_(a)=0, v_(b)≥ 0. 
In the first case, since 
    ϖ__e_ ^n=ϖ^⌈n+1/2⌉_ϖ^⌊n+1/2⌋_ϖ^⌊n+1/2⌋+1_ϖ^⌈n+1/2⌉_
 we must have

    a(v/v'-1)≡ 0 ϖ^⌈n+1/2⌉,   vbD-v'^-1bD'≡ 0 ϖ^⌊n+1/2⌋ +1.

The second equation is automatic as v v'=D'/D. From the first equation and the computation for v_(v/v'-1) above, we get v_(a)≥⌈l+1/2⌉.
Using similar argument, one can easily show that it is impossible for k'^-1tk∈^× K__e_(n) when v_(a)=0, v_(b)≥ 0.


Suppose that ≃ are ramified, c(θ)=2n, w_π=1, ϵ(π_×χ)=1, and 0<c(θχ)=2l≤ 2n with n-l even. Then

    I(φ_new,χ)=1/(q-1)q^⌈c(θ)/2 e_⌉-11/q^⌊ l/2⌋ (1+θχ(√(D)))^2    .

In particular it is either 0 or asymptotically 1/C(π×π_χ)^1/4.


We already know that |φ_v,φ_v'|=1/q^⌊ l/2⌋. By the previous lemma the support of the integral also has volume 1/q^⌊ l/2⌋, while the integrand  |π(t)φ_v,φ_v'χ(t)|≤ 1. Then  π(t)φ_v,φ_v'χ(t) must be some constant γ on the whole support with |γ|=1. To detect this constant we just have to take t=01D0. Then

    γ=Φ_φ_00v'^-1vD0χ(√(D))=θ(v'^-1√(D'))χ(√(D))=θχ(√(D)).

In the last equality we have used θ|_^×=1 so that 
    θ(v'^-1√(D'))=θv'^-1√(D')/√(D)√(D)=θ(√(D)).
 Note that θχ(√(D))=± 1.
The last statement is easy to check.









§ A SPECIAL EXAMPLE FROM ARITHMETIC GEOMETRY

Now we specialize to the case required in the proof of <cit.>. 
We shall first review the global arithmetic setting, and use the subscripts to indicate the local components.

For an arbitrary nonzero integer n, let E_n be the elliptic curve defined by the affine equation x^3+y^3=n. Then E_n has complex multiplication by the field K=(√(-3)). For a prime p≡ 4,7 9, the elliptic curve E_p is closely related to the well-known Sylvester conjecture. Let π be the automorphic representation of _2() corresponding to E_9 and π_3 the 3-adic part of π. Then c(π_3)=5. 
Assume that f_3 is the standard newform of π_3. Let χ:(K̅/K)_K^× be the character given by χ(σ)=(√(3p))^σ-1. We also view χ as a Hecke character on _K by the Artin map.
Then c(χ_3)=4. 
The following normalized Waldspurger's period integral

    β^0_3(f_3, f_3)
    =∫_t∈_3^×\ K_3^× (π(t)f_3,f_3)/(f_3,f_3)χ_3(t)dt

appears in the proof of the explicit Gross-Zagier formula for E_p in <cit.>. Here K is embedded into M_2() as in <cit.> which linearly extends the following map:

    √(-3)↦4p+17+72/p-8p/9-4-18/p18p+72+288/p-4p-17-72/p=:a3^-2b3^3c-a

with 3||a if p≡ 4 9, 9||a if p≡ 7 9, b≡ p 9 and c≡ -1 9.  Then (√(-3))=-a^2-3bc=3.
 Note here that K is embedded differently from the fixed embedding we have been using in (<ref>). We choose the notation β_3^0  to reflect this difference, and to be consistent with the notation in <cit.>. We shall  work out the relation between these two embeddings later on.

 




To apply the results in the previous section to compute (<ref>), we take ϖ=3=q, D=-3, K_3≃≃≃(√(-3))_3, c(θ_3)=c(χ_3)=4, n=2.

By Lemma <ref> we have the minimal vector φ_0=(ϖ^-2U_(1)) in the Kirillov model. As seen from previous sections, more accurate information is needed. 


 §.§ Local characters associated to the arithmetic information


First of all we make use of the arithmetic information to give the local characters explicitly. Recall that K=(√(-3)) is an imaginary quadratic field and _K=[ω] is its ring of integers with ω=-1+√(-3)/2. Let Θ:K^×\_K^×^× be the unitary Hecke character associated to the base-changed CM elliptic curve E_9_/K. 

Then Θ has conductor 9_K. For any place v of K, let Θ_v be the local component of Θ at the place v. Then Θ_v is the character used to construct the local Weil-Deligne representation in Theorem <ref>. We denote Θ_3 the 3-part of Θ. Then π_3 is the local representation of _2(_3) corresponding to Θ_3. Note

    _K,3^×/(1+9_K,3)≃⟨± 1⟩ ^/2×⟨ 1+√(-3)⟩^/3×⟨ 1-√(-3)⟩^/3×⟨ 1+3√(-3)⟩^/3.


We have c(Θ_3)=4. Its values are

given explicitly by

    Θ_3(-1)=-1,  Θ_3(1+√(-3))=-1-√(-3)/2,  Θ_3(√(-3))=i,


    Θ_3(1-√(-3))=-1+√(-3)/2,  Θ_3(1+3√(-3))=-1+√(-3)/2.


    



It is well-known that Θ_∞(x)=||x||/x, (see for example <cit.> and note that we normalize it to make it unitary) and Θ is unramfied when 3∤ v. Note

    Θ_∞(-1)Θ_3(-1)=1,  Θ_∞(-1)=-1.

So Θ_3(-1)=-1.

Let =(a) be a prime of K  relatively prime to 6, with the unique generator a≡ 2 3.
By <cit.>, we have 

    Θ()=-_^-1/2(-3/a)_6a.

Where (·/a)_6 is the sixth power residue symbol and _ is the norm of .
If =(5), then

    Θ_5(5)=-(-3/5)_6=-1.


By

    Θ_∞(10)Θ_2(10)Θ_3(10)Θ_5(10)=1,

we get Θ_2(2)=-1. Since Θ_2 is unramified and 1+√(-3) is another uniformizer of (2), we see Θ_2(1±√(-3))=Θ_2(2)=-1.
By

    Θ_∞(1±√(-3))Θ_2(1±√(-3))Θ_3(1±√(-3))=1,  Θ_∞(1±√(-3))=2/1±√(-3),

we get

    Θ_3(1±√(-3))=-1∓√(-3)/2.

The other evaluations can be obtained in a similar way.












The local  character  χ_3 has conductor _3^×(1+9_K,3), and hence it is in fact a character of the quotient group _K,3^×/_3^×(1+9_K,3). Note that

    _K,3^×/_3^×(1+9_K,3)≃⟨ 1+√(-3)⟩^/3×⟨ 1+3√(-3)⟩^/3.

We have the following, 
 
c(χ_3)=4 and χ_3|__3^×=1. Its values are given explicitly by the following table:



p 9   χ_3(1+√(-3))   χ_3(1+3√(-3))   χ_3(√(-3))

4   ω   ω   1

7   ω^2   ω   1







This follows directly from the explicit local class field theory.
We note that all the elements in 1+9_K,3 is a cube in K_3. 







Hence for any t∈ K_3^×,

    χ_3(t)=(√(3p))^σ_t-1=t,3p/K_3;3={t,12/K_3;3,      p≡ 4 9;
    t,21/K_3;3,      p≡ 7 9.
    .

Recall σ_t is the  image of t under the the Artin map, and ·,·/K_3;3 denotes the 3-rd Hilbert symbol over K_3^× as in <cit.>. Using the local and global principal, it is straight-forward to compute the values of χ_3 as in the above table. For example,  

    χ_3(1+√(-3))=1+√(-3),3/K_3;31+√(-3),4/K_3;3=1+√(-3),3/K_2;3^-11+√(-3),4/K_2;3^-1=ω,

where the second equality uses the fact that the product of the Hilbert symbol at all places equals 1 and the Hilbert symbol we considered is unramified outside the place 2 and 3,  the third
equality uses the formula in <cit.>. 




 §.§ Local period integral



From Section <ref>, we see that the test vector issue for Waldspurger's period integral is closely related to c(θ_3χ_3) or c(θ_3χ_3) and some further details like α_θ_3χ_3. We can work out these  by using Lemma <ref>, <ref>, and the relation between θ_3 and Θ_3 as in Theorem <ref>.


If p≡ 4 9,
the local character Θ_3χ_3 is given explicitly by



    Θ_3χ_3(-1)=-1,  Θ_3χ_3(1+√(-3))=ω,


    Θ_3χ_3(1-√(-3))=ω^2,  Θ_3χ_3(1+3√(-3))=1,  Θ_3χ_3(√(-3))=i.



If p≡ 7 9,
the local character Θ_3χ_3 is given explicitly by



    Θ_3χ_3(-1)=-1,  Θ_3χ_3(1+√(-3))=1,


    Θ_3χ_3(1-√(-3))=1,  Θ_3χ_3(1+3√(-3))=1,  Θ_3χ_3(√(-3))=i.



Now we can prove the following key Lemma in our special case.

When p≡ 7 9, we have θ_3χ_3=1. When p≡ 4 9, we have c(θ_3χ_3)=2 and α_θ_3χ_3=1/3√(-3).


Let ψ_3 be the additive character such that ψ_3(x)=e^2π i ι(x) where ι:_3→_3/_3 ⊂/ is the map given by x↦ -x_3 which is compatible with the choice in <cit.>. Let ψ_K_3(x)=ψ_3∘_K_3/_3(x), be the additive character of K_3. 

Recall that α_Θ_3 is the number associated to Θ_3 as in Lemma <ref> so that

    Θ_3(1+x)=ψ_K_3(α_Θ_3 x),

for any x satisfying v_K_3(x)≥ c(Θ_3)/2=2. By the definition of ψ_K_3 and Lemma <ref>, we know that α_Θ_3=1/9√(-3).  Now let η_3 be the quadratic character associated to the quadratic field extension K_3/_3.

Then by <cit.>, λ_K_3/_3(ψ')=τ(η_3,ψ_3')/√(3)=-i, here τ(η_3,ψ_3') is the Gauss sum and ψ_3'(x)=ψ_3(x/3) is the additive character of level one. By <cit.>, λ_K_3/_3(ψ_3)=η_3(3)λ_K_3/_3(ψ_3')=-i. Then Δ_θ_3 is the unique level one character of K_3 such that Δ_θ_3|__3^×=η_3 and

    Δ_θ_3(√(-3))=η((√(-3))^3α_Θ_3)λ_K_3/_3(ψ_3)^3=-i.

Recall that θ_3=Θ_3Δ_θ_3. Then by Corollary <ref> we can easily check that: 



  * If p≡ 7 9, θ_3χ_3 is the trivial character. 

  * If p≡ 4 9, θ_3χ_3 is of level 2 and by definition we can choose α_θ_3χ_3=1/3√(-3).




Suppose (_3^×\_K,3^×)=1 so that (_3^×\ K_3^×)=2.
For f_3 being the newform, K being embedded as in (<ref>) and θ_3,χ_3 as given above, we have

    β^0_3(f_3,f_3)=
     1,     if p≡ 7 9
    
     1/2,     if p≡ 4 9.




We may assume f_3 to be L^2-normalized. To evaluate f_3 for the embedding in (<ref>) is equivalent to use the standard embedding (<ref>) of  and use different translate of the newform.
In particular the embedding in (<ref>) is conjugate to the standard embedding by the following. 

    a3^-2b3^3c-a=-9ca/301^-101D0-9ca/301.

Thus  we have

    β^0_3(f_3,f_3)   =∫_^×\^×π_3-9ca/301^-1t-9ca/301f_3,f_3χ(t)dt
       =∫_^×\^×π_3t-9ca/301f_3,π_3-9ca/301f_3χ(t)dt,

which is by definition π_3-9ca/301f_3,π_3-9ca/301f_3 for the bilinear pairing as in (<ref>) and the standard embedding as in (<ref>).

Note that by Corollary <ref>

    π_3-9ca/301f_3
    =1/√(2)∑_x∈ (O_v/ϖ O_v)^×π_31a/301x001φ_0

where φ_0 is the minimal test vector.

Now there are two cases. When p≡ 7 9, 9|| a and the action of 1a/301 on φ_x=π_3x001φ_0 is by a simple character. By the l=0 case in Section <ref>, we have a unique xϖ for which φ_x,φ_x is nonvanishing. According to Proposition <ref>
 there are no off-diagonal terms, and we have

    β^0_3f_3,f_3 =1/(q-1)q^⌈c(θ_3)/2 e_⌉-1φ_x,φ_x=1/2· 2=1.


When p≡ 4 9, we have 3||a and u=a/3. By Lemma <ref>, this is the case l=1 and n-l=1 is odd. By the choice in Section <ref>, 

    D'=1/α_θ_3^2ϖ_^2c(θ_3)=-3.


By Lemma <ref>, α_θ_3χ_3^-1=1/3√(-3) in this case, and we have 

    Δ(u)   = 4ϖ^nα_θ_3χ_3√(D)ϖ^nα_θ_3χ_3√(D)-2√(D/D')+4D/D'Du^2 
       ≡ 4· 9·1/3√(-3)·√(-3)· (-2)+4 · (-3)a^2/9ϖ^2
       ≡  -8· 3-4· 3 ϖ^2
       ≡ 0 ϖ^2.

Δ(u) is indeed  congruent to a square. Then we can get a unique solution of vϖ from (<ref>), and again by Proposition <ref>,

    β^0_3f_3,f_3=1/(q-1)q^⌈c(θ_3)/2 e_⌉-11/q^⌊ l/2⌋=1/2.



Let f' be the admissible test vector of (π, χ) which is as defined in <cit.>. By definition, the 3-adic part f_3' is χ_3^-1-eigen under the action of K_3^×. The following corollary is most directly used in <cit.>.


For the admissible test vector f'_3 and the newform f_3 we have 

    β_3^0(f'_3,f'_3)/β_3^0(f_3,f_3)=
     2,     if p≡ 7 9,
    
     4,     if p≡ 4 9.



Keep the normalization of the volumes in Proposition <ref>. By definition of f', we have β_3^0(f'_3,f'_3)=(_3^×\ K_3^×)=2. Then the corollary follows from Proposition <ref>.


alpha

