Open Access

Cuspidal part of an Eisenstein series restricted to an index 2 subfield

Research in Number Theory20162:33

DOI: 10.1007/s40993-016-0061-7

Received: 28 September 2016

Accepted: 3 October 2016

Published: 25 November 2016

Abstract

Let \({\mathbb {E}}\) be a quadratic algebra over a number field \({\mathbb {F}}\). Let E(gs) be an Eisenstein series on \(GL_2({\mathbb {E}})\), and let F be a cuspidal automorphic form on \(GL_2({\mathbb {F}})\). We will consider in this paper the following automorphic integral:
$$\begin{aligned} \int \limits _{Z_{{\mathbb {A}}}{\text {GL}}_{2}({\mathbb {F}})\backslash {\text {GL}}_{2}({\mathbb {A}}_{{\mathbb {F}}})} F(g)E(g,s) dg. \end{aligned}$$
This is in some sense the complementary case to the well-known Rankin–Selberg integral and the triple product formula. We will approach this integral by Waldspurger’s formula, giving a criterion about when the integral is automatically zero, and otherwise the L-functions it represents. We will also calculate the local integrals at some ramified places, where the level of the ramification can be arbitrarily large.

1 Background

In this paper we are interested in the cuspidal part of an Eisenstein series restricted to an index 2 subfield. More specifically, let \({\mathbb {E}}\) be a quadratic algebra over a number field \({\mathbb {F}}\). Let F be a cusp form of a cuspidal automorphic representation \(\pi \) on \({\text {GL}}_2({\mathbb {A}}_{\mathbb {F}})\). Let E(gs) be an Eisenstein series over \({\mathbb {E}}\), defined from two characters \(\chi _1\) and \(\chi _2\) over \({\mathbb {E}}^*\). (see (2.4) for more details of the definition) It is well-known that such Eisenstein series is in the continuous spectrum for \(L^2({\text {GL}}_2({\mathbb {E}})\backslash {\text {GL}}_2({\mathbb {A}}_{\mathbb {E}}))\). Its integral against a cusp form on \({\text {GL}}_2({\mathbb {A}}_{\mathbb {E}})\) will simply be zero.

But we are interested in the spectral decomposition of E(gs) when we restrict it to \({\text {GL}}_2({\mathbb {A}}_{\mathbb {F}})\). In particular we consider the following integral:
$$\begin{aligned} {\mathbb {I}}(E,F,s)=\int \limits _{Z_{{\mathbb {A}}}{\text {GL}}_{2}({\mathbb {F}})\backslash {\text {GL}}_{2}({\mathbb {A}}_{{\mathbb {F}}})} F(g)E(g,s) dg. \end{aligned}$$
(1.1)
This integral is not necessarily zero. We would like to see when this integral is automatically zero and otherwise how \({\mathbb {I}}(E,F,s)\) depends on s.

In addition to its own interest, this automorphic integral is in some sense the complementary case to the well-known Rankin–Selberg integral and triple product formula. It’s also a special case of the automorphic integral related to arithmetic height pairing on certain Shimura varieties according to the main theorem in the work of Bruinier, Kudla and Yang in [4]. The work in this paper may shed some light on how to understand that integral in general.

Let \(w_\pi \) denote the central character of \(\pi \). To avoid triviality, we will assume throughout this paper that
$$\begin{aligned} w_\pi \cdot (\chi _1\chi _2)|_{{\mathbb {A}}_{\mathbb {F}}^*}=1. \end{aligned}$$
(1.2)
Under this assumption, we will relate \({\mathbb {I}}(E,F,s)\) to certain L-functions and special values of L-functions. This is not surprising as we have already seen many examples relating automophic integrals and L-functions.

1.1 Automorphic integrals and L-functions

Integral is an important tool to study L-functions, as in the earliest example of the integral representation for the Riemann zeta function. It is used to show, for example, the functional equation and the analytic continuation of the L- functions. Tate in his thesis gave the first adelic version of the story. (see [2] as a reference). Let \(\mu \) be a Hecke character on \({\mathbb {A}}_{\mathbb {F}}^*\) and \(f\in S({\mathbb {A}}_{\mathbb {F}})\) be a Schwartz function. Tate showed that the integral
$$\begin{aligned} \int \limits _{{\mathbb {A}}_{\mathbb {F}}^*}f(x)\mu (x)|x|^sd^*x \end{aligned}$$
(1.3)
represents the L-function of the Hecke character \(L(\mu ,s)\). His work provided the basic idea to relate the automorphic integrals with the L-functions in general: write the automorphic integral as a product of local integrals, then identify the local integrals with the corresponding local L-factors for unramified places. The local integral at ramified places could be different from expectation. It depends on, for example, the choice of the Schwartz functions. Thus the global integral could differ from the L-function by factors at the set of ramified places, which is finite.

We introduce here two more examples which are similar to (1.1).

1.1.1 Rankin–Selberg integral

Let \(F_i\) be cusp forms over \({\mathbb {F}}\), coming from automorphic cupidal representations \(\pi _i\) for \(i=1,2\). Let E(gs) be the Eisenstein series over \({\mathbb {F}}\) (not over \({\mathbb {E}}\)) associated to two Hecke characters \(\chi _1\) and \(\chi _2\) of \({\mathbb {A}}_{\mathbb {F}}^*\). Then the integral
$$\begin{aligned} \int \limits _{Z_{{\mathbb {A}}}{\text {GL}}_{2}({\mathbb {F}})\backslash {\text {GL}}_{2}({\mathbb {A}}_{{\mathbb {F}}})} F_1(g)F_2(g)E(g,s) dg \end{aligned}$$
(1.4)
represents (see for example [2])
$$\begin{aligned} L(\pi _1\times \pi _2,\chi _1,s). \end{aligned}$$
If we specify \(\chi _1\) to be the trivial character, then we get the standard Rankin–Selberg L-function \(L(\pi _1\times \pi _2,s)\). The Rankin–Selberg method can be applied to more general reductive groups. For a survey on this subject, see for example [3].

1.1.2 Triple product formula

Let \({\mathbb {B}}\) be a quaternion algebra. Let \(\pi _i\) for \(i=1,2,3\) be three irreducible unitary cuspidal automorphic representations of \({\mathbb {B}}^*\). Let \(F_i\in \pi _i\) be cusp forms for \(i=1,2,3\). Let \(\Pi \) denote \(\pi _1\otimes \pi _2\otimes \pi _3\) in this subsection. Consider the integral
$$\begin{aligned} \int \limits _{Z_{{\mathbb {A}}}{\mathbb {B}}^*({\mathbb {F}})\backslash {\mathbb {B}}^*({\mathbb {A}})} F_1(g)F_2(g)F_3(g) dg. \end{aligned}$$
(1.5)
This integral gives an element of \(\mathrm{Hom}_{{\mathbb {B}}^*({\mathbb {A}})}(\Pi ,{\mathbb {C}})\), which is at most one dimensional. Prasad in his thesis [20] gave a criterion in terms of local epsilon factors for the local component of \(\mathrm{Hom}_{{\mathbb {B}}^*({\mathbb {A}})}(\Pi ,{\mathbb {C}})\) to be nonzero. Jacquet then conjectured that the central value
$$\begin{aligned} L(\pi _1\otimes \pi _2\otimes \pi _3,1/2) \end{aligned}$$
(1.6)
of the triple product L-function does not vanish if and only if there exists a quaternion algebra \({\mathbb {B}}\) and the corresponding \(F_i\)’s such that (1.5) does not vanish. This conjecture was first proved by Harris and Kudla in [10, 11] using an integral representation of triple product L-function (see [6, 18]) and the regularized Siegel–Weil formula (see [17]). Later on, more explicit formulae relating (1.5) and (1.6) were given in [1, 8, 25] for some special cases. Ichino then generalized the above results in [14], where he considered \(\Pi \) as an irreducible unitary cuspidal automorphic representations over an étale cubic algebra \({\mathbb {K}}\) (this in particular includes the case \(\Pi =\pi _1\otimes \pi _2\otimes \pi _3\) when \({\mathbb {K}}\) is just \({\mathbb {F}}\oplus {\mathbb {F}}\oplus {\mathbb {F}}\)). He showed that a pairing of integral (1.5)
$$\begin{aligned} \int \limits _{Z_{{\mathbb {A}}}{\mathbb {B}}^*({\mathbb {F}})\backslash {\mathbb {B}}^*({\mathbb {A}})} F_1(g)F_2(g)F_3(g)dg\int \limits _{Z_{{\mathbb {A}}}{\mathbb {B}}^*({\mathbb {F}})\backslash {\mathbb {B}}^*({\mathbb {A}})} F_1'(g)F_2'(g)F_3'(g)dg \end{aligned}$$
represents
$$\begin{aligned} \frac{L(\Pi ,1/2)}{L(\Pi ,Ad,1)}. \end{aligned}$$

1.1.3 Comparison

Now we compare the integrals (1.1), (1.4) and (1.5). For simplicity, let \({\mathbb {B}}\) be the matrix algebra for (1.5). We first consider the case when \({\mathbb {E}}={\mathbb {F}}\oplus {\mathbb {F}}\) for (1.1), so the Eisenstein series there is a product of two Eisenstein series over \({\mathbb {F}}\). Then (1.1), (1.4) and (1.5) give a complete list of integrals of possible products of three automorphic forms, either cusp form or Eisenstein series, over \(Z_{{\mathbb {A}}}{\text {GL}}_{2}({\mathbb {F}})\backslash {\text {GL}}_{2}({\mathbb {A}}_{{\mathbb {F}}})\).

In general for the Rankin–Selberg integral, we can start with a cusp form defined over a quadratic algebra \({\mathbb {E}}\), restrict it to the base field and integrate it against an Eisenstein series over \({\mathbb {F}}\). When \({\mathbb {E}}\) is a quadratic field extension, the integral represents Asai L-function([16]). Similarly for the triple product formula, we can start with a cusp form defined over an étale cubic algebra \({\mathbb {K}}\), and integrate it over the diagonal \(Z_{{\mathbb {A}}}{\text {GL}}_{2}({\mathbb {F}})\backslash {\text {GL}}_{2}({\mathbb {A}}_{{\mathbb {F}}})\). So we have the following table:

Degree of the algebra that the cusp form is defined over

Degree of the algebra that the Eisenstein series is defined over

L-functions represented

3

No Eisenstein series

Triple product L-function

2

1

Rankin–Selberg L-function or Asai L-function

1

2

To be solved in this paper

Note that we need at least one cusp form to guarantee convergence. So our work on (1.1) is a complementary case to the Rankin–Selberg integral and the triple product formula.

Despite their similarity, we won’t follow, for example, Ichino’s method directly, as cusp forms and Eisenstein series are somewhat different in nature. It turns out that our integral is more closely related to Waldspurger’s period integral (see Sect. 2.4 for its definition and properties).

1.2 Main results and organization

If we write \({\mathbb {E}}={\mathbb {F}}(\sqrt{D})\), then we can embed \({\mathbb {E}}\) into the matrix algebra by
$$\begin{aligned} t=a+b\sqrt{D}\mapsto \begin{pmatrix} a &{} \quad b \\ bD &{} \quad a \end{pmatrix}. \end{aligned}$$
(1.7)
Let \(\eta \) be a quadratic character associated to the quadratic extension \({\mathbb {E}}/{\mathbb {F}}\). Recall that the Eisenstein series E(gs) is associated to two characters \(\chi _1\), \(\chi _2\). For \(t\in {\mathbb {E}}^*\), define the character \(\Omega \) such that
$$\begin{aligned} \Omega (t)=\chi _1(\bar{t})\chi _2(t). \end{aligned}$$
(1.8)
Define
$$\begin{aligned} \chi =\frac{\chi _1}{\chi _2}. \end{aligned}$$
(1.9)
Let \(\Pi \) be the base change of \(\pi \) to \({\mathbb {E}}\) in this subsection.

The first goal of this paper is to prove the following theorem:

Theorem 1.1

  1. (1)

    If \(\mathrm{Hom}_{{\mathbb {A}}_{\mathbb {E}}^*}(\pi \otimes \Omega ,{\mathbb {C}})=0\) or \(L(\Pi \otimes \Omega ,1/2)=0\), then \({\mathbb {I}}(E,F,s)=0\).

     
  2. (2)
    Otherwise, we can fix \(F_1\in \hat{\pi }\) such that
    $$\begin{aligned} C=\int \limits _{Z_{{\mathbb {A}}}{\mathbb {E}}^{*}\backslash {\mathbb {A}}_{\mathbb {E}}^*}F_{1}(t_{1})\Omega ^{-1}(t_{1})dt_{1}\ne 0 \end{aligned}$$
    It is independent of s and
    $$\begin{aligned} \frac{C\cdot {\mathbb {I}}(E,F,s)}{(F_1,F)}=\frac{\zeta (2)L (\Pi \otimes \Omega ,1/2)L (\pi \otimes \chi _{1}|_{{\mathbb {F}}^*},2s+1/2)}{2L(\pi ,Ad,1)L (\chi ,2s+1)} \prod \limits _{v} {\mathbb {P}}^0_v. \end{aligned}$$
    (1.10)
     

Here local integral \({\mathbb {P}}^0_v\) is given as in (4.20).

The second goal of this paper is to work out the local integrals \({\mathbb {P}}_v^0\) at some ramified places when the global integral is not trivially zero.

Remark 1.2

Using Waldspurger’s result, one can at least determine |C| once we specify the choices of \(F_1\), F. It may seem that one still cannot get an explicit formula even if we compute \({\mathbb {P}}_v^0\) explicitly. However one can divide this formula by Waldspurger’s formula (see Theorem 2.22 and Corollary 2.24) and get
$$\begin{aligned} \frac{{\mathbb {I}}(E,F,s)}{\displaystyle \int \limits _{Z_{{\mathbb {A}}}{\mathbb {E}}^{*}\backslash {\mathbb {A}}_{\mathbb {E}}^*}F(t)\Omega (t)dt}=\frac{L (\pi \otimes \chi _{1}|_{{\mathbb {F}}^*},2s+1/2)}{L (\chi ,2s+1)} \prod \limits _{v} \frac{{\mathbb {P}}^0_v}{P^0_v}. \end{aligned}$$
(1.11)
Here \(P^0_v\) is as in (2.32). Now explicit results on the local integrals \({\mathbb {P}}_v^0\) and \(P_v^0\) will allow us to compare our period integrals \({\mathbb {I}}(E,F,s)\) with Waldspurger’s period integral
$$\begin{aligned} \int \limits _{Z_{{\mathbb {A}}}{\mathbb {E}}^{*}\backslash {\mathbb {A}}_{\mathbb {E}}^*}F(t)\Omega (t)dt \end{aligned}$$
explicitly. This will be used in our future work on span of restriction of Hecke Eisenstein series with levels. (1.11) also allows us to see the main L-functions more clearly.

We will mostly discuss the disjoint ramifications, where the test vectors are chosen to be Gross–Prasad test vectors. But we will also consider a case (Case 5 in Sect. 5) where there are joint ramifications. As a result, locally \(\mathrm{Hom}_{{\mathbb {E}}^*}(\pi \otimes \Omega ,{\mathbb {C}})\ne 0\) in this case and the local new form is a proper test vector. We shall also compute the archimedean place for a special situation in Sect. 6. As a very special result from these local calculations, we have

Corollary 1.3

Let \({\mathbb {E}}={\mathbb {Q}}(\sqrt{D})\) be a real quadratic extension of \({\mathbb {Q}}\) for a square-free integer D. Let \(N=\prod _p p^{c_p}>0\) be an integer such that for any p|N, p is inert in \({\mathbb {E}}\). Let \(F\in \pi \) \(F_1\in \hat{\pi }\) be anti-holomorphic cuspidal new forms of weight \(-2k\). Let E be a holomorphic new Eisenstein series of parallel weight (k,k), defined by two characters \(\chi _1\) \(\chi _2\). Suppose that \(\pi \), its central character \(w_\pi \), \(\chi _1\), \(\chi _1|_{\mathbb {Q}}\) all have finite conductor N and \(\chi _2\) has finite conductor 1. Then
$$\begin{aligned} \frac{C\cdot {\mathbb {I}}(E,F,s)}{(F_1,F)}=\frac{\zeta (2) L (\Pi \otimes \Omega ,1/2)L (\pi \otimes \chi _{1}|_{{\mathbb {F}}^*},2s+1/2)}{2L(\pi ,Ad,1)L (\chi ,2s+1)} {\mathbb {P}}_\infty \prod \limits _{p|N} {\mathbb {P}}^0_p, \end{aligned}$$
(1.12)
where
$$\begin{aligned} {\mathbb {P}}_\infty= & {} {\frac{4 \pi }{2k-1}}{\frac{1}{(1+D)^k}}.\\ {\mathbb {P}}^0_p= & {} \frac{1}{(p+1)(p^{{c_{p}}-1})\chi _1(\sqrt{D})}. \end{aligned}$$
Here \(\chi _{1,p}\) is the p-component of the Hecke character \(\chi _1\).

We refer the readers to Case 5 in Sect. 5 for specific choice of local component of the Eisenstein series in this result. Note that the L-functions here are the standard L-functions (not completed). We have also used the relation between the Tamagawa measure and the measure we shall use for local computations (see (2.7)). The condition about the finite conductor is just to make sure that only Case 5 show up in ramifications. The period integral C also depends on the choice of D, which decides the embedding of \({\mathbb {E}}\) into the matrix algebra.

The tools developed in the local calculation of this paper turns out to be useful for other automorphic integrals. In particular it facilitates the study of the local integral for triple product formula which has direct arithmetic applications to subconvexity bound and equidistribution problems. See [12] and [13] for more details.

The arrangement of this paper is as follows: Sect. 2 will cover some basic definitions, facts and well-known theories. In particular we will review the Weil representations (following [24]), Shimizu’s lifting (see [21]). We will also discuss some special elements in the Weil representation and their properties. In Sect. 2.4 we will review Gross and Prasad’s test vector, and also two formulations of Waldspurger’s formula, one in terms of Shimizu’s lifting which we shall use most of time, the other one in terms of matrix coefficient.

In Sect. 3, we will use the standard technique of folding and unfolding to rewrite \({\mathbb {I}}(E,F,s)\) as
$$\begin{aligned} \int \limits _{{\mathbb {A}}_{\mathbb {E}}^*\backslash {\text {GL}}_{2}({\mathbb {A}})} \Phi _s(\gamma _0 g)\int \limits _{Z_{{\mathbb {A}}}{\mathbb {E}}^{*}\backslash {\mathbb {A}}_{\mathbb {E}}^*}F(tg)\Omega (t)dt\,dg. \end{aligned}$$
(1.13)
This is actually a weighted integral of Waldspurger’s period integral. As a corollary,
$$\begin{aligned} \left\{ \begin{array}{c} \mathrm{Hom}_{{\mathbb {A}}_{\mathbb {E}}^*}(\pi \otimes \Omega ,{\mathbb {C}})=0\\ \text {or } L(\Pi \otimes \Omega ,1/2)=0 \end{array}\right\} \Longrightarrow {\mathbb {I}}(E,F,s)=0, \end{aligned}$$
which is part (1) of Theorem 1.1. When this doesn’t happen, we can pair (1.13) with a fixed period integral and apply Waldspurger’s formula in Theorem 2.22. The simplifying observation is that the weighted integral can be combined with the inner integral of Waldspurger’s formula. This leads us to the main identity of Theorem 1.1. It can be formulated in terms of Shimizu lifting or matrix coefficient.

In Sect. 4, we will compute the local integral arising from Sect. 3 for unramified places, i.e. when locally \(\pi _v\) is unramified, \({\mathbb {E}}_v/{\mathbb {F}}_v\) is either inert or split, and \(\Phi _s\) is unramified (which in turn implies that \(\chi _{1,v}\) and \(\chi _{2,v}\) are unramified). We will see in Proposition 4.5 and Proposition 4.6 that the local integral gives the expected L-factors and \({\mathbb {P}}^0_v=1\) for unramified places. The work in Sects. 3 and 4 completes the proof of Theorem 1.1.

In Sect. 5, we will do local computations for other non-archimedean places. We will specify certain patterns of ramifications, but the levels of the ramification for \(\pi _v\) and \(\Phi _{s,v}\) can be arbitrary. We will also make sure that the local components of \(F_1\) and F are either Gross and Prasad’s test vectors or local new forms, and keep the calculations easier at the possible cost of using somewhat complicated choice of \(\Phi _s\).

In Sect. 6, we compute the local integral at real places for the special setting as in Corollary 1.3.

In Appendix we will prove Proposition A.1 which gives better description of the Kirillov model of a supercuspidal representation. This proposition is a key ingredient in the local calculation in Sect. 5.3. It is also important in the local calculations of the triple product formula in [12] and [13].

2 Notations and preliminary results

2.1 Definitions and basic facts

Let \({\mathbb {F}}\) denote a number field. Let \(\pi \) be an automorphic cuspidal representation of \({\text {GL}}_2\) over \({\mathbb {F}}\) with the central character \(w_\pi \). Let \({\mathbb {B}}\) be a quaternion algebra over \({\mathbb {F}}\), and \({\mathbb {E}}/{\mathbb {F}}\) be a quadratic algebra which is embedded in \({\mathbb {B}}\). Let \({\mathbb {A}}_{\mathbb {F}}\) and \({\mathbb {A}}_{\mathbb {E}}\) be the corresponding adelic rings of \({\mathbb {F}}\) and \({\mathbb {E}}\). Without loss of generality we can write \({\mathbb {E}}\) as \({\mathbb {F}}(\sqrt{D})\) for \(D\in {\mathbb {F}}\) an algebraic integer. (If \({\mathbb {E}}\simeq {\mathbb {F}}\oplus {\mathbb {F}}\), just take \(D=1\).)

In this paper we will be mostly interested in the case when \({\mathbb {B}}\) is the matrix algebra. In that case, we fix the embedding \({\mathbb {E}}\hookrightarrow {\mathbb {B}}\) as follows:
$$\begin{aligned} t=a+b\sqrt{D}&\mapsto \begin{pmatrix} a &{} \quad b \\ bD &{} \quad a \end{pmatrix}. \end{aligned}$$
(2.1)
Note that the quadratic norm is consistent with the determinant of matrices for this embedding.
Let \(\chi _1\) and \(\chi _2\) be two Hecke characters on \({\mathbb {E}}^*\backslash {\mathbb {A}}_{\mathbb {E}}^*\) such that
$$\begin{aligned} w_\pi \cdot (\chi _1\chi _2)|_{{\mathbb {A}}_{\mathbb {F}}^*}=1. \end{aligned}$$
(2.2)
Define
$$\begin{aligned} \chi =\frac{\chi _1}{\chi _2}. \end{aligned}$$
(2.3)
Let \(\Phi _s\) be a section of the induced representation \(Ind_B^{{\text {GL}}_2}(\chi _1,\chi _2,s)\), where B is the Borel subgroup of \({\text {GL}}_2\). So \(\Phi _s\) satisfies
$$\begin{aligned} \Phi _s\left( \begin{pmatrix} a_1 &{} \quad n \\ 0 &{} \quad a_2 \end{pmatrix} g\right) =\chi _1(a_1)\chi _2(a_2)\left| \frac{a_1}{a_2}\right| _{{\mathbb {A}}_{\mathbb {E}}}^{s+1/2}\Phi _s(g) \end{aligned}$$
for all \(\begin{pmatrix} a_1 &{} \quad n \\ 0 &{} \quad a_2 \end{pmatrix}\in B({\mathbb {A}}_{\mathbb {E}})\) and \(g\in {\text {GL}}_2({\mathbb {A}}_{\mathbb {E}})\).
Let
$$\begin{aligned} E(g,s)=\sum \limits _{\gamma \in B({\mathbb {E}})\backslash {\text {GL}}_{2}({\mathbb {E}})}\Phi _s(\gamma g) \end{aligned}$$
(2.4)
be the associated Eisenstein series.
Let \({\mathbb {F}}_v\) be the corresponding local field of \({\mathbb {F}}\) at a place v. Let \(K_v\) denote the standard maximal compact subgroup of \({\text {GL}}_2({\mathbb {F}}_v)\), and
$$\begin{aligned} K=\prod _vK_v. \end{aligned}$$
(2.5)
We will call an element of a local representation spherical if it is invariant under \(K_v\). For unramified representations, there is a unique up to constant spherical element.
When v is a finite place, let \(\varpi _v\) denote a uniformizer of \({\mathbb {F}}_v\). Let \(O_F\) be the ring of integers of the local field \({\mathbb {F}}_v\), and \(O_E\) be the ring of integers for \({\mathbb {E}}_v\). Let v(x) denote the valuation of \(x\in F_v^*\). Let \(q^{-1}=|\varpi _v|_v\). For an integer \(c>0\), define:
$$\begin{aligned} K_1(\varpi _v^c)=\left\{ k\in K_v| \quad k\equiv \begin{pmatrix} * &{} \quad * \\ 0 &{} \quad 1 \end{pmatrix} \mod (\varpi _v^c)\right\} . \end{aligned}$$
(2.6)
Similarly denote by \(K_0(\varpi _v^c)\) for those congruent to \(\begin{pmatrix} * &{} \quad * \\ 0 &{} \quad * \end{pmatrix}\mod (\varpi _v^c)\) and \(K_1^1(\varpi _v^c)\) for those congruent to \(\begin{pmatrix} 1 &{} \quad * \\ 0 &{} \quad 1 \end{pmatrix}\mod (\varpi _v^c)\).
We shall pick the Haar measure dg on \(Z_{{\mathbb {A}}}{\text {GL}}_{2}({\mathbb {F}})\backslash {\text {GL}}_{2}({\mathbb {A}}_{{\mathbb {F}}})\) to be the Tamagawa measure. For simplicity we shall pick the Haar measure \(dg_v\) at non-archimedean places to be such that the volume of \(K_v\) is 1, and the Haar measure at real places to be
$$\begin{aligned} \frac{1}{2\pi } \frac{da \, dm \,d\theta }{a^2} \end{aligned}$$
where we write
$$\begin{aligned} {\mathbb {R}}^*\backslash {\text {GL}}_2({\mathbb {R}})=\left\{ \begin{pmatrix} a &{} \quad m \\ 0 &{} \quad 1 \end{pmatrix}|a\in {\mathbb {R}}^*,m\in {\mathbb {R}}\right\} K \end{aligned}$$
for \(K=SO(2)\). Then
$$\begin{aligned} dg=|\Delta _{\mathbb {F}}|^{-3/2}\zeta _{\mathbb {F}}^{-1}(2)\prod \limits _{v}dg_v \end{aligned}$$
(2.7)
Now we describe the integrals on \({\text {GL}}_2({\mathbb {F}}_v)\) when v is finite. These results are easy and probably known by experts.

Lemma 2.1

For every positive integer c,
$$\begin{aligned} {\text {GL}}_2({\mathbb {F}}_v)=\coprod \limits _{0\le i\le c} B\begin{pmatrix} 1 &{} \quad 0 \\ \varpi _v^i &{} \quad 1 \end{pmatrix}K_1(\varpi _v^c). \end{aligned}$$

Normalize the Haar measure on \({\text {GL}}_2({\mathbb {F}}_v)\) such that \(K_v\) has volume 1. Then we have the following:

Lemma 2.2

Locally let f be a \(K_1(\varpi _v^c)\)-invariant function, on which the center acts trivially. Then
$$\begin{aligned} \int \limits _{F_v^*\backslash {\text {GL}}_2({\mathbb {F}}_v)}f(g)dg=\sum \limits _{0\le i\le c}A_i\int \limits _{{\mathbb {F}}_v^*\backslash B({\mathbb {F}}_v)}f\left( b\begin{pmatrix} 1 &{} \quad 0 \\ \varpi _v^i &{} \quad 1 \end{pmatrix}\right) db. \end{aligned}$$
(2.8)
Here db is the left Haar measure on \({\mathbb {F}}_v^*\backslash B({\mathbb {F}}_v)\), and
$$\begin{aligned} A_0=\frac{q}{q+1}, \quad A_c=\frac{1}{(q+1)q^{c-1}}\quad \text { and }\quad A_i=\frac{q-1}{(q+1)q^i} \quad \text { for } \ 0<i<c. \end{aligned}$$

Proof

For \(0\le j\le c\), let \(f_j\) be the characteristic function of \(K_0(\varpi ^j)\). \(f_0\) is just the characteristic function of K. Clearly they are all right-invariant under \(K_1(\varpi ^c)\). The integral of these functions just give the volume of these compact subgroups. Suppose that the Haar measure on \({\text {GL}}_2\) are so normalized that the volumes of K and \(B(O_F)=B\cap K\) are 1. The volume of \(K_0(\varpi ^j)\) is \(\frac{1}{(q+1)q^{j-1}}\) for \(j>0\). On the other hand, we can evaluate the integral by the right hand side of (2.8).
$$\begin{aligned} f_j\left( b\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) = {\left\{ \begin{array}{ll} 1,&{} \text { if }b\in B(O_F)\text { and}\;\;j\le i;\\ 0, &{} \text { otherwise}. \end{array}\right. } \end{aligned}$$
So
$$\begin{aligned} \frac{1}{(q+1)q^{j-1}}=\int \limits _{g\in {\text {GL}}_2}f_j(g)dg=\sum \limits _{0\le i\le c}A_i\int \limits _{b\in B}f_j\left( b\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) db=\sum \limits _{j\le i\le c}A_i \end{aligned}$$
for \(0<j\le c\). When \(j=0\), we get
$$\begin{aligned} 1=\sum \limits _{0\le i\le c}A_i. \end{aligned}$$
Then it’s easy to see that the values of the coefficients \(A_i\) in the lemma are the only choice. \(\square \)

We also record here some easy results about integrals for additive and multiplicative characters.

Lemma 2.3

Let \(\psi _v\) be an unramified additive character at v. Then
$$\begin{aligned} \int \limits _{v(m)=j}\psi _v(m)dm= {\left\{ \begin{array}{ll} 0,&{} \quad \text { if }j<-1;\\ -1,&{} \quad \text { if }j=-1;\\ q^{-j}(1-q^{-1}),&{} \quad \text { if }j\ge 0. \end{array}\right. } \end{aligned}$$
(2.9)

Lemma 2.4

Suppose that \(\mu \) is a character of level \(k>0\) on \({\mathbb {F}}_v^*\). Then
$$\begin{aligned} \int \limits _{x\in O_F^*}\mu (1+\varpi ^i x)dx= {\left\{ \begin{array}{ll} 0, &{} \quad \text { if }i<k-1;\\ -q^{-1},&{} \quad \text { if }i=k-1;\\ 1-q^{-1},&{} \quad \text { if }i\ge k. \end{array}\right. } \end{aligned}$$
(2.10)

2.2 The Weil representation

The Weil representation can be defined for more general reductive group pairs, but we will focus on the following setting as in [24]:

Fix \(\psi \) a nontrivial additive character of \({\mathbb {F}}\). Let \({\mathbb {B}}\) be a quaternion algebra over \({\mathbb {F}}\) with \(\iota : x\mapsto \bar{x}\) being the main involution. We can define the reduced norm on \({\mathbb {B}}\) via
$$\begin{aligned} Q(x)=x\;\iota (x). \end{aligned}$$
We will focus on the case when \({\mathbb {B}}\) is the matrix algebra \(M_2({\mathbb {F}})\) later on. Denote by \(GO({\mathbb {B}})\) the orthogonal similitude group of \({\mathbb {B}}\), with the similitude character \(\nu \). An element \((g_1,g_2)\) in \({\mathbb {B}}^*\times {\mathbb {B}}^*\) acts on \({\mathbb {B}}\) via \((g_1,g_2)\cdot x=g_1x\ g_2^{-1}\). This actually give us a short exact sequence
$$\begin{aligned} 1\rightarrow {\mathbb {F}}^*\rightarrow ({\mathbb {B}}^*\times {\mathbb {B}}^*)\rtimes \{1,\iota \} \rightarrow GO({\mathbb {B}})\rightarrow 1. \end{aligned}$$
(2.11)
Here \({\mathbb {F}}^*\) is embedded into the group in the middle by \(x\mapsto (x,x)\rtimes 1\). \(\iota \) acts on \({\mathbb {B}}^*\times {\mathbb {B}}^*\) by \((g_1,g_2)\mapsto (\iota (g_2)^{-1},\iota (g_1)^{-1})\). We will simply write \((g_1,g_2)\) for \((g_1,g_2)\rtimes 1\) when considered as an element of \(GO({\mathbb {B}})\).

Definition 2.5

The Weil representation for the similitude group pair \({\text {GL}}_2\times GO({\mathbb {B}})\) on the space of Schwartz functions \(S({\mathbb {B}}\times {\mathbb {F}}^*)\) is defined as follows: for \(f(x,u)\in S({\mathbb {B}}\times {\mathbb {F}}^*)\), \(\alpha ,\delta \in {\mathbb {F}}^*\), \(\beta \in {\mathbb {F}}\), \(g\in GO({\mathbb {B}})\),
  1. (i)

    \(r'\left( \begin{pmatrix} 1 &{} \quad \beta \\ 0 &{} \quad 1 \end{pmatrix}\right) f(x,u)=\psi _u(\beta Q(x))f(x,u)\),

     
  2. (ii)

    \(r'\left( \begin{pmatrix} 0 &{} \quad 1 \\ -1 &{} \quad 0 \end{pmatrix}\right) f(x,u)=\gamma [\psi _u,q]\int \limits _{{\mathbb {B}}} f(y,u)\psi _u(<x,y>)dy\),

     
  3. (iii)

    \(r'\left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad \alpha ^{-1} \end{pmatrix}\right) f(x,u)=|\alpha |^2f(\alpha x,u)\),

     
  4. (iv)

    \(r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ 0 &{} \quad \delta \end{pmatrix}\right) f(x,u)=|\delta |^{-1}f(x,\delta ^{-1}u)\),

     
  5. (v)

    \(r''(g)f(x,u)=f(g^{-1}\cdot x,u\nu (g))\).

     
Here \(\gamma [\psi _u,q]\) equal to 1 if \({\mathbb {B}}\) is the matrix algebra and \(-1\) is \({\mathbb {B}}\) is a division algebra. \(\psi _u(x)=\psi (ux)\). \(\langle x,y \rangle =Q(x+y)-Q(x)-Q(y)\) in (ii).

Remark 2.6

For \((g_1,g_2)\in GO({\mathbb {B}})\), we have \(\nu (g_1,g_2)=Q(g_1)Q(g_2)^{-1}\), and
$$\begin{aligned} r''(g_1,g_2)f(x,u)=f(g_1^{-1}xg_2,uQ(g_1)Q(g_2)^{-1}). \end{aligned}$$
(2.12)
Also by combining (iii) and (iv), we can get
$$\begin{aligned} r'\left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) f(x,u)=|\alpha |f(\alpha x,\alpha ^{-1}u). \end{aligned}$$
(2.13)
We will use these simple facts later.

2.2.1 Special elements in the Weil representation

For a finite place v, now we specify \({\mathbb {B}}_v=M_2({\mathbb {F}}_v)\). We will discuss explicitly some special elements in the Weil representation \(S(M_2({\mathbb {F}}_v)\times {\mathbb {F}}_v^*)\) as given above.

According to (v) of Definition 2.5,
$$\begin{aligned} r''(1,g)f(x,u)=f\left( xg,\frac{u}{\det g}\right) . \end{aligned}$$
We will say a Schwartz function is invariant under the right action of (or just right-invariant), for example, \(K_1(\varpi _v^c)\), if \(r''(1,g)f(x,u)=f(x,u)\) for all \(g\in K_1(\varpi _v^c)\).
Denote by \(\omega \) the matrix \(\begin{pmatrix} 0 &{} \quad 1 \\ -1 &{} \quad 0 \end{pmatrix}\). Assume that the local additive character \(\psi _v\) is unramified. Let \(x=\begin{pmatrix} x_1 &{} \quad x_2 \\ x_3 &{} \quad x_4 \end{pmatrix}, y=\begin{pmatrix} y_1 &{} \quad y_2 \\ y_3 &{} \quad y_4 \end{pmatrix}\in M_2({\mathbb {F}}_v)\). By definition,
$$\begin{aligned} Q(x)=\det x=x_1x_4-x_2x_3, \end{aligned}$$
and
$$\begin{aligned} \langle x,y \rangle =x_1y_4+x_4y_1-x_2y_3-x_3y_2. \end{aligned}$$
So
$$\begin{aligned} r'(\omega )f(x,u)=\int \psi _v(u(x_1y_4+x_4y_1-x_2y_3-x_3y_2))f(y,u)dy. \end{aligned}$$
(2.14)

Lemma 2.7

Let \(f=char\left( \begin{pmatrix} O_F &{} \quad O_F \\ O_F &{} \quad O_F \end{pmatrix}\right) (x)\times char(O_F^*)(u)\in S(M_2({\mathbb {F}}_v)\times {\mathbb {F}}_v^*)\). It is invariant by \(K_v\) under both the right action and the Weil representation \(r'\).

Proof

One can check directly. \(\square \)

Remark 2.8

For the conciseness of notations, later on we will simply write, for example, \(f=char\left( \begin{pmatrix} O_F &{} \quad O_F \\ O_F &{} \quad O_F \end{pmatrix}\right) \times char(O_F^*)\) for functions in \(S(M_2({\mathbb {F}}_v)\times {\mathbb {F}}_v^*)\).

Lemma 2.9

Let \(f=char\left( \begin{pmatrix} O_F &{} \quad O_F \\ \varpi _v^cO_F &{} \quad O_F \end{pmatrix}\right) \times char(O_F^*)\), for integer \(c>0\).
  1. (i)

    It is invariant by \(K_1(\varpi _v^c)\) under both the right action and the Weil representation.

     
  2. (ii)
    For \(n\in {\mathbb {F}}_v^*\) with \(0\le v(n)=j\le c\),
    $$\begin{aligned} r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ n &{} \quad 1 \end{pmatrix}\right) f(x,u)= & {} q^{j-c}char\left( \begin{pmatrix} O_F &{} \quad \varpi _v^{j-c}O_F \\ \varpi _v^{j}O_F &{} \quad O_F \end{pmatrix}\right) \psi _v(- ux_2x_3n^{-1}) \nonumber \\&\times char(O_F^*). \end{aligned}$$
    (2.15)
    This function is still right \(K_1(\varpi _v^c)\)-invariant.
     

Proof

We will prove the formula in (ii) directly. The rest are easy to check. Note that \(\begin{pmatrix} 1 &{} \quad 0 \\ n &{} \quad 1 \end{pmatrix}=-\omega \begin{pmatrix} 1 &{} \quad -n \\ 0 &{} \quad 1 \end{pmatrix}\omega \). Then by definition
$$\begin{aligned} r'(\omega )f(x,u)= & {} q^{-c}char\left( \begin{pmatrix} O_F &{} \quad \varpi _v^{-c}O_F \\ O_F &{} \quad O_F \end{pmatrix}\right) \times char(O_F^*),\\ r'\left( \begin{pmatrix} 1 &{} \quad -n \\ 0 &{} \quad 1 \end{pmatrix}\omega \right) f(y,u)= & {} q^{-c}char\left( \begin{pmatrix} O_F &{} \quad \varpi _v^{-c}O_F \\ O_F &{} \quad O_F \end{pmatrix}\right) \psi _v(- un\det y)\\ \nonumber&\times char(O_F^*). \end{aligned}$$
Note that \(\psi _v(- un\det y)=\psi _v(uny_2y_3)\) for \(y\in \begin{pmatrix} O_F &{} \quad \varpi _v^{-c}O_F \\ O_F &{} \quad O_F \end{pmatrix}\). For another action of \(\omega \), the integral in \(y_1y_4\) is very easy. Now we focus on the following integral:
$$\begin{aligned} q^{-c}\int \limits _{y_2\in \varpi _v^{-c}O_F}\int \limits _{y_3\in O_F}\psi _v(uy_3(ny_2-x_2))\psi _v(-ux_3y_2)dy_3dy_2. \end{aligned}$$
Let \(x_2\) be fixed. For the integral in \(y_3\) to be non-zero, we need \(y_2\in n^{-1} x_2+\varpi _v^{-j}O_F\) as \(v(n)=j\). Then the integral becomes
$$\begin{aligned} q^{-c}\int \limits _{y_2\in \varpi _v^{-c}O_F\cap n^{-1} x_2+\varpi _v^{-j}O_F}\psi _v(-ux_3y_2)dy_2. \end{aligned}$$
Note that \(\varpi _v^{-j}O_F\subseteq \varpi _v^{-c}O_F\). The domain of the integral is not empty iff \(x_2\in \varpi _v^{j-c}O_F\). In that case, the integral becomes
$$\begin{aligned} q^{-c}\int \limits _{y_2\in n^{-1} x_2+\varpi _v^{-j}O_F}\psi _v(-ux_3y_2)dy_2=q^{j-c}\psi _v(- ux_2x_3n^{-1}) \text { if }x_3\in \varpi _v^j O_F. \end{aligned}$$
So we get \(r'\left( \omega \begin{pmatrix} 1 &{} \quad -n \\ 0 &{} \quad 1 \end{pmatrix}\omega \right) =q^{j-c}char\left( \begin{pmatrix} O_F &{} \quad \varpi _v^{j-c}O_F \\ \varpi _v^{j}O_F &{} \quad O_F \end{pmatrix}\right) \psi _v(- ux_2x_3n^{-1})\times char(O_F^*)\). Then just note that the action of \(-1\) will not change this function. \(\square \)

Now we consider a slightly different type of Schwartz functions.

Lemma 2.10

Let \(b_1,b_2\in O_F\) and c be an integer. Define \(f=char\left( \begin{pmatrix} b_1+\varpi _v^cO_F &{} \quad O_F \\ b_2+\varpi _v^cO_F &{} \quad O_F \end{pmatrix}\right) \times char(O_F^*)\).
  1. (i)

    f is \(K_1^1(\varpi _v^c)\)-invariant under the Weil representation \(r'\) and the right action.

     
  2. (ii)
    For \(n\in {\mathbb {F}}_v^*\) with \(0\le v(n)=j\le c\),
    $$\begin{aligned} r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ n &{} \quad 1 \end{pmatrix}\right) f= & {} q^{2(j-c)}char\left( \begin{pmatrix} b_1+\varpi _v^jO_F &{} \quad \varpi _v^{j-c}O_F \\ b_2+\varpi _v^jO_F &{} \quad \varpi _v^{j-c}O_F \end{pmatrix}\right) \nonumber \\&\times \psi _v(un^{-1}[(x_1-b_1)x_4-x_2(x_3-b_2)])char(O_F^*). \end{aligned}$$
    (2.16)
    This function is still right \(K_1^1(\varpi _v^c)\)-invariant.
     

Proof

Similar to the proof in the last lemma. \(\square \)

Remark 2.11

If \(f=char\left( \begin{pmatrix} b_1+\varpi _v^cO_F &{} \quad O_F \\ b_2+\varpi _v^cO_F &{} \quad O_F \end{pmatrix}\right) \times char(\beta +\varpi _v^c O_F)\) with \(\beta \in (O_F/\varpi _v^cO_F)^*\), one has a similar result.

2.3 Shimizu’s lifting

Now we review briefly Shimizu’s lifting (see [21] for more details). For the dual group pair \({\text {GL}}_2\times GO({\mathbb {B}})\), we can use the Theta lifting to give an automorphic representation of \(GO({\mathbb {B}})\) corresponding to a given automorphic representation \(\pi '\) of \({\text {GL}}_2\). One can lift this representation further by the exact sequence (2.11) to an automorphic representation \(\Theta (\pi ')\) for \({\mathbb {B}}^*\times {\mathbb {B}}^*\).

In particular, let \(f\in S({\mathbb {B}}({\mathbb {A}})\times {\mathbb {A}}_F^*)\) be an element of the Weil representation defined above, \(g_1,g_2\in {\mathbb {B}}^*({\mathbb {A}})\), \(\varphi \in \pi '\) be a cusp form, and \(h\in {\text {GL}}_2({\mathbb {A}}_{{\mathbb {F}}})\). The theta kernel is
$$\begin{aligned} \theta (f,h,g_1,g_2)=\sum \limits _{x\in {\mathbb {B}}({\mathbb {F}}),u\in {\mathbb {F}}^*}r'(h)r''(g_1,g_2)f(x,u). \end{aligned}$$
(2.17)
The global Theta lifting is
$$\begin{aligned} \theta (f,\varphi ,g_1,g_2)=\int \limits _{{\text {GL}}_2({\mathbb {F}})\backslash {\text {GL}}_2({\mathbb {A}}_{{\mathbb {F}}})}\varphi (h)\theta (f,h,g_1,g_2)dh. \end{aligned}$$
(2.18)
The integral is absolutely convergent since \(\varphi \) is a cusp form. Then \(\Theta (\pi ')\) is just the collection of all such \(\theta (f,\varphi ,g_1,g_2)\) for all possible \(\varphi \in \pi '\) and \(f\in S({\mathbb {B}}({\mathbb {A}})\times {\mathbb {A}}_{\mathbb {F}}^*)\).

Theorem 2.12

(Shimizu’s lifting) Let \(\pi '\) be a cuspidal automorphic representation of \({\text {GL}}_2\).
  1. (i)

    If \(\pi '\) doesn’t appear in the image of Jacquet–Langlands correspondence, then \(\Theta (\pi ')=0\).

     
  2. (ii)

    Otherwise, let \(\sigma \) be an automorphic representation of \({\mathbb {B}}^*\) such that \(JL(\sigma )=\pi '\). Then \(\Theta (\pi ')=\sigma \otimes \hat{\sigma }\).

     

Remark 2.13

In particular this theorem applies to the case when \({\mathbb {B}}\) is the matrix algebra. In this case, \({\mathbb {B}}^*\simeq {\text {GL}}_2\) and \(\sigma \simeq \pi '\).

2.4 Period integral, test vectors and Waldspurger’s formula

2.4.1 Waldspurger’s period integral

Let \(F_1\) be an element of \(\sigma \), which is an automorphic representation of \({\mathbb {B}}^*\) with the central characters \(w_{\sigma }\). Let \(\Omega \) be a Hecke character over the quadratic algebra \({\mathbb {E}}\) such that \(\Omega |_{{\mathbb {A}}^*_{\mathbb {F}}}=w_{\sigma }\). Waldspurger studied in [24] the following period integral
$$\begin{aligned} \int \limits _{Z_{\mathbb {A}}{\mathbb {E}}^*\backslash {\mathbb {A}}_{\mathbb {E}}^*}F_1(t)\Omega ^{-1}(t)dt. \end{aligned}$$
(2.19)
This period integral actually gives an element in \(\mathrm{Hom}_{{\mathbb {A}}_{\mathbb {E}}^*}(\sigma \otimes \Omega ^{-1},{\mathbb {C}})\). But it’s not necessary that this space is non-zero.

Now we discuss the local obstruction for this integral to be nonzero. We first need some definitions.

The Hasse invariant \(\epsilon ({\mathbb {B}}_v)\) of a local quaternion algebra \({\mathbb {B}}_v\) is defined to be 1 if \({\mathbb {B}}_v\simeq M_2({\mathbb {F}}_v)\), and \(-1\) if it’s a division algebra. Let \(\pi '\) be the image of \(\sigma \) under Jacquet–Langlands correspondence. Then one can define the local root number \(\epsilon (\frac{1}{2},\Pi _{\pi ',v}\otimes \Omega _v^{-1})\) where \(\Pi _{\pi ',v}\) is the base change of \(\pi '_v\) to \({\mathbb {E}}_v\). In general the local root number would depend on the chosen additive character \(\psi _v\). The condition \(\Omega |_{{\mathbb {A}}_{\mathbb {F}}^*}=w_{\sigma }\) will guarantee that this local root number is independent of \(\psi _v\) and only takes values \(\pm 1\). See [23].

The following theorem is due to Tunnell and Saito ([22, 23]).

Theorem 2.14

The space \(\mathrm{Hom}_{{\mathbb {E}}_v^*}(\sigma _v\otimes \Omega _v^{-1},{\mathbb {C}})\) is at most one-dimensional. It is nonzero if and only if
$$\begin{aligned} \epsilon \left( \frac{1}{2},\Pi _{\pi ',v}\otimes \Omega _v^{-1}\right) =\Omega _v^{-1}(-1) \epsilon ({\mathbb {B}}_v). \end{aligned}$$
(2.20)

Example 2.15

Suppose that \(\Omega _v\) is unramified and \({\mathbb {B}}_v\simeq M_2({\mathbb {F}}_v)\). So \(\pi _v'=JL(\sigma _v)\simeq \sigma _v\). Let \(n(\pi '_v)\) denote the level of \(\pi '_v\). If \({\mathbb {E}}_v\) is split over \({\mathbb {F}}_v\), then \(\epsilon (\frac{1}{2},\Pi _{\pi ',v}\otimes \Omega _v^{-1})\) is always 1, and \(\mathrm{Hom}_{{\mathbb {E}}_v^*}(\pi '_v\otimes \Omega _v^{-1},{\mathbb {C}})\) is non-zero. If \({\mathbb {E}}_v\) is inert over \({\mathbb {F}}_v\), \(\epsilon (\frac{1}{2},\Pi _{\pi ',v}\otimes \Omega _v^{-1})=1\) if and only if \(n(\pi '_v)\) is even. As a result, \(\mathrm{Hom}_{{\mathbb {E}}_v^*}(\pi '_v\otimes \Omega _v^{-1},{\mathbb {C}})\) is non-zero if and only if \(n(\pi '_v)\) is even (see [7] Proposition 6.3.).

2.4.2 Gross and Prasad’s test vector

If \(\mathrm{Hom}_{{\mathbb {E}}_v^*}(\sigma _v\otimes \Omega _v^{-1},{\mathbb {C}})\) is non-zero for a non-archimedean place, let l be a non-zero element of it. Gross and Prasad in [9] gave a choice of test vector \(F_{1,v}\in \sigma _v\) such that \(l(F_{1,v})\ne 0\), under the hypothesis that either \(\pi _v'\) or \(\Omega _v\) is unramified. This hypothesis implies that the central character is always unramified.

We first assume that \(\Omega _v\) is unramified. On \({\mathbb {B}}_v\) we have a Trace map defined to be
$$\begin{aligned} Tr(\alpha )=\alpha +\iota (\alpha ), \end{aligned}$$
where \(\iota \) is the main involution on \({\mathbb {B}}_v\). An order R of \({\mathbb {B}}_v\) is defined to be a subring of \({\mathbb {B}}_v\) containing \(O_F\) which is a free \(O_F\)-module of rank 4 (equivalently, \(R\otimes _{O_F}{\mathbb {F}}_v={\mathbb {B}}_v\)). Its dual is defined to be
$$\begin{aligned} R^\perp =\{\beta \in {\mathbb {B}}_v|Tr(\alpha \beta )\in O_F \text { for all }\alpha \in R\}. \end{aligned}$$
Recall \(q=|\varpi _v|^{-1}\). Define the reduced discriminant d(R) of R to be the integer such that
$$\begin{aligned} \sharp (R^\perp / R)=q^{2d(R)}. \end{aligned}$$
See [7] for more details.

Let \(R_c\) be an order of reduced discriminant \(c=n(\pi _v')\) which contains \(O_E\) under the embedding \({\mathbb {E}}_v\hookrightarrow {\mathbb {B}}_v\). It is unique up to conjugacy by \({\mathbb {E}}_v^*\). Let \(R_c^*\) denote its units.

Proposition 2.16

Assume that \(\Omega _v\) is unramified and \(\mathbb {B}_v\simeq M_2(\mathbb {F}_v)\). If \(n(\pi _v')\ge 2\), further assume that \({\mathbb {E}}_v/{\mathbb {F}}_v\) is unramified.

When \(\mathrm{Hom}_{{\mathbb {E}}_v^*}(\pi '_v\otimes \Omega _v^{-1},{\mathbb {C}})\ne 0\), let l be a non-trivial element of it. Let \(F_{v}\in \pi _v'\) be the unique (up to constant) element fixed by \(R_c^*\). Then \(l(F_{v})\ne 0\).

Remark 2.17

Proposition 2.16 has statements on the other side of the Jacquet–Langlands correspondence when \(\epsilon (\frac{1}{2},\Pi _{\pi ',v}\otimes \Omega _v^{-1})=-\Omega _v^{-1}(-1)\). But we won’t record them here as we don’t need them.

Example 2.18

Suppose that \({\mathbb {B}}_v\simeq M_2({\mathbb {F}}_v)\). Suppose that \({\mathbb {E}}_v/ {\mathbb {F}}_v\) is inert and can be written as \({\mathbb {F}}_v(\sqrt{D})\). Recall that \({\mathbb {E}}_v\) can be embedded into \(M_2({\mathbb {F}}_v)\) via
$$\begin{aligned} a+b\sqrt{D}\mapsto \begin{pmatrix} a &{} \quad b \\ bD &{} \quad a \end{pmatrix}. \end{aligned}$$
By Example 2.15, \(\pi '_v\simeq \sigma _v\) should be of even level \(c=2k\). Then we can choose
$$\begin{aligned} R_c=\left\{ \begin{pmatrix} a+\varpi _v^k O_F &{} \quad b+\varpi _v^k O_F \\ bD+\varpi _v^k O_F &{} \quad a+\varpi _v^k O_F \end{pmatrix}|a+b\sqrt{D}\in O_E\right\} . \end{aligned}$$
(2.21)

Example 2.19

When \({\mathbb {E}}_v/{\mathbb {F}}_v\) is split, \({\mathbb {B}}_v\) must be the matrix algebra and \(\pi '_v\simeq \sigma _v\). Suppose that 2 is a unit for the local field. For a split place, fix an element \(\sqrt{D}\in {\mathbb {F}}_v\) such that \(\sqrt{D}^2=D\). One can easily check that
$$\begin{aligned} \begin{pmatrix} 1 &{} \quad -\frac{1}{\sqrt{D}} \\ \sqrt{D} &{} \quad 1 \end{pmatrix}^{-1}\begin{pmatrix} a &{} \quad b \\ bD &{} \quad a \end{pmatrix}\begin{pmatrix} 1 &{} \quad \frac{1}{-\sqrt{D}} \\ \sqrt{D} &{} \quad 1 \end{pmatrix}=\begin{pmatrix} a+b{\sqrt{D}} &{} \quad 0 \\ 0 &{} \quad a-b\sqrt{D} \end{pmatrix}. \end{aligned}$$
(2.22)
We can pick
$$\begin{aligned} R_c=\begin{pmatrix} 1 &{} \quad -\frac{1}{\sqrt{D}} \\ \sqrt{D} &{} \quad 1 \end{pmatrix}\begin{pmatrix} O_F &{} \quad O_F \\ \varpi _v^c O_F &{} \quad O_F \end{pmatrix}\begin{pmatrix} 1 &{} \quad -\frac{1}{\sqrt{D}} \\ \sqrt{D} &{} \quad 1 \end{pmatrix}^{-1}. \end{aligned}$$
(2.23)
The element fixed by \(R_c^*\) is just the image of the new form under the action of \(\pi _v'\left( \begin{pmatrix} 1 &{} \quad -\frac{1}{\sqrt{D}} \\ \sqrt{D} &{} \quad 1 \end{pmatrix}\right) \).

Now we assume that \(\pi '_v\) is unramified and \(\Omega _v\) is ramified of level c. This already implies that \({\mathbb {B}}_v\simeq M_2({\mathbb {F}}_v)\) and \(\pi '_v\) is an unramified principal series. Let \(O_c=O_F+\varpi _v^c O_E\). Let R be a maximal order in \(M_2({\mathbb {F}}_v)\) which optimally contains the order \(O_c\). This just means that R is maximal and \(R\cap {\mathbb {E}}_v=O_c\). Such maximal order is unique up to conjugacy by \({\mathbb {E}}_v^*\). Similarly we have the following result:

Proposition 2.20

Assume that \(\pi '_v\) is unramified and \(\Omega _v\) is ramified of level c.

When \(\mathrm{Hom}_{{\mathbb {E}}_v^*}(\pi '_v\otimes \Omega _v^{-1},{\mathbb {C}})\ne 0\), let l be a non-trivial element of it. Let \(F_{v}\in \pi _v'\) be the unique (up to constant) element fixed by \(R^*\). Then \(l(F_{v})\ne 0\).

Example 2.21

Suppose that \({\mathbb {E}}_v/{\mathbb {F}}_v\) is inert, \(v(D)=0\) and \({\mathbb {B}}_v\simeq M_2({\mathbb {F}}_v)\). Then we can pick
$$\begin{aligned} R=\left\{ \begin{pmatrix} O_F &{} \quad \varpi _v^cO_F \\ \varpi _v^{-c}O_F &{} \quad O_F \end{pmatrix}\right\} . \end{aligned}$$

2.4.3 Waldspurger’s formula

Denote by \(\Delta \) the modulus function for \({\text {GL}}_2\) such that
$$\begin{aligned} \Delta \left( \begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad a_2 \end{pmatrix} k\right) =\left| \frac{a_1}{a_2}\right| ^{1/2}. \end{aligned}$$

Theorem 2.22

(Waldspurger’s formula) Let \(F_{1}\in \sigma \), \(F_{2}\in \hat{\sigma }\). Let \(\varphi \in \pi '\) such that \(\theta (f,\varphi ,g_{1},g_{2})=F_{1}(g_{1})F_{2}(g_{2})\) under the Shimizu lifting. Let \(\Omega \) be a Hecke character of \({\mathbb {E}}^*\) such that \(\Omega |_{{\mathbb {A}}_{\mathbb {F}}^*}=w_\sigma \). Then
$$\begin{aligned}&\int \limits _{Z_{{\mathbb {A}}}{\mathbb {E}}^{*}\backslash {\mathbb {A}}_{\mathbb {E}}^*} F_{1}(t_{1}g_{1}) \Omega ^{-1}(t_{1})dt_{1}\int \limits _{Z_{{\mathbb {A}}}{\mathbb {E}}^{*}\backslash {\mathbb {A}}_{\mathbb {E}}^*} F_{2}(t_{2}g_{2})\Omega (t_{2}) dt_{2} \nonumber \\&\quad =L(\eta ,1)\int \limits _{N_{{\mathbb {A}}}Z_{{\mathbb {A}}}\backslash {\text {GL}}_{2}({\mathbb {A}})}\int \limits _{{\mathbb {A}}_{\mathbb {E}}^*} W_{\varphi }^-(h)\Delta (h)^{w-1/2}r'(h)r''(g_{1},g_{2})f(t,Q(t)^{-1})\Omega (t)dt\,dh|_{w=1/2} \nonumber \\&\quad =L(\Pi _{\pi '}\otimes \Omega ^{-1},1/2) \prod \limits _{v\in S}P_0(f_v,\Omega _v,1/2), \end{aligned}$$
(2.24)
where \(W_{\varphi }^-\) is the Whittaker function corresponding to \(\varphi \) with respect to \(\psi ^-(x)=\psi (-x)\). \(\eta \) is the quadratic Hecke character associated to \({\mathbb {E}}/{\mathbb {F}}\). S is the finite set of ramified places. \(P_0(f_v,\Omega _v,w)\) is defined as
$$\begin{aligned} P_0(f_v,\Omega _v,w)= & {} \frac{L_v(\eta _v,w+1/2)}{L_v(\Pi _{\pi ',v}\otimes \Omega _v^{-1},w/2+1/4)}\nonumber \\&\times \int \limits _{NZ\backslash {\text {GL}}_{2}({\mathbb {F}}_v)}\int \limits _{{\mathbb {E}}_v^{*}} W_{\varphi }^-(h)\Delta (h)^{w-1/2}r'(h)r''(g_{1},g_{2})f(t,Q(t)^{-1})\Omega (t)dt\,dh.\nonumber \\ \end{aligned}$$
(2.25)

One result of Waldspurger’s formula is the following:

Corollary 2.23

There exists \(F\in \sigma \) such that
$$\begin{aligned} \int \limits _{Z_{{\mathbb {A}}}{\mathbb {E}}^{*}\backslash {\mathbb {A}}_{\mathbb {E}}^*} F(t) \Omega ^{-1}(t)dt\ne 0, \end{aligned}$$
if and only if \(\mathrm{Hom}_{{\mathbb {A}}_{\mathbb {E}}^*}(\sigma \otimes \Omega ^{-1},{\mathbb {C}})\ne 0\) and \(L(\Pi _{\pi '}\otimes \Omega ^{-1},1/2)\ne 0\).
Waldspurger further considered the integral
$$\begin{aligned} B(f_v,s)=\int \limits _{NZ\backslash {\text {GL}}_{2}({\mathbb {F}}_v)}\int \limits _{{\mathbb {F}}_v^{*}} W_{\varphi }^-(h)\Delta (h)^{s-1}r'(h)f(x,x^{-2})w_\sigma (x)d^*x\,dh. \end{aligned}$$
(2.26)
He found that there exists a bilinear pairing between \(\sigma _v\) and its dual, such that
$$\begin{aligned} B(f_v,1)=\langle F_{1,v},F_{2,v}\rangle . \end{aligned}$$
(2.27)
And at unramified places,
$$\begin{aligned} B(f_v,s)=\frac{L(\pi ,Ad,s)}{\zeta (2s)}. \end{aligned}$$
(2.28)
The global pairing can be written as
$$\begin{aligned} (F_1,F_2)=\int \limits _{{\mathbb {B}}^*({\mathbb {F}})Z({\mathbb {A}})\backslash {\mathbb {B}}^*({\mathbb {A}})}F_1(g)\overline{F_2(g)}dg=\int \limits _{{\mathbb {B}}^*({\mathbb {F}})Z({\mathbb {A}})\backslash {\mathbb {B}}^*({\mathbb {A}})}\theta (f,h,g,g)w_\sigma ^{-1}(g)dg. \end{aligned}$$
(2.29)
For the local pairings defined above, we have the following formula
$$\begin{aligned} (F_1,F_2)=\frac{2L(\pi ,Ad,1)}{\zeta (2)}\prod \limits _{v}B^0(f_v,1), \end{aligned}$$
(2.30)
where \(B^0(f_v,1)=\frac{\zeta (2)}{L(\pi ,Ad,1)} B(f_v,1).\)
Now the point is that when the local integral in Theorem 2.22 is absolutely convergent, it can be rewritten as
$$\begin{aligned} P(f_v,\Omega _v,w)=\int \limits _{{\mathbb {F}}_v^*\backslash {\mathbb {E}}_v^*} \langle F_{1,v},\hat{\sigma }_v(e)F_{2,v} \rangle \Omega _{{v}}(e)de. \end{aligned}$$
By combining Theorem 2.22 with (2.30), one can get

Corollary 2.24

For notations as in Theorem 2.22, we have
$$\begin{aligned}&\frac{\int \nolimits _{Z_{{\mathbb {A}}}{\mathbb {E}}^{*}\backslash {\mathbb {A}}_{\mathbb {E}}^*} F_{1}(t_{1}g_{1}) \Omega ^{-1}(t_{1})dt_{1}\int \nolimits _{Z_{{\mathbb {A}}}{\mathbb {E}}^{*}\backslash {\mathbb {A}}_{\mathbb {E}}^*} F_{2}(t_{2}g_{2})\Omega (t_{2}) dt_{2}}{(F_1,F_2)} \nonumber \\&\quad =\frac{\zeta (2)L(\Pi _{\pi '}\otimes \Omega ^{-1},1/2)}{2L(\pi ,Ad,1)}\prod \nolimits _{v}P^0_v. \end{aligned}$$
(2.31)
where
$$\begin{aligned} P^0_v=\frac{L(\pi ,Ad,1)L_v(\eta _v,1)}{\zeta (2)L_v(\Pi _{\pi ',v}\otimes \Omega _v^{-1},1/2)} \frac{\int \nolimits _{{\mathbb {F}}_v^*\backslash {\mathbb {E}}_v^*} \langle \sigma _v(g_{1,v}) F_{1,v},\hat{\sigma }_v(eg_{2,v})F_{2,v} \rangle \Omega _{v}(e)de}{ \langle F_{1,v},F_{2,v} \rangle }. \end{aligned}$$
(2.32)

Note that the local terms on the right-hand side are now independent of the normalization of local pairings. So instead of the local pairing defined by (2.27), one can use whichever local pairing that facilitates computations. The local integral is essentially an integral of matrix coefficients.

3 Global analysis

In this paper we are interested in the following automorphic integral:
$$\begin{aligned} {\mathbb {I}}(E,F,s)=\int \limits _{Z_{{\mathbb {A}}}{\text {GL}}_{2}({\mathbb {F}})\backslash {\text {GL}}_{2}({\mathbb {A}}_{{\mathbb {F}}})} F(g)E(g,s) dg, \end{aligned}$$
(3.1)
where F is an automorphic cusp form over \({\mathbb {F}}\) and E(gs) is an Eisenstein series defined over a quadratic algebra \({\mathbb {E}}\) as in (2.4). We write \({\mathbb {E}}={\mathbb {F}}(\sqrt{D})\) for some algebraic integer D.

Lemma 3.1

$$\begin{aligned} {\mathbb {I}}(E,F,s)=\int \limits _{Z_{{\mathbb {A}}}{\mathbb {E}}^{*}\backslash {\text {GL}}_{2}({\mathbb {A}})} \Phi _s\left( \begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix} g\right) F(g)dg. \end{aligned}$$
(3.2)

Proof

We first decide the double coset representatives of \(B({\mathbb {E}})\backslash {\text {GL}}_{2}({\mathbb {E}})/{\text {GL}}_2({\mathbb {F}})\). By Bruhat decomposition,
$$\begin{aligned} {\text {GL}}_{2}({\mathbb {E}})=B({\mathbb {E}})\cup \left( \bigcup \limits _{n\in {\mathbb {E}}}B({\mathbb {E}})\omega \begin{pmatrix} 1 &{} \quad n \\ 0 &{} \quad 1 \end{pmatrix}\right) =B({\mathbb {E}}) \cup \left( \bigcup \limits _{m\in {\mathbb {E}}}B({\mathbb {E}})\begin{pmatrix} 1 &{} \quad 0 \\ m &{} \quad 1 \end{pmatrix}\omega \right) \end{aligned}$$
for \(\omega =\begin{pmatrix} 0 &{} \quad 1 \\ -1 &{} \quad 0 \end{pmatrix}\). Note that \(\omega \in {\text {GL}}_{2}({\mathbb {F}})\). The relation
$$\begin{aligned} B({\mathbb {E}})\begin{pmatrix} 1 &{} \quad 0 \\ m_1 &{} \quad 1 \end{pmatrix}{\text {GL}}_2({\mathbb {F}})= B({\mathbb {E}})\begin{pmatrix} 1 &{} \quad 0 \\ m_2 &{} \quad 1 \end{pmatrix}{\text {GL}}_2({\mathbb {F}}) \end{aligned}$$
is equivalent to
$$\begin{aligned} \begin{pmatrix} 1 &{} \quad 0 \\ m_1 &{} \quad 1 \end{pmatrix}\begin{pmatrix} a &{} \quad b \\ c &{} \quad d \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ -m_2 &{} \quad 1 \end{pmatrix}\in B({\mathbb {E}}) \end{aligned}$$
for some \(\begin{pmatrix} a &{} \quad b \\ c &{} \quad d \end{pmatrix}\in {\text {GL}}_2({\mathbb {F}})\). By equating the lower left element of the product to 0, we get the following condition:
$$\begin{aligned} m_{2}=\frac{am_{1}+c}{bm_{1}+d}. \end{aligned}$$
From this one can figure out the double coset representatives and the stabilizers of the right \({\text {GL}}_2({\mathbb {F}})\) action for each representative:
  1. (i)

    Case \(m=0\), the stabilizer is \(\{c=0\}=N({\mathbb {F}})\), the unipotent subgroup. The corresponding orbit is negligible.

     
  2. (ii)

    Case \(m=\sqrt{D}\), the stabilizer is \(\{d=a,c=bD\}= \left\{ aI+b\begin{pmatrix} 0 &{} \quad 1 \\ D &{} \quad 0 \end{pmatrix} \right\} \), which can be further identified with \({\mathbb {E}}^*\).

     
As a result, we can rewrite (3.1) as
$$\begin{aligned} {\mathbb {I}}(E,F,s)&=\int \limits _{Z_{{\mathbb {A}}}{\text {GL}}_2({\mathbb {F}})\backslash {\text {GL}}_{2}({\mathbb {A}})} \left( \sum \limits _{\alpha \in N({\mathbb {F}})\backslash {\text {GL}}_2({\mathbb {F}})}\Phi _s \left( \begin{pmatrix} 1 &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\alpha g \right) \right. \nonumber \\&\quad +\left. \sum \limits _{\alpha \in {\mathbb {E}}^*\backslash {\text {GL}}_2({\mathbb {F}})}\Phi _s \left( \begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix}\alpha g\right) \right) F(g)dg \nonumber \\ {}&=\int \limits _{Z_{{\mathbb {A}}}N({\mathbb {F}})\backslash {\text {GL}}_{2}({\mathbb {A}})} \Phi _s(g)F(g)dg+\int \limits _{Z_{{\mathbb {A}}}{\mathbb {E}}^*\backslash {\text {GL}}_{2}({\mathbb {A}})} \Phi _s \left( \begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix}g \right) F(g)dg.\end{aligned}$$
(3.3)
One just has to see that the first term is 0 since F is a cusp form. This is why the corresponding orbit is called negligible. \(\square \)
Denote
$$\begin{aligned} \gamma _0 =\begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix}. \end{aligned}$$
For \(t=\begin{pmatrix} a &{} \quad b \\ bD &{} \quad a \end{pmatrix}\in {\mathbb {A}}_{\mathbb {E}}^*\), one can check that
$$\begin{aligned} \gamma _0 t\gamma _0^{-1}=\begin{pmatrix} a-b\sqrt{D} &{} \quad b \\ 0 &{} \quad a+b\sqrt{D} \end{pmatrix} \end{aligned}$$
is actually upper triangular. Recall that \(\Phi _s\) satisfies
$$\begin{aligned} \Phi _{s} \left( \begin{pmatrix} a &{} \quad b \\ 0 &{} \quad d \end{pmatrix}x\right) =\chi _{1}(a)\chi _{2}(d) \left| \frac{a}{d}\right| _{{\mathbb {A}}_{\mathbb {E}}}^{s+1/2}\Phi _{s}(x). \end{aligned}$$
From now on we fix our notation for \(\Omega \) as follows:

Definition 3.2

Define for \(t\in {\mathbb {A}}_{\mathbb {E}}^*\)
$$\begin{aligned} \Omega (t)=\chi _1(\overline{t})\chi _2(t)=\chi _{1}(a-b\sqrt{D})\chi _{2}(a+b\sqrt{D}). \end{aligned}$$
(3.4)

Lemma 3.3

With notations as above, we have \(\Phi _s (\gamma _0 tg)=\Phi _s (\gamma _0 g) \Omega (t)\) for any \(g\in {\text {GL}}_{2}({\mathbb {A}})\) and \(t\in {\mathbb {A}}_{\mathbb {E}}^*\).

Now we can further write (3.2) as
$$\begin{aligned} {\mathbb {I}}(E,F,s)=\int \limits _{{\mathbb {A}}_{\mathbb {E}}^*\backslash {\text {GL}}_{2}({\mathbb {A}})} \Phi _s(\gamma _0 g)\int \limits _{Z_{{\mathbb {A}}}{\mathbb {E}}^{*}\backslash {\mathbb {A}}_{\mathbb {E}}^*}F(tg)\Omega (t)dt\,dg. \end{aligned}$$
(3.5)
Note that the interior part of the integral is Waldspurger’s period integral to which one can apply Theorem 2.22. The whole integral can be thought of as a weighted integral of Waldspurger’s period integral.

To fit into Theorem 2.22, take the quaternion algebra \({\mathbb {B}}\) there to be \(M_2({\mathbb {F}})\). Pick \(F_{2}=F\in \hat{\sigma }=\pi \). Then \(F_1\in \sigma \simeq \hat{\pi }\) and \(\varphi \in \pi ' \simeq \hat{\pi }\). Pick \(\Omega \) as the one we defined above, and pick \(g_1\equiv 1\), \(g_2=g\) in (2.25). Then there are two possible situations:

First, if \(\int \limits _{Z_{{\mathbb {A}}}{\mathbb {E}}^{*}\backslash {\mathbb {A}}_{\mathbb {E}}^*}F(tg)\Omega (t)dt=0\) for any g, then \({\mathbb {I}}(E,F,s)=0\). In particular we have the following corollary:

Corollary 3.4

Let \(\Pi \) be the base change of \(\pi \) to \({\mathbb {E}}\). If \(\mathrm{Hom}_{{\mathbb {A}}_{\mathbb {E}}^*}(\pi \otimes \Omega ,{\mathbb {C}})=0\) or \(L(\Pi \otimes \Omega ,1/2)=0\), then \({\mathbb {I}}(E,F,s)=0\).

Secondly, if \(\int \limits _{Z_{{\mathbb {A}}}{\mathbb {E}}^{*}\backslash {\mathbb {A}}_{\mathbb {E}}^*}F(tg)\Omega (t)dt\) is not identically zero, one can fix \(F_1\in \hat{\pi }\) such that the period integral \( C=\int \limits _{Z_{{\mathbb {A}}}{\mathbb {E}}^{*}\backslash {\mathbb {A}}_{\mathbb {E}}^*}F_{1}(t_{1})\Omega ^{-1}(t_{1})dt_{1}\) is not zero.

Note that since \(\Omega \) does not depend on s, this fixed period integral C is also independent of s. Then by Theorem 2.22 we have the following relation:
$$\begin{aligned} C\cdot {\mathbb {I}}(E,F,s)&=\int \limits _{{\mathbb {A}}_{\mathbb {E}}^*\backslash {\text {GL}}_{2}({\mathbb {A}})} \Phi _s(\gamma _0 g)\int \limits _{Z_{{\mathbb {A}}}{\mathbb {E}}^{*}\backslash {\mathbb {A}}_{\mathbb {E}}^*}F_{1}(t_{1})\Omega ^{-1}(t_{1})dt_{1}\int \limits _{Z_{{\mathbb {A}}}{\mathbb {E}}^{*}\backslash {\mathbb {A}}_{\mathbb {E}}^*}F(t_2g)\Omega (t_2)dt_2\,dg \nonumber \\&=L(\eta ,1)\int \limits _{{\mathbb {A}}_{\mathbb {E}}^*\backslash {\text {GL}}_{2}({\mathbb {A}})} \Phi _s(\gamma _0 g)\int \limits _{Z_{{\mathbb {A}}}N_{{\mathbb {A}}}\backslash {\text {GL}}_{2}({\mathbb {A}})}\int \limits _{{\mathbb {A}}_{\mathbb {E}}^*} W_{\varphi }^-(h)\Delta (h)^{w-1/2}r'(h)r''(1,g) \nonumber \\&\quad \times f(t,Q(t)^{-1})\Omega (t)dt\,dh\, dg|_{w=1/2}. \end{aligned}$$
(3.6)
Recall \(\Phi _s (\gamma _0 tg)=\Phi _s (\gamma _0 g) \Omega (t)\). By the definition of the Weil representation, in particular by formula (2.12), we have \(r''(1,g)f(t,Q(t)^{-1})=f(tg,\det (tg)^{-1})\). Then we can actually combine the integrals in t and g. This is why we were applying Waldspurger’s work in a slightly different way. Using Corollary 2.24, we have

Proposition 3.5

Denote
$$\begin{aligned} {\mathbb {I}}(E,F,s)=\int \limits _{Z_{{\mathbb {A}}}{\text {GL}}_{2}({\mathbb {F}})\backslash {\text {GL}}_{2}({\mathbb {A}}_{{\mathbb {F}}})} F(g)E(g,s) dg, \end{aligned}$$
(3.7)
where F is an automorphic cusp form over \({\mathbb {F}}\) and E(gs) is an Eisenstein series defined over a quadratic algebra \({\mathbb {E}}\) as in (2.4).
  1. (1)

    If \(\mathrm{Hom}_{{\mathbb {A}}_{\mathbb {E}}^*}(\pi \otimes \Omega ,{\mathbb {C}})=0\) or \(L(\Pi \otimes \Omega ,1/2)=0\), then \({\mathbb {I}}(E,F,s)=0\).

     
  2. (2)
    Otherwise, we can fix \(F_1\in \hat{\pi }\), such that
    $$\begin{aligned} C=\int \limits _{Z_{{\mathbb {A}}}{\mathbb {E}}^{*}\backslash {\mathbb {A}}_{\mathbb {E}}^*}F_{1}(t_{1})\Omega ^{-1}(t_{1})dt_{1} \end{aligned}$$
    is not zero, independent of s. Let \(\varphi \in \hat{\pi }\) and f be a Schwarz function such that \(\theta (f,\varphi ,g_{1},g_{2})=F_{1}(g_{1})F(g_{2})\) under the Shimizu lifting. Then we have the following euler products of local integrals:
    $$\begin{aligned} C\cdot {\mathbb {I}}(E,F,s)= & {} L(\eta ,1)\prod \limits _{v} \int \limits _{ZN\backslash {\text {GL}}_{2}({\mathbb {F}}_v)}\int \limits _{{\text {GL}}_2({\mathbb {F}}_v)} W_{\varphi ,v}^-(h)\Delta (h)^{w-1/2}r'(h)\nonumber \\&\times \, f_v(g,\det (g)^{-1})\Phi _{s,{v}}(\gamma _0 g)dgdh |_{w=1/2}, \end{aligned}$$
    (3.8)
    or equivalently
    $$\begin{aligned} \frac{C\cdot {\mathbb {I}}(E,F,s)}{(F_1,F)}&=\frac{\zeta (2)L(\eta ,1)}{2L(\pi ,Ad,1)}\prod \limits _{v} \frac{L_v(\pi ,Ad,1)}{\zeta _v(2)} \nonumber \\&\quad \times \frac{\int \limits _{ZN\backslash {\text {GL}}_{2}({\mathbb {F}}_v)}\int \limits _{{\text {GL}}_2({\mathbb {F}}_v)} W_{\varphi ,v}^-(h)\Delta (h)^{w-1/2}r'(h)f_v(g,\det (g)^{-1})\Phi _{s,{v}}(\gamma _0 g)dgdh |_{w=1/2}}{\int \limits _{ZN\backslash {\text {GL}}_{2}({\mathbb {F}}_v)}\int \limits _{{\mathbb {F}}_v^*} W_{\varphi ,v}^-(h)\Delta (h)^{w-1/2}r'(h)f_v(x,x^{-2})w_{\hat{\pi }}(x)d^*x\,dh |_{w=1/2}}\nonumber \\&=\frac{\zeta (2)L(\eta ,1)}{2L(\pi ,Ad,1)}\prod \limits _{v} \frac{L_v(\pi ,Ad,1)}{\zeta _v(2)} \frac{\int \limits _{{\mathbb {F}}_v^*\backslash {\text {GL}}_2({\mathbb {F}}_v)} \langle F_{1,v},\pi _v(g)F_{v} \rangle \Phi _{s,{v}}(\gamma _0g)dg}{\langle F_{1,v},F_{v} \rangle }. \end{aligned}$$
    (3.9)
     
For the following, we shall denote
$$\begin{aligned} {\mathbb {P}}(s,w,f,\Phi _s)=\int \limits _{ZN\backslash {\text {GL}}_{2}({\mathbb {F}}_v)}\int \limits _{{\text {GL}}_2({\mathbb {F}}_v)} W_{\varphi ,v}^-(h)\Delta (h)^{w-1/2}r'(h)f_v(g,\det (g)^{-1})\Phi _{s,{v}}(\gamma _0 g)dg\,dh. \end{aligned}$$
(3.10)
When \(w=\frac{1}{2}\), we also have
$$\begin{aligned} {\mathbb {P}}\left( s,\frac{1}{2},f,\Phi _s\right) =\int \limits _{{\mathbb {F}}_v^*\backslash {\text {GL}}_2({\mathbb {F}}_v)}\langle F_{1,v},\pi _v(g)F_{v}\rangle \Phi _{s,{v}}(\gamma _0g)\,dg. \end{aligned}$$
(3.11)
For most local calculations, in particular for unramified places, we shall use the first expression. But in some cases we shall also use the second expression when calculations can be made easier.

Remark 3.6

In general, \(\varphi \) is not necessarily a newform to have \(\theta (f,\varphi ,g_{1},g_{2})=F_{1}(g_{1})F(g_{2})\). But \(\varphi \) is always a linear combination of translates of a newform, and we can always make a change of variable in the global Theta lifting to incorporate the translates. So we can choose without loss of generality that \(\varphi \) is a newform at the cost of f being possibly more complicated. This explains why we didn’t take \(\varphi \) as a variable for the integral \({\mathbb {P}}\).

One potential shortage for the second formulation is that \((F_1,F)\) could be zero while \(C{\mathbb {I}}(E,F,s)\) is nonzero. We will avoid this problem by making \((F_1,F)\ne 0\) for all the local calculations in the following sections.

4 Local calculations at unramified places

In the rest sections we will mostly focus on the local integrals, so we will suppress the subscript v to simplify the notations. In this section we will compute the local integral at unramified places. In particular we shall follow the first formula of (3.9) to do computation. Recall that the denominator of (3.9) was already computed by Waldspurger and reviewed in (2.28). So we shall mainly focus on computing \({\mathbb {P}}(s,w,f,\Phi _s)\).

We specify here what we mean by an unramified place: the quadratic extension \({\mathbb {E}}\) over \({\mathbb {F}}\) is either inert or split at this place; \(\pi \) is unramified and the corresponding Whittaker function \(W^-_\varphi (h)\) is right K-invariant and normalized so that \(W^-_\varphi (1)=1\); \(\chi _{i}\) is unramified for \(i=1,2\); \(\Phi _s\) is right K-invariant and \(\Phi _s(1)=1\); f is the Schwartz function
$$\begin{aligned} f=char\left( \begin{pmatrix} O_F &{} \quad O_F \\ O_F &{} \quad O_F \end{pmatrix}\right) \times char(O_{F}^{*}). \end{aligned}$$
We always fix an unramified additive character \(\psi \) for any non-archimedean places.
We will show in Propositions 4.5 and 4.6 that at unramified places,
$$\begin{aligned} {\mathbb {P}}(s,1/2,f,\Phi _s)=\frac{L (\Pi \otimes \Omega ,1/2)L (\pi \otimes \chi _{1}|_{{\mathbb {F}}^*},2s+1/2)}{L (\eta ,1)L (\chi ,2s+1)}, \end{aligned}$$
(4.1)
where \(\Pi \) is the base change of \(\pi \), \(\eta \) is the character associated to the quadratic extension, and \(\chi \) as in (2.3) is over \({\mathbb {E}}\) (so is the corresponding L-function).
We introduce here a few more notations before we start. When \(\pi \) is unramified at v, let \(\pi =\pi (\mu _1,\mu _2)\). Then \(\varphi \in \hat{\pi }\simeq \pi (\mu _1^{-1},\mu _2^{-1})\) and the central character of \(\hat{\pi }\) satisfies \(w_{\hat{\pi }}=\chi _{1,s}\chi _{2,s}|_{\mathbb {F}}\). For any multiplicative character \(\chi \), we simply write \(\chi \) for \(\chi (\varpi )\) when there is no confusion. We will also write
https://static-content.springer.com/image/art%3A10.1007%2Fs40993-016-0061-7/MediaObjects/40993_2016_61_Equ332_HTML.gif
Then by the definition of \(\Phi _s\),
$$\begin{aligned} \Phi _s \left( \begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad a_2 \end{pmatrix}g \right) =\chi _{1,s}(a_1)\chi _{2,s}(a_2)\Phi _s(g). \end{aligned}$$
We shall also assume without loss of generality that if \(\mathbb {E}_v/\mathbb {F}_v\) is unramified, D is a unit in the local field.

Remark 4.1

First of all, D is a unit for almost all places.

Secondly, suppose everything else are unramified but we use \(D'=D a^2\) for some non-unit a. Recall one formulation of the local integral is
$$\begin{aligned} {\mathbb {P}}\left( s,\frac{1}{2},f,\Phi _s\right) '=\int \limits _{{\mathbb {F}}^*\backslash {\text {GL}}_2({\mathbb {F}})} \langle F_{1},\pi (g)F \rangle \Phi _{s}(\gamma _0'g)dg. \end{aligned}$$
(4.2)
The only influence of a different D is now
$$\begin{aligned} \gamma _0'=\begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D'} &{} \quad 1 \end{pmatrix}=\begin{pmatrix} a^{-1} &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix} \begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix}\begin{pmatrix} a &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}. \end{aligned}$$
Denote \(\begin{pmatrix} a &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\) by m(a). Then we have
$$\begin{aligned} {\mathbb {P}}\left( s,\frac{1}{2},f,\Phi _s \right) '&=\int \limits _{{\mathbb {F}}^*\backslash {\text {GL}}_2({\mathbb {F}})} \langle F_{1},\pi (g)F \rangle \Phi _{s}(\gamma _0'g)dg \nonumber \\&=\int \limits _{{\mathbb {F}}^*\backslash {\text {GL}}_2({\mathbb {F}})} \langle F_{1},\pi (g)F \rangle \Phi _{s}(m(a)^{-1}\gamma _0m(a)g)dg \nonumber \\&=\chi _{1,s}(a^{-1})\int \limits _{{\mathbb {F}}^*\backslash {\text {GL}}_2({\mathbb {F}})} \langle \hat{\pi }(m(a))F_{1},\pi (g m(a))F \rangle \Phi _{s}(\gamma _0gm(a))dg. \end{aligned}$$
(4.3)
So if we require \(\hat{\pi }(m(a))F_{1},\pi ( m(a))F,\Phi _{s}(\cdot m(a))\) to be newforms, we will get exactly the same integral as the D being a unit case. This argument also applies to the ramified cases.
Since \(W^-_\varphi \) and f are both right K-invariant at unramified places,
$$\begin{aligned} {\mathbb {P}}(s,w,f,\Phi _s)= & {} \int \limits _{{\mathbb {F}}^{*}}W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) |\alpha |^{w/2-1/4}\nonumber \\&\times \int \limits _{{\text {GL}}_{2}({\mathbb {F}})}r' \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) f(g,\det (g)^{-1})\Phi _{s}(\gamma _0 g)dg|\alpha |^{-1}d^*\alpha . \end{aligned}$$
(4.4)
By the definition of the Weil representation, in particular by equation (2.13),
$$\begin{aligned}&\int \limits _{{\text {GL}}_{2}({\mathbb {F}})}r'\left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) f(g,\det (g)^{-1})\Phi _{s}(\gamma _0 g)dg\\&\quad =|\alpha |\int \limits _{{\text {GL}}_{2}({\mathbb {F}})}f(\alpha g,\alpha ^{-1}\det (g)^{-1})\Phi _{s}(\gamma _0 g)dg. \end{aligned}$$
By substituting \(\alpha g\rightarrow g\), we get
$$\begin{aligned}&|\alpha | \int \limits _{{\text {GL}}_{2}({\mathbb {F}})}f(\alpha g,\alpha ^{-1}\det (g)^{-1})\Phi _{s}(\gamma _0 g)dg\\&\quad =|\alpha |\Phi _{s}(\alpha )^{-1} \int \limits _{{\text {GL}}_{2}({\mathbb {F}})}f(g,\alpha \det (g)^{-1})\Phi _{s}(\gamma _0 g)dg. \end{aligned}$$
To be precise, \(\Phi _s(\alpha )\) here should be understood as \(\chi _1\chi _2(\alpha )\) which is actually independent of s. Then the local integral becomes
$$\begin{aligned} \int \limits _{{\mathbb {F}}^{*}}W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) |\alpha |^{w/2-1/4} \Phi _{s}(\alpha )^{-1} \int \limits _{{\text {GL}}_{2}({\mathbb {F}})}f(g,\alpha \det (g)^{-1})\Phi _{s}(\gamma _0 g)dg\,d^*\alpha . \end{aligned}$$
(4.5)
Denote
$$\begin{aligned} I(\alpha ,f,\Phi _s)=\int \limits _{{\text {GL}}_{2}({\mathbb {F}})}f(g,\alpha \det (g)^{-1})\Phi _{s}(\gamma _0 g)dg. \end{aligned}$$
(4.6)
At unramified places f and \(\Phi _{s}\) are both right K- invariant, so we just have to do the integral over
$$\begin{aligned} B({\mathbb {F}})=\left\{ \begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad a_2 \end{pmatrix}\right\} \end{aligned}$$
for \(I(\alpha ,f,\Phi _s)\). Denote
$$\begin{aligned} n=v(a_1), \quad k=v(m), \quad l=v(a_2), \end{aligned}$$
(4.7)
where v(x) means the valuation of x. By the definition of f, \(\left( \begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad a_2 \end{pmatrix}, \frac{\alpha }{a_1a_2}\right) \) is in the support of f if and only if \(n,l,k\ge 0\) and \(\frac{\alpha }{a_1a_2}\in O_F^*\). The latter implies \(l+n=v(\alpha )\). So
$$\begin{aligned} I(\alpha ,f,\Phi _s)=\int \limits _{0\le n\le v(\alpha )}\int \limits _{k\ge 0}\int \limits _{l=v(\alpha )-n}\Phi _s \left( \begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix}\begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad a_2 \end{pmatrix}\right) d^*a_2|a_1|^{-1}dm\,d^*a_1.\nonumber \\ \end{aligned}$$
(4.8)
Here we have used that the left Haar measure for the Borel subgroup is
$$\begin{aligned} d^*a_2|a_1|^{-1}dm\,d^*a_1. \end{aligned}$$
Recall \(\gamma _0=\begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix}.\) One can easily check that
$$\begin{aligned} \begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix}\begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad a_2 \end{pmatrix}=\begin{pmatrix} a_1 &{} \quad m \\ a_1\sqrt{D} &{} \quad a_2+m\sqrt{D} \end{pmatrix}. \end{aligned}$$

Lemma 4.2

  1. (1)

    If \(v(a_2+m\sqrt{D})\ge v(a_1\sqrt{D})\), then \(\Phi _s \left( \begin{pmatrix} a_1 &{} \quad m \\ a_1\sqrt{D} &{} \quad a_2+m\sqrt{D} \end{pmatrix}\right) =\chi _{1,s}\left( \frac{a_2}{\sqrt{D}}\right) \chi _{2,s}(a_1\sqrt{D})\).

     
  2. (2)

    If \(v(a_2+m\sqrt{D})\le v(a_1\sqrt{D})\), then \(\Phi _s\left( \begin{pmatrix} a_1 &{} \quad m \\ a_1\sqrt{D} &{} \quad a_2+m\sqrt{D} \end{pmatrix}\right) =\chi _{1,s}\left( \frac{a_1a_2}{a_2+m\sqrt{D}}\right) \chi _{2,s}(a_2+m\sqrt{D})\).

     

Proof

  1. (1)
    When \(v(a_2+m\sqrt{D})\ge v(a_1\sqrt{D})\),
    $$\begin{aligned} \begin{pmatrix} a_1 &{} \quad m \\ a_1\sqrt{D} &{} \quad a_2+m\sqrt{D} \end{pmatrix}=\begin{pmatrix} \frac{a_2}{\sqrt{D}} &{} \quad a_1 \\ 0 &{} \quad a_1\sqrt{D} \end{pmatrix}\begin{pmatrix} 0 &{} \quad -1 \\ 1 &{} \quad \frac{a_2+m\sqrt{D}}{a_1\sqrt{D}} \end{pmatrix}. \end{aligned}$$
     
  2. (2)
    When \(v(a_2+m\sqrt{D})\le v(a_1\sqrt{D})\),
    $$\begin{aligned} \begin{pmatrix} a_1 &{} \quad m \\ a_1\sqrt{D} &{} \quad a_2+m\sqrt{D} \end{pmatrix}=\begin{pmatrix} \frac{a_1a_2}{a_2+m\sqrt{D}} &{} \quad m \\ 0 &{} \quad a_2+m\sqrt{D} \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \frac{a_1\sqrt{D}}{a_2+m\sqrt{D}} &{} \quad 1 \end{pmatrix}. \end{aligned}$$
     
Then the statements follow from the definition of \(\Phi _s\) and its right K-invariance. \(\square \)

Now we have to consider the inert places separately from the split places.

4.1 Inert places

In this subsection we assume that v is an inert place. As a result, \(v(a_2+m\sqrt{D})=\min \{l,k\}\). Note that for this place \(\sqrt{D}\) is a unit in the local field. Then by the above lemma, we get

Lemma 4.3

  1. 1.
    If \(0\le n \le \frac{v(\alpha )}{2}\), then \(l=v(\alpha )-n\ge n\).
    1. (1i)

      If \(k\ge n\), \(\Phi _s\left( \begin{pmatrix} a_1 &{} \quad m \\ a_1\sqrt{D} &{} \quad a_2+m\sqrt{D} \end{pmatrix}\right) =\chi _{1,s}\left( \frac{a_2}{\sqrt{D}}\right) \chi _{2,s}(a_1\sqrt{D})=\chi _{1,s}^{v(\alpha )-n}\chi _{2,s}^{n}\).

       
    2. (1ii)

      If \(0\le k< n\), \(\Phi _s\left( \begin{pmatrix} a_1 &{} \quad m \\ a_1\sqrt{D} &{} \quad a_2+m\sqrt{D} \end{pmatrix}\right) =\chi _{1,s}\left( \frac{a_1a_2}{a_2+m\sqrt{D}}\right) \chi _{2,s}(a_2+m\sqrt{D})=\chi _{1,s}^{v(\alpha )-k}\chi _{2,s}^{k}\).

       
     
  2. 2.
    If \(\frac{v(\alpha )}{2}\le n\le v(\alpha )\), then \(l=v(\alpha )-n\le n\).
    1. (2i)

      If \(k\ge l\), \(\Phi _s\left( \begin{pmatrix} a_1 &{} \quad m \\ a_1\sqrt{D} &{} \quad a_2+m\sqrt{D} \end{pmatrix}\right) =\chi _{1,s}\left( \frac{a_1a_2}{a_2+m\sqrt{D}}\right) \chi _{2,s}(a_2+m\sqrt{D})=\chi _{1,s}^n\chi _{2,s}^{v(\alpha )-n}\).

       
    2. (2ii)

      If \(k<l\), \(\Phi _s \left( \begin{pmatrix} a_1 &{} \quad m \\ a_1\sqrt{D} &{} \quad a_2+m\sqrt{D} \end{pmatrix}\right) =\chi _{1,s}\left( \frac{a_1a_2}{a_2+m\sqrt{D}}\right) \chi _{2,s}(a_2+m\sqrt{D})=\chi _{1,s}^{v(\alpha )-k}\chi _{2,s}^{k}\).

       
     
As \(\int \limits _{O^*_F}d^*a=1\) , \(\int \limits _{O^*_F}dm=1-q^{-1}\) and \(|a_1|^{-1}=q^n\), one can rewrite the integral (4.8) as a summation
$$\begin{aligned} I(\alpha ,f,\Phi _s)= & {} \sum \limits _{0\le n\le \frac{v(\alpha )}{2}}\left[ \sum \limits _{0\le k< n}\chi _{1,s}^{v(\alpha )-k}\chi _{2,s}^{k}q^{n-k}(1-q^{-1}) \right. \nonumber \\&\quad \left. +\sum \limits _{n\le k<\infty }\chi _{1,s}^{v(\alpha )-n}\chi _{2,s}^{n}q^{n-k}(1-q^{-1})\right] \nonumber \\&\quad +\sum \limits _{\frac{v(\alpha )}{2}< n\le v(\alpha )}\left[ \sum \limits _{0\le k< v(\alpha )-n}\chi _{1,s}^{v(\alpha )-k}\chi _{2,s}^{k}q^{n-k}(1-q^{-1})\right. \nonumber \\&\quad \left. + \sum \limits _{v(\alpha )-n\le k<\infty }\chi _{1,s}^{n}\chi _{2,s}^{v(\alpha )-n}q^{n-k}(1-q^{-1})\right] . \end{aligned}$$
(4.9)
Then it’s a tedious process of summation and combining terms. We will skip the process and give the conclusion directly:

Lemma 4.4

Let v be an inert place. When \(v(\alpha )<0\), \(I(\alpha ,f,\Phi _s)=0\). When \(v(\alpha )\ge 0\),
$$\begin{aligned} I(\alpha ,f,\Phi _s)&=\chi _{1,s}^{v(\alpha )}\frac{1-q^{v(\alpha )+1}}{1-q}\frac{1-q^{-1}}{1-\frac{\chi _{2,s}}{q\chi _{1,s}}} +\chi _{1,s}^{v(\alpha )}\frac{1-\left( \frac{\chi _{2,s}}{\chi _{1,s}}\right) ^{b+1}}{1-\frac{\chi _{2,s}}{\chi _{1,s}}}\frac{q^{-1}\left( 1-\frac{\chi _{2,s}}{\chi _{1,s}}\right) }{1-\frac{\chi _{2,s}}{q\chi _{1,s}}}\nonumber \\ {}&\quad +(q^{-1}\chi _{2,s})^{v(\alpha )}\frac{\left( \frac{q^2\chi _{1,s}}{\chi _{2,s}}\right) ^{b+1}-\left( \frac{q^2\chi _{1,s}}{\chi _{2,s}}\right) ^{v(\alpha )+1}}{1-\frac{q^2\chi _{1,s}}{\chi _{2,s}}}\frac{q^{-1}\left( 1-\frac{\chi _{2,s}}{\chi _{1,s}}\right) }{1-\frac{\chi _{2,s}}{q\chi _{1,s}}}\nonumber \\ {}&\quad \quad \quad \text {if}\; v(\alpha )=2b,2b+1 \\ {}&=-\frac{q(1+q)\frac{\chi _{1,s}}{\chi _{2,s}}}{1-q^2\frac{\chi _{1,s}}{\chi _{2,s}}}\chi _{1,s}^{v(\alpha )}q^{v(\alpha )} \nonumber \\ {}&\quad +\frac{1}{1-q^2\frac{\chi _{1,s}}{\chi _{2,s}}} {\left\{ \begin{array}{ll} \left( 1+q\frac{\chi _{1,s}}{\chi _{2,s}}\right) \chi _{1,s}^b\chi _{2,s}^b,&{}\text {if } v(\alpha )=2b;\\ (1+q)\chi _{1,s}^{b+1}\chi _{2,s}^b,&{}\text {if } v(\alpha )=2b+1. \end{array}\right. } \nonumber \end{aligned}$$
(4.10)
Now we return to the integral (4.5). For a spherical element \(\varphi \in \hat{\pi }\simeq \pi (\mu _1^{-1},\mu _2^{-1})\), we have
$$\begin{aligned} W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) = {\left\{ \begin{array}{ll} |\alpha |^{1/2}\frac{\mu _1^{-1}(\varpi \alpha )-\mu _2^{-1}(\varpi \alpha )}{\mu _1^{-1}(\varpi )-\mu _2^{-1}(\varpi )},&{}\text {if }v(\alpha )\ge 0;\\ 0,&{}\text {otherwise}. \end{array}\right. } \end{aligned}$$
(4.11)
By the condition (1.2), we have \(\mu _1\mu _2\chi _1\chi _2=1\). Denote
$$\begin{aligned} \delta =q^{-(\frac{w}{2}+\frac{1}{4})}. \end{aligned}$$
We again skip tedious calculations and show results directly:
$$\begin{aligned} {\mathbb {P}}(s,w,f,\Phi _s)=\frac{(1+\delta ^2)\left( 1-q\frac{\chi _{1,s}}{\chi _{2,s}}\delta ^2\right) +(\mu _1+\mu _2)\chi _{1,s}\delta (1-q\delta ^2)}{(1-q\mu _1\chi _{1,s}\delta )(1-q\mu _2\chi _{1,s}\delta )(1-\mu _1^2\chi _{1,s}\chi _{2,s}\delta ^2)(1-\mu _2^2\chi _{1,s}\chi _{2,s}\delta ^2)}. \end{aligned}$$
For a character \(\chi \) of \({\mathbb {F}}^*\), define \(s(\chi )\) to be the real number such that \(|\chi (x)| =|x| ^{s(\chi )}\). We have following proposition for the inert case:

Proposition 4.5

Let v be a non-archimedean inert place for \({\mathbb {E}}/{\mathbb {F}}\). Suppose that \(\mathrm{Re}(s)\ge (s(\chi _2)-s(\chi _1))/4\).
  1. (i)

    There exists \(\epsilon >0\) such that, the integral \({\mathbb {P}}(s,w,f,\Phi _s)\) converges uniformly in any compact subset of \(D=\{w\in {\mathbb {C}};\mathrm{Re}(w)>1/2-\epsilon \}\). It’s holomorphic in D.

     
  2. (ii)
    For an unramified place we have:
    $$\begin{aligned} {\mathbb {P}}(s,w,f,\Phi _s)=\frac{(1+\delta ^2)\left( 1-q\frac{\chi _{1,s}}{\chi _{2,s}}\delta ^2\right) +(\mu _1+\mu _2)\chi _{1,s}\delta (1-q\delta ^2)}{(1-q\mu _1\chi _{1,s}\delta )(1-q\mu _2\chi _{1,s}\delta )(1-\mu _1^2\chi _{1,s}\chi _{2,s}\delta ^2)(1-\mu _2^2\chi _{1,s}\chi _{2,s}\delta ^2)}. \end{aligned}$$
    where \(\delta =q^{-(\frac{w}{2}+\frac{1}{4})}\). If we evaluate at \(w=1/2\), and write out variable s explicitly, we get
    $$\begin{aligned} {\mathbb {P}}(s,1/2,f,\Phi _s)&=\frac{1+q^{-1}}{(1-\mu _1^2\chi _{1}\chi _{2}q^{-1})(1-\mu _2^2\chi _{1}\chi _{2}q^{-1})}\nonumber \\ {}&\quad \times \frac{1-\frac{\chi _{1}}{\chi _{2}}q^{-(4s+2)}}{(1-\mu _1\chi _{1}q^{-(2s+1/2)})(1-\mu _2\chi _{1}q^{-(2s+1/2)})}\\&=\frac{L (\Pi \otimes \Omega ,1/2)L (\pi \otimes \chi _{1}|_{{\mathbb {F}}^*},2s+1/2)}{L (\eta ,1)L (\chi ,2s+1)}. \end{aligned}$$
    Recall \(\chi =\frac{\chi _1}{\chi _2}\). \(L (\chi ,2s+1) \) here is a product of L factors over all places of \({\mathbb {E}}\) above v. In this case there is only one place with the order of the residue field being \(q^2\).
     

Proof

For part (1), one can easily imitate Waldspurger’s proof for his local integral in [24]. Part (2) follows directly from the calculation above. \(\square \)

4.2 Split places

Now we consider the case when v splits into two places \(v_1\) and \(v_2\) of \({\mathbb {E}}\). We will use superscript (1) (2) to denote the component at each of these two places. For simplicity we assume that 2 is a unit, or equivalently \(2\not \mid v\). D is now a square in the local field \({\mathbb {F}}\). Fix one of its square roots and denote it by \(\sqrt{D}\) and call the other one \(-\sqrt{D}\).

We write
https://static-content.springer.com/image/art%3A10.1007%2Fs40993-016-0061-7/MediaObjects/40993_2016_61_Equ333_HTML.gif
where \(\Phi _{s}^{(i)}\left( \begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad a_2 \end{pmatrix}g\right) =\chi ^{(i)}_{1,s}(a_1)\chi ^{(i)}_{2,s}(a_2)\Phi _{s}^{(i)}(g)\) and \(\Phi _{s}^{(i)}(1)=1\). Here we denote https://static-content.springer.com/image/art%3A10.1007%2Fs40993-016-0061-7/MediaObjects/40993_2016_61_IEq576_HTML.gif , https://static-content.springer.com/image/art%3A10.1007%2Fs40993-016-0061-7/MediaObjects/40993_2016_61_IEq577_HTML.gif as in the inert case. In this setting we write
$$\begin{aligned} \gamma _0=\begin{pmatrix} 1 &{} \quad 0 \\ (\sqrt{D},-\sqrt{D}) &{} \quad 1 \end{pmatrix}. \end{aligned}$$
Similarly
$$\begin{aligned} \Omega (t)&=\chi _1(a-b\sqrt{D})\chi _2(a+b\sqrt{D}) \nonumber \\&=\chi ^{(1)}_1(a-b\sqrt{D})\chi ^{(2)}_1(a+b\sqrt{D})\chi ^{(1)}_2(a+b\sqrt{D})\chi ^{(2)}_2(a-b\sqrt{D})\ \nonumber \\&=\chi ^{(1)}_1\chi ^{(2)}_2(a-b\sqrt{D})\chi ^{(1)}_2\chi ^{(2)}_1(a+b\sqrt{D}). \end{aligned}$$
(4.12)
We start with \(I(\alpha , f,\Phi _s)=\int \limits _{{\text {GL}}_{2}({\mathbb {F}})}f(g,\alpha \det (g)^{-1})\Phi _{s}(\gamma _0 g)dg\). Recall
$$\begin{aligned} n=v(a_1), k=v(m), l=v(a_2). \end{aligned}$$
Note that f is left invariant by \(\begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix}\). By substituting \(\begin{pmatrix} 1 &{} \quad 0 \\ -\sqrt{D} &{} \quad 1 \end{pmatrix} g\mapsto g\) we get
$$\begin{aligned} I(\alpha ,f,\Phi _s)&=\int \limits _{{\text {GL}}_{2}({\mathbb {F}})}f(g,\alpha \det (g)^{-1})\Phi _{s}^{(1)}\left( \begin{pmatrix} 1 &{} \quad 0 \\ 2\sqrt{D} &{} \quad 1 \end{pmatrix} g\right) \Phi _{s}^{(2)}(g)dg \nonumber \\&=\int \limits _{0\le n\le v(\alpha )}\int \limits _{k\ge 0}\int \limits _{l=v(\alpha )-n}\Phi _{s}^{(1)}\left( \begin{pmatrix} a_1 &{} \quad m \\ 2a_1\sqrt{D} &{} \quad a_2+2m\sqrt{D} \end{pmatrix}\right) \chi ^{(2)}_{1,s}(a_1) \nonumber \\&\quad \times \chi ^{(2)}_{2,s}(a_2)d^*a_2|a_1|^{-1}\,dm\,d^*a_1.\end{aligned}$$
(4.13)
Then we can apply Lemma 4.2 for \(\Phi _{s}^{(1)}\left( \begin{pmatrix} a_1 &{} \quad m \\ 2a_1\sqrt{D} &{} \quad a_2+2m\sqrt{D} \end{pmatrix}\right) \), as \(2\sqrt{D}\) is still a unit. One can further expect Lemma 4.3 to hold mostly, with one exception: according to case (2i) of Lemma 4.3, we expect
$$\begin{aligned} \Phi _{s}^{(1)}\left( \begin{pmatrix} a_1 &{} \quad m \\ 2a_1\sqrt{D} &{} \quad a_2+2m\sqrt{D} \end{pmatrix}\right) =\left( \chi _{1,s}^{(1)}\right) ^n\left( \chi _{2,s}^{(1)}\right) ^{v(\alpha )-n}. \end{aligned}$$
But when \(v(a_2)=v(m)\) in this case, \(v(a_2+2m\sqrt{D})\) could be larger than \(v(a_2)\) or v(m), resulting in a different value for \(\Phi _s^{(1)}\).
We introduce here a correction term \(\Delta I\) for \(I(\alpha ,f,\Phi _s)\):
$$\begin{aligned} I(\alpha ,f,\Phi _s)=I'+\Delta I. \end{aligned}$$
Here \(I'\) is the result one would get if we follow Lemma 4.3 completely. As an analogue of the first expression of (4.10), we have (skipping some tedious steps):
$$\begin{aligned} I'&=\left( \chi ^{(1)}_{1,s}\chi ^{(2)}_{2,s}\right) ^{v(\alpha )} \frac{1-\left( \frac{q\chi ^{(2)}_{1,s}}{\chi ^{(2)}_{2,s}}\right) ^{v(\alpha )+1}}{1-\frac{q\chi ^{(2)}_{1,s}}{\chi ^{(2)}_{2,s}}} \frac{1-q^{-1}}{1-\frac{\chi ^{(1)}_{2,s}}{q\chi ^{(1)}_{1,s}}} \nonumber \\&\quad +\left( \chi ^{(1)}_{1,s}\chi ^{(2)}_{2,s}\right) ^{v(\alpha )} \frac{1-\left( \frac{\chi ^{(2)}_{1,s}\chi ^{(1)}_{2,s}}{\chi ^{(1)}_{1,s}\chi ^{(2)}_{2,s}}\right) ^{b+1}}{1-\frac{\chi ^{(2)}_{1,s}\chi ^{(1)}_{2,s}}{\chi ^{(1)}_{1,s}\chi ^{(2)}_{2,s}}} \frac{q^{-1}\left( 1-\frac{\chi ^{(1)}_{2,s}}{\chi ^{(1)}_{1,s}}\right) }{1-\frac{\chi ^{(1)}_{2,s}}{q\chi ^{(1)}_{1,s}}}\nonumber \\&\quad +\left( q^{-1}\chi ^{(1)}_{2,s}\chi ^{(2)}_{2,s}\right) ^{v(\alpha )} \frac{\left( \frac{q^2\chi ^{(1)}_{1,s}\chi ^{(2)}_{1,s}}{\chi ^{(1)}_{2,s}\chi ^{(2)}_{2,s}}\right) ^{b+1}-\left( \frac{q^2\chi ^{(1)}_{1,s}\chi ^{(2)}_{1,s}}{\chi ^{(1)}_{2,s}\chi ^{(2)}_{2,s}}\right) ^{v(\alpha )+1}}{1-\frac{q^2\chi ^{(1)}_{1,s}\chi ^{(2)}_{1,s}}{\chi ^{(1)}_{2,s}\chi ^{(2)}_{2,s}}} \frac{q^{-1}\left( 1-\frac{\chi ^{(1)}_{2,s}}{\chi ^{(1)}_{1,s}}\right) }{1-\frac{\chi ^{(1)}_{2,s}}{q\chi ^{(1)}_{1,s}}} \end{aligned}$$
(4.14)
for \(v(\alpha )=2b,2b+1\).
We give here a more detailed description of the correction term. Fix n such that \(\frac{v(\alpha )}{2}< n\le v(\alpha )\) and fix m such that \(k=v(m)=l\). Consider the integration in \(d^*a_2\) for (4.13), that is,
$$\begin{aligned} \int \limits _{v(a_2)=v(\alpha )-n}\Phi _{s}^{(1)}\left( \begin{pmatrix} a_1 &{} \quad m \\ 2a_1\sqrt{D} &{} \quad a_2+2m\sqrt{D} \end{pmatrix}\right) \chi ^{(2)}_{1,s}(a_1)\chi ^{(2)}_{2,s}(a_2)d^*a_2. \end{aligned}$$
(4.15)
For a subset of measure \(\frac{q-2}{q-1}\) in \(a_2\),
$$\begin{aligned} v(a_2+2m\sqrt{D})=k \;\;{\text {and}}\;\;\Phi _{s}^{(1)}\left( \begin{pmatrix} a_1 &{} \quad m \\ 2a_1\sqrt{D} &{} \quad a_2+2m\sqrt{D} \end{pmatrix}\right) =\left( \chi _{1,s}^{(1)}\right) ^n\left( \chi _{2,s}^{(1)}\right) ^{v(\alpha )-n}; \end{aligned}$$
For a subset of measure \(\frac{1}{q-1}\frac{q-1}{q}=\frac{1}{q}\) in \(a_2\),
$$\begin{aligned}&v(a_2+2m\sqrt{D})=k+1 \quad {\text { and }} \\&\Phi _{s}^{(1)}\left( \begin{pmatrix} a_1 &{} \quad m \\ 2a_1\sqrt{D} &{} \quad a_2+2m\sqrt{D} \end{pmatrix}\right) =\left( \chi _{1,s}^{(1)}\right) ^{n-1}\left( \chi _{2,s}^{(1)}\right) ^{v(\alpha )-n+1}; \end{aligned}$$
and etc. But once \(v(a_2+2m\sqrt{D})\ge n\), the value of \(\Phi _{s}^{(1)}\) will just remain to be \((\chi ^{(1)}_{1,s})^{v(\alpha )-n}(\chi ^{(1)}_{2,s})^{n}\). Then the integral (4.15) becomes
$$\begin{aligned} \left( \chi ^{(1)}_{1,s}\chi ^{(2)}_{1,s}\right) ^n\left( \chi ^{(1)}_{2,s}\chi ^{(2)}_{2,s}\right) ^{v(\alpha )-n}&\left\{ \frac{q-2}{q-1}+\frac{1}{q}\frac{\chi ^{(1)}_{2,s}}{\chi ^{(1)}_{1,s}}+\frac{1}{q^2}\left( \frac{\chi ^{(1)}_{2,s}}{\chi ^{(1)}_{1,s}}\right) ^2+\cdots \right. \nonumber \\&\quad +\left. \frac{q}{q^{2n-v(\alpha )}(q-1)}\left( \frac{\chi ^{(1)}_{2,s}}{\chi ^{(1)}_{1,s}}\right) ^{2n-v(\alpha )} \right\} . \end{aligned}$$
Comparing with the supposed value \((\chi ^{(1)}_{1,s}\chi ^{(2)}_{1,s})^n(\chi ^{(1)}_{2,s}\chi ^{(2)}_{2,s})^{v(\alpha )-n}\), we get the correction
$$\begin{aligned} -\left( \chi ^{(1)}_{1,s}\chi ^{(2)}_{1,s}\right) ^n\left( \chi ^{(1)}_{2,s}\chi ^{(2)}_{2,s}\right) ^{v(\alpha )-n}\frac{1-\frac{\chi ^{(1)}_{2,s}}{\chi ^{(1)}_{1,s}}}{1-\frac{\chi ^{(1)}_{2,s}}{q\chi ^{(1)}_{1,s}}}\frac{1}{q-1}\left( 1-\left( \frac{\chi ^{(1)}_{2,s}}{q\chi ^{(1)}_{1,s}}\right) ^{2n-v(\alpha )}\right) . \end{aligned}$$
Integrating this in m and \(a_1\) would then give
$$\begin{aligned} \Delta I&=-\sum \limits _{\frac{v(\alpha )}{2}<n\le v(\alpha )}\left( q^{-1}\chi ^{(1)}_{2,s}\chi ^{(2)}_{2,s}\right) ^{v(\alpha )}\left( \frac{q^2\chi ^{(1)}_{1,s}\chi ^{(2)}_{1,s}}{\chi ^{(1)}_{2,s}\chi ^{(2)}_{2,s}}\right) ^n\frac{q^{-1}\left( 1-\frac{\chi ^{(1)}_{2,s}}{\chi ^{(1)}_{1,s}}\right) }{1-\frac{\chi ^{(1)}_{2,s}}{q\chi ^{(1)}_{1,s}}}\nonumber \\&\quad \times \left( 1-\left( \frac{\chi ^{(1)}_{2,s}}{q\chi ^{(1)}_{1,s}}\right) ^{2n-v(\alpha )}\right) .\end{aligned}$$
(4.16)
Then one can check that
$$\begin{aligned} I(\alpha ,f,\Phi _s)= & {} \left( \chi ^{(1)}_{1,s}\chi ^{(2)}_{2,s}\right) ^{v(\alpha )} \frac{1-\left( \frac{q\chi ^{(2)}_{1,s}}{\chi ^{(2)}_{2,s}}\right) ^{v(\alpha )+1}}{1-\frac{q\chi ^{(2)}_{1,s}}{\chi ^{(2)}_{2,s}}} \frac{\frac{\chi ^{(1)}_{1,s}}{\chi ^{(1)}_{2,s}}(1-q)}{1-q\frac{\chi ^{(1)}_{1,s}}{\chi ^{(1)}_{2,s}}} \nonumber \\&+\left( \chi ^{(1)}_{1,s}\chi ^{(2)}_{2,s}\right) ^{v(\alpha )} \frac{1-\left( \frac{\chi ^{(2)}_{1,s}\chi ^{(1)}_{2,s}}{\chi ^{(1)}_{1,s}\chi ^{(2)}_{2,s}}\right) ^{v(\alpha )+1}}{1-\frac{\chi ^{(2)}_{1,s}\chi ^{(1)}_{2,s}}{\chi ^{(1)}_{1,s}\chi ^{(2)}_{2,s}}} \frac{q^{-1}\left( 1-\frac{\chi ^{(1)}_{2,s}}{\chi ^{(1)}_{1,s}}\right) }{1-\frac{\chi ^{(1)}_{2,s}}{q\chi ^{(1)}_{1,s}}}. \end{aligned}$$
(4.17)
The rest story will be the same as in the inert case, so we will skip some steps and show the results directly. By the assumption on the central character, we have \(\mu _1\mu _2\chi ^{(1)}_{1,s}\chi ^{(1)}_{2,s}\chi ^{(2)}_{1,s}\chi ^{(2)}_{2,s}=1\). Recall \(\delta =q^{-(\frac{w}{2}+\frac{1}{4})}\). Then
$$\begin{aligned}&{\mathbb {P}}(s,w,f,\Phi _s) \nonumber \\&\quad =\frac{1-\delta ^2-(\mu _1+\mu _2)\chi ^{(1)}_{1,s}\chi ^{(2)}_{1,s}\delta (1-q\delta ^2)+(1-q)\left( \frac{\chi ^{(1)}_{1,s}}{\chi ^{(1)}_{2,s}}+\frac{\chi ^{(2)}_{1,s}}{\chi ^{(2)}_{2,s}}\right) \delta ^2+\frac{q\chi ^{(1)}_{1,s}\chi ^{(2)}_{1,s}}{\chi ^{(1)}_{2,s}\chi ^{(2)}_{2,s}}\delta ^2(1-\delta ^2)}{(1-\mu _1\chi ^{(1)}_{2,s}\chi ^{(2)}_{1,s}\delta )(1-\mu _1\chi ^{(1)}_{1,s}\chi ^{(2)}_{2,s}\delta )(1-\mu _2\chi ^{(1)}_{2,s}\chi ^{(2)}_{1,s}\delta )(1-\mu _2\chi ^{(1)}_{1,s}\chi ^{(2)}_{2,s}\delta )}\\&\qquad \times \frac{1}{(1-q\mu _1\chi ^{(1)}_{1,s}\chi ^{(2)}_{1,s}\delta )(1-q\mu _2\chi ^{(1)}_{1,s}\chi ^{(2)}_{1,s}\delta )}. \end{aligned}$$
Recall in the beginning of this subsection we have rewritten \(\Omega =\chi ^{(1)}_1\chi ^{(2)}_2(a-b\sqrt{D})\chi ^{(1)}_2\chi ^{(2)}_1(a+b\sqrt{D})\). Define \(s(\Omega )=s(\chi ^{(1)}_{1,s}\chi ^{(2)}_{2,s})-s(\chi ^{(1)}_{2,s}\chi ^{(2)}_{1,s})=s(\chi ^{(1)}_{1}\chi ^{(2)}_{2})-s(\chi ^{(1)}_{2}\chi ^{(2)}_{1})\). This is independent of s.

Then we have the following proposition:

Proposition 4.6

Let v be a non-archimedean split place for \({\mathbb {E}}/{\mathbb {F}}\) and \(\delta =q^{-(\frac{w}{2}+\frac{1}{4})}\). Suppose that \(\mathrm{Re}(s)>(s(\chi ^{(1)}_{2}\chi ^{(2)}_{2})-s(\chi ^{(1)}_{1}\chi ^{(2)}_{1}))/4\),
  1. (i)

    There exists an \(\epsilon ' >0\) such that, the integral \({\mathbb {P}}(s,w,f,\Phi _s)\) converges uniformly in any compact subset of \(D'=\{w\in {\mathbb {C}};\mathrm{Re}(w)>1/2+|s(\Omega )|-\epsilon ' \}\). It’s holomorphic in \(D'\).

     
  2. (ii)
    For an unramified place we have:
    $$\begin{aligned}&{\mathbb {P}}(s,w,f,\Phi _s)\nonumber \\&\quad =\frac{1-\delta ^2-(\mu _1+\mu _2)\chi ^{(1)}_{1,s}\chi ^{(2)}_{1,s}\delta (1-q\delta ^2)+(1-q)\left( \frac{\chi ^{(1)}_{1,s}}{\chi ^{(1)}_{2,s}}+\frac{\chi ^{(2)}_{1,s}}{\chi ^{(2)}_{2,s}}\right) \delta ^2+\frac{q\chi ^{(1)}_{1,s}\chi ^{(2)}_{1,s}}{\chi ^{(1)}_{2,s}\chi ^{(2)}_{2,s}}\delta ^2(1-\delta ^2)}{(1-\mu _1\chi ^{(1)}_{2,s}\chi ^{(2)}_{1,s}\delta )(1-\mu _1\chi ^{(1)}_{1,s}\chi ^{(2)}_{2,s}\delta )(1-\mu _2\chi ^{(1)}_{2,s}\chi ^{(2)}_{1,s}\delta )(1-\mu _2\chi ^{(1)}_{1,s}\chi ^{(2)}_{2,s}\delta )}\\&\qquad \times \frac{1}{\left( 1-q\mu _1\chi ^{(1)}_{1,s}\chi ^{(2)}_{1,s}\delta \right) \left( 1-q\mu _2\chi ^{(1)}_{1,s}\chi ^{(2)}_{1,s}\delta \right) }. \end{aligned}$$
    When \(|s(\Omega )|\) is small enough, we can evaluate at \(w=1/2\) and write out s explicitly:
    $$\begin{aligned}&{\mathbb {P}}(s,1/2,f,\Phi _s) \nonumber \\ {}&=\frac{(1-q^{-1})}{\left( 1-\mu _1\chi ^{(1)}_{2}\chi ^{(2)}_{1}q^{-1/2}\right) \left( 1-\mu _1\chi ^{(1)}_{1}\chi ^{(2)}_{2}q^{-1/2}\right) \left( 1-\mu _2\chi ^{(1)}_{2}\chi ^{(2)}_{1}q^{-1/2}\right) \left( 1-\mu _2\chi ^{(1)}_{1}\chi ^{(2)}_{2}q^{-1/2}\right) }\nonumber \\&\quad \times \frac{\left( 1-\frac{\chi ^{(1)}_{1}}{\chi ^{(1)}_{2}}q^{-(2s+1)}\right) \left( 1-\frac{\chi ^{(2)}_{1}}{\chi ^{(2)}_{2}}q^{-(2s+1)}\right) }{\left( 1-\mu _1\chi ^{(1)}_{1}\chi ^{(2)}_{1}q^{-(2s+1/2)}\right) \left( 1-\mu _2\chi ^{(1)}_{1}\chi ^{(2)}_{1}q^{-(2s+1/2)}\right) }\nonumber \\&=\frac{L (\Pi \otimes \Omega ,1/2)L (\pi \otimes \chi _{1}|_{{\mathbb {F}}^*},2s+1/2)}{L (\eta ,1)L (\chi ,2s+1)}. \end{aligned}$$
    (4.18)
    Recall \(\chi =\frac{\chi _1}{\chi _2}\). \(L (\chi ,2s+1) \) here is a product of L factors over all places of \({\mathbb {E}}\) above v. In the split case there are two places over v, thus two factors.
     

Remark 4.7

Again the proof of part (1) will be very similar to Waldspurger’s original proof.

Theorem 4.8

For notations as in Proposition 3.5, we have
$$\begin{aligned} \frac{C\cdot {\mathbb {I}}(E,F,s)}{(F_1,F)}=\frac{\zeta (2)L (\Pi \otimes \Omega ,1/2)L (\pi \otimes \chi _{1}|_{{\mathbb {F}}^*},2s+1/2)}{2L(\pi ,Ad,1)L (\chi ,2s+1)} \prod \limits _{v} {\mathbb {P}}^0_v(s,1/2,f_v,\Phi _{s,v}),\nonumber \\ \end{aligned}$$
(4.19)
where for non-archimedean places,
$$\begin{aligned} {\mathbb {P}}^0_v(s,1/2,f_v,\Phi _{s,v})= \frac{L_v(\pi ,Ad,1)L_v (\eta ,1)L_v (\chi ,2s+1)}{\zeta _v(2)L_v (\Pi \otimes \Omega ,1/2)L _v(\pi \otimes \chi _{1}|_{{\mathbb {F}}^*},2s+1/2)} \frac{{\mathbb {P}}_v(s,1/2,f_v,\Phi _{s,v})}{\langle F_{1,v},F_{v} \rangle },\nonumber \\ \end{aligned}$$
(4.20)
for archimedean places,
$$\begin{aligned} {\mathbb {P}}^0_v(s,1/2,f_v,\Phi _{s,v})=\frac{{\mathbb {P}}_v(s,1/2,f_v,\Phi _{s,v})}{\langle F_{1,v},F_{v} \rangle }, \end{aligned}$$
(4.21)
where \({\mathbb {P}}_v(s,1/2,f_v,\Phi _{s,v})\) is as in (3.10) or alternatively (3.11).

The product in v is a finite product since \({\mathbb {P}}^0_v(s,1/2,f_v,\Phi _{s,v})=1\) for almost all places due to Propositions 4.5, 4.6 and the local unramified calculations done in [24].

5 Local calculations for other non-archimedean places

In this section we will compute \({\mathbb {P}}^0(s,w,f,\Phi _s)\) for some ramified non-archimedean places. The additive character \(\psi \) is assumed without loss of generality to be unramified, as the formulation of the local integral (3.11) is completely independent of the choice of \(\psi \). As explained in Remark 4.1, we assume \(v(D)=0\) if \(\mathbb {E}_v/\mathbb {F}_v\) is unramified, and \(v(D)=1\) otherwise. Before we start, we first list the ramified cases we are going to consider in this section.

Case

\(\pi \)

\(\chi _1\) and \(\chi _2\)

\({\mathbb {E}}/{\mathbb {F}}\)

1

Unramified

Unramified

Ramified

2

Unramified special

Unramified

Split

3

Highly ramified of level c

Unramified

Split

4

Unramified

\(\chi _{1}\) level c

Inert

5

\(\mu _2\) level c

\(\chi _1\) level c

Inert

6

Highly ramified of even level c

Unramified

Inert

The characters not mentioned (that is, \(\mu _1\) and \(\chi _2\)) in Cases 4 and 5 are all unramified. This implies that \(\chi _{1}|_{{\mathbb {F}}^*}\) is unramified in Case 4 and is of level c in Case 5. Here by \(\pi \) highly ramified, we mean \(\pi \) is of level \(c\ge 2\).

This section will be organized to solve these cases one by one. It may seem that Case 6 should be done earlier. But we are going to use a different approach, so we leave it to the last.

In Case 1 to Case 5, we will also need to keep track of the denominator of the local integral using (2.27) that
$$\begin{aligned} \langle F_1,F \rangle =\int \limits _{NZ\backslash {\text {GL}}_{2}({\mathbb {F}}_v)}\int \limits _{{\mathbb {F}}_v^{*}} W_{\varphi }^-(h)r'(h)f(x,x^{-2})w_{\hat{\pi }}(x)d^*x\,dh. \end{aligned}$$
(5.1)
We are going to get the following table of normalized local integral:

Case

Choice of \(F_1\), F

\({\mathbb {P}}_v^0(s,1/2,f,\Phi _s)\)

1

G-P test vector

1

2

G-P test vector

\(\displaystyle \frac{1}{q(1-\chi ^{(2)}q^{-(2s+1)})}\)

3

G-P test vector

\(\displaystyle \frac{1}{q^c}\frac{L(\pi , Ad,1)}{1-\chi ^{(2)}q^{-(2s+1)}}\)

4

G-P test vector

\(\displaystyle \frac{{\mathbb {P}}'(1/2)}{1+q^{-1}}\) for \({\mathbb {P}}'(w)\) given in (5.66)

5

new form

\(\displaystyle \frac{1}{(q+1){q^{c-1}}\chi _{1}(\sqrt{D})}\)

6

G-P test vector

\(\displaystyle \frac{L(\pi ,Ad,1)}{q^c(1-\chi q^{-(4s+2)})} \)

Recall here that \( \chi =\frac{\chi _1}{\chi _2}.\)

Remark 5.1

  1. 1.

    When c is odd and the quadratic extension is inert, the local integral is automatically zero according to Example 2.15. Thus Case 2,3,6 will cover all situations when only \(\pi \) is ramified. When \(\chi _1\) and/or \(\chi _2\) are ramified, the situation could be very complicated. So we restrict ourselves to Case 4 and 5 only.

     
  2. 2.

    \(\varphi \) is always a newform. G-P test vector above refers to the Gross and Prasad’s test vectors as discussed in Subsect. 2.4.2. Note that from Case 1 to Case 5 we are going to use Theta lifting for a Schwartz function f and in this table we claim that we are using G-P test vector/new form. This is simply because the Schwartz function f we will choose has corresponding left/right invariance, and G-P test vector/new form is unique up to constant by such invariance. For the specific choices of f and \(\Phi _s\), we refer the readers to each subsections. We are going to keep \(F_1\) F and calculations as simple as possible, at the cost of sometimes complicated choices of \(\Phi _s\).

     
  3. 3.

    \(L(\pi , Ad,1)\) is not given explicitly in Cases 3 and 6 as it could be different for highly ramified principal series and supercuspidal representations, while we wish to keep the formula uniform.

     

One corollary for Case 5 above is the following new result after Gross and Prasad’s work:

Corollary 5.2

Suppose that the local field extension \({\mathbb {E}}/{\mathbb {F}}\) is inert and that \(\pi \), \(w_\pi \), \(\Omega \), \(\Omega |_{\mathbb {F}}\) have same levels. Then
$$\begin{aligned} \mathrm{Hom}_{{\mathbb {E}}^*}(\pi \otimes \Omega ,{\mathbb {C}})\ne 0, \end{aligned}$$
and the local new form for \(\pi \) is a test vector for any nonzero element from this space.

5.1 \({\mathbb {E}}/{\mathbb {F}}\) ramified

Here we consider the case when \(\pi \) and \(\Phi _{s}\) are both unramified, but \({\mathbb {E}}/{\mathbb {F}}\) is a ramified local field extension. Let \(\varpi _{\mathbb {E}}\) be a uniformizer of \({\mathbb {E}}\) such that \(\varpi _{\mathbb {E}}^2=\varpi \). For simplicity, we still let \(v(\varpi )=1\) and write \(\mu \) or \(\chi \) in short for \(\mu (\varpi )\) or \(\chi (\varpi )\). We suppose that \(v(\sqrt{D})=\frac{1}{2}\). We will prove in this subsection the following result

Proposition 5.3

Suppose that \(\pi \) and \(\Phi _{s}\) are both unramified at v, \({\mathbb {E}}/{\mathbb {F}}\) is ramified with \(v(\sqrt{D})=1/2\). We pick f and \(\Phi _s\) as in the unramified case. Then
$$\begin{aligned} {\mathbb {P}}(s,w,f,\Phi _s)=\frac{[1+(\mu _1^2\chi _{1,s}\chi _{2,s})(\varpi _E)\delta +(\mu _2^2\chi _{1,s}\chi _{2,s})(\varpi _E)\delta +\delta ^2](1-q\delta ^2\frac{\chi _{1,s}}{\chi _{2,s}}(\varpi _E))}{(1-q\mu _1\chi _{1,s}\delta )(1-q\mu _2\chi _{1,s}\delta )(1-\mu _1^2\chi _{1,s}\chi _{2,s}\delta ^2)(1-\mu _2^2\chi _{1,s}\chi _{2,s}\delta ^2)}\nonumber \\ \end{aligned}$$
(5.2)
for \(\delta =q^{-(\frac{w}{2}+\frac{1}{4})}\). When \(w=\frac{1}{2}\),
$$\begin{aligned} {\mathbb {P}}(s,\frac{1}{2},f,\Phi _s)&=\frac{1}{(1-(\mu ^2_{1}\chi _1\chi _2)(\varpi _E)q^{-\frac{1}{2}})(1-(\mu ^2_{2}\chi _1\chi _2)(\varpi _E)q^{-\frac{1}{2}})}\nonumber \\ {}&\quad \times \frac{1-\frac{\chi _{1}}{\chi _{2}}(\varpi _E)q^{-(2s+1)}}{(1-\mu _1\chi _{1}q^{-(2s+1/2)})(1-\mu _2\chi _{1}q^{-(2s+1/2)})} \end{aligned}$$
(5.3)
is just as expected. Thus \({\mathbb {P}}^0(s,1/2,f,\Phi _s)=1\).

To compare with the unramified case, we write \(\chi _{i,s}^{\frac{1}{2}}\) to mean \(\chi _{i,s}(\varpi _E)\). As in the inert case, we can start with Eqs. (4.5) and (4.8). Lemma 4.2 still holds. Then we have the following lemma as an analogue of Lemma 4.3:

Lemma 5.4

  1. (1)
    If \(0\le n < \frac{v(\alpha )}{2}\), then \(l> n\).
    1. (1i)

      If \(k\ge n\), \(\Phi _s\left( \begin{pmatrix} a_1 &{} \quad m \\ a_1\sqrt{D} &{} \quad a_2+m\sqrt{D} \end{pmatrix}\right) =\chi _{1,s}\left( \frac{a_2}{\sqrt{D}}\right) \chi _{2,s}(a_1\sqrt{D})=\chi _{1,s}^{v(\alpha )-n-\frac{1}{2}}\chi _{2,s}^{n+\frac{1}{2}}\).

       
    2. (1ii)

      If \(0\le k< n\), \(\Phi _s\left( \begin{pmatrix} a_1 &{} \quad m \\ a_1\sqrt{D} &{} \quad a_2+m\sqrt{D} \end{pmatrix}\right) =\chi _{1,s}\left( \frac{a_1a_2}{a_2+m\sqrt{D}}\right) \chi _{2,s}(a_2+m\sqrt{D})=\chi _{1,s}^{v(\alpha )-k-\frac{1}{2}}\chi _{2,s}^{k+\frac{1}{2}}\).

       
     
  2. (2)
    If \(\frac{v(\alpha )}{2}\le n\le v(\alpha )\), then \(l\le n\).
    1. (2i)

      If \(k\ge l\), \(\Phi _s\left( \begin{pmatrix} a_1 &{} \quad m \\ a_1\sqrt{D} &{} \quad a_2+m\sqrt{D} \end{pmatrix}\right) =\chi _{1,s}\left( \frac{a_1a_2}{a_2+m\sqrt{D}}\right) \chi _{2,s}(a_2+m\sqrt{D})=\chi _{1,s}^n\chi _{2,s}^{v(\alpha )-n}\).

       
    2. (2ii)

      If \(k<l\), \(\Phi _s\left( \begin{pmatrix} a_1 &{} \quad m \\ a_1\sqrt{D} &{} \quad a_2+m\sqrt{D} \end{pmatrix}\right) =\chi _{1,s}\left( \frac{a_1a_2}{a_2+m\sqrt{D}}\right) \chi _{2,s}(a_2+m\sqrt{D})=\chi _{1,s}^{v(\alpha )-k-\frac{1}{2}}\chi _{2,s}^{k+\frac{1}{2}}\).

       
     

Proof

One just need to use Lemma 4.2, and note that \(v(a_2+m\sqrt{D})=Min\{l,k+\frac{1}{2}\}\), \(v(a_1\sqrt{D})=n+\frac{1}{2}\). Then the result is clear. \(\square \)

Now to compute \(I(\alpha ,f,\Phi _s)\) for this case, we compare Lemma 5.4 to Lemma 4.3. We see that the values of \(\Phi _{s}\) differ by \((\frac{\chi _{2,s}}{\chi _{1,s}})^{\frac{1}{2}}\) except the case (2i). Denote by I the formula (4.10) for the inert case, and we get the relation
$$\begin{aligned}&I(\alpha ,f,\Phi _s)\nonumber \\&\quad =\left( \frac{\chi _{2,s}}{\chi _{1,s}}\right) ^{\frac{1}{2}}\left[ I+\left( \left( \frac{\chi _{2,s}}{\chi _{1,s}}\right) ^{-\frac{1}{2}}-1\right) \sum \limits _{\frac{v(\alpha )}{2}\le n\le v(\alpha )} \sum \limits _{v(\alpha )-n\le k<\infty }\chi _{1,s}^{n}\chi _{2,s}^{v(\alpha )-n}q^{n-k}(1-q^{-1})\right] .\nonumber \\ \end{aligned}$$
(5.4)
As a result,
$$\begin{aligned} I(\alpha ,f,\Phi _s)=\left( \frac{\chi _{2,s}}{\chi _{1,s}}\right) ^{\frac{1}{2}}I+\left( 1-\left( \frac{\chi _{2,s}}{\chi _{1,s}}\right) ^{\frac{1}{2}}\right) \chi _{2,s}^{v(\alpha )}q^{-v(\alpha )}\frac{\left( q^2\frac{\chi _{1,s}}{\chi _{2,s}}\right) ^{v(\alpha )-b}{-}\left( q^2\frac{\chi _{1,s}}{\chi _{2,s}}\right) ^{v(\alpha )+1}}{1-q^2\frac{\chi _{1,s}}{\chi _{2,s}}}\nonumber \\ \end{aligned}$$
(5.5)
for \(v(\alpha )=2b,2b+1\).
Then one can follow the same steps to get:
$$\begin{aligned} {\mathbb {P}}(s,w,f,\Phi _s)=\frac{\left( 1+\mu _1(\chi _{1,s}\chi _{2,s})^{\frac{1}{2}}\delta +\mu _2(\chi _{1,s}\chi _{2,s})^{\frac{1}{2}}\delta +\delta ^2\right) \left( 1-q\delta ^2\left( \frac{\chi _{2,s}}{\chi _{1,s}}\right) ^{\frac{1}{2}}\right) }{(1-\mu _1^2\chi _{1,s}\chi _{2,s}\delta ^2)(1-\mu _2^2\chi _{1,s}\chi _{2,s}\delta ^2)(1-q\mu _1\chi _{1,s}\delta )(1-q\mu _2\chi _{1,s}\delta )},\nonumber \\ \end{aligned}$$
(5.6)
where \(\delta =q^{-(\frac{w}{2}+\frac{1}{4})}\). Recall that \(\mu _1\mu _2\chi _1\chi _2=1\). When \(w=\frac{1}{2}\), \(\delta =q^{-\frac{1}{2}}\), so
$$\begin{aligned} {\mathbb {P}}(s,\frac{1}{2},f,\Phi _s)&=\frac{(1+(\mu ^2_{1}\chi _1\chi _2)(\varpi _E)q^{-\frac{1}{2}})(1+(\mu ^2_{2}\chi _1\chi _2)(\varpi _E)q^{-\frac{1}{2}})}{(1-\mu _1^2\chi _{1}\chi _{2}q^{-1})(1-\mu _2^2\chi _{1}\chi _{2}q^{-1})}\nonumber \\ {}&\quad \times \frac{1-\frac{\chi _{1}}{\chi _{2}}(\varpi _E)q^{-(2s+1)}}{(1-\mu _1\chi _{1}q^{-(2s+1/2)})(1-\mu _2\chi _{1}q^{-(2s+1/2)})}. \end{aligned}$$
(5.7)
This is exactly the expected L-factor
$$\begin{aligned} \frac{L_v (\Pi \otimes \Omega ,1/2)L _v(\pi \otimes \chi _{1}|_{{\mathbb {F}}^*},2s+1/2)}{L_v (\eta ,1)L_v (\chi ,2s+1)}. \end{aligned}$$
The local pairing \(\langle F_1,F \rangle \) is the same as in the unramified case, as we choose same Schwartz function f, and the quadratic extension is not related to this calculation.

5.2 Unramified special representation

In this subsection we consider the case when \(\pi =\sigma (\mu _1,\mu _2)\) is an unramified special representation. Then \(\varphi \) and its corresponding Whittaker function \(W_\varphi ^-\) belong to \(\hat{\pi }=\sigma (\mu _1^{-1},\mu _2^{-1})\). Since \(\sigma (\mu _1^{-1},\mu _2^{-1})\) is equivalent to \(\sigma (\mu _2^{-1},\mu _1^{-1})\), we can assume without loss of generality that \(\sigma (\mu _1^{-1},\mu _2^{-1})\) is an irreducible subrepresentation of \(\pi (\mu _1^{-1},\mu _2^{-1})\). This implies that https://static-content.springer.com/image/art%3A10.1007%2Fs40993-016-0061-7/MediaObjects/40993_2016_61_IEq718_HTML.gif . We shall do the computation in the split case, as the inert case would fail the Tunnell-Saito criterion as seen in Example 2.15. Write \(\Phi _s=\Phi _s^{(1)}\Phi _s^{(2)}\) as in Sect. 4.2.

Pick
$$\begin{aligned} f=char\left( \begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix}\begin{pmatrix} O_F &{} \quad O_F \\ \varpi O_F &{} \quad O_F \end{pmatrix}\right) \times char(O_F^*). \end{aligned}$$
Pick \(\Phi _s^{(2)}\) to be the unique right \(K_1(\varpi )\)-invariant function supported on \(BK_1(\varpi )\) such that \(\Phi _s^{(2)}(1)=1\), and \(\Phi _s^{(1)}\) just to be the standard right K-invariant function. The property of f implies that \(F_1\) is the Gross–Prasad test vector while F is the local newform, which is different from what we claimed in the beginning of this section. We shall now do a simple trick. Let
$$\begin{aligned} f'=char\left( \begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix}\begin{pmatrix} O_F &{} \quad O_F \\ \varpi O_F &{} \quad O_F \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ -\sqrt{D} &{} \quad 1 \end{pmatrix}\right) \times char(O_F^*), \end{aligned}$$
(5.8)
and
$$\begin{aligned} \Phi _s'(g)=\Phi _s\left( g\begin{pmatrix} 1 &{} \quad 0 \\ -\sqrt{D} &{} \quad 1 \end{pmatrix}\right) . \end{aligned}$$
(5.9)
for \(\Phi _s\) defined above. The matrix here should be thought of as an element of \({\text {GL}}_2({\mathbb {F}})\) diagonally embedded in \({\text {GL}}_2({\mathbb {E}})\).
By a simple change of variable, we have
$$\begin{aligned} {\mathbb {P}}(s,w,f,\Phi _s)={\mathbb {P}}(s,w,f',\Phi _s'), \end{aligned}$$
(5.10)
and the Theta lifting of \(f'\) will be Gross–Prasad test vectors for both \(F_1\) and F.

Proposition 5.5

Suppose that \(\chi _1\) and \(\chi _2\) are unramified, and \({\mathbb {E}}/{\mathbb {F}}\) is split. Suppose that \(\pi =\sigma (\mu _1,\mu _2)\) is an unramified special representation such that https://static-content.springer.com/image/art%3A10.1007%2Fs40993-016-0061-7/MediaObjects/40993_2016_61_IEq735_HTML.gif . Further assume that 2 is a unit. Then
$$\begin{aligned} {\mathbb {P}}(s,w,f',\Phi _s')&={\mathbb {P}}(s,w,f,\Phi _s)\nonumber \\&=\frac{1-q^{-1}}{(q+1)^2} \frac{1-\frac{\chi _{1,s}^{(1)}}{\chi _{2,s}^{(1)}}}{(1-\delta \chi ^{(1)}_{2,s}\chi ^{(2)}_{1,s}\mu _2)(1-\delta \chi ^{(1)}_{1,s}\chi ^{(2)}_{2,s}\mu _2)(1-q\delta \chi ^{(1)}_{1,s}\chi ^{(2)}_{1,s}\mu _2)}.\nonumber \\ \end{aligned}$$
(5.11)
At \(w=1/2\), we have
$$\begin{aligned}&{\mathbb {P}}(s,1/2,f',\Phi _s')\nonumber \\&\quad =\frac{1}{(q+1)^2} \frac{(1-q^{-1})\left( 1-\frac{\chi ^{(1)}_{1}}{\chi ^{(1)}_{2}}q^{-(2s+1)}\right) }{(1{-}\mu _2\chi ^{(1)}_{2}\chi ^{(2)}_{1}q^{-1/2})(1{-}\mu _2\chi ^{(1)}_{1}\chi ^{(2)}_{2}q^{-1/2})(1{-}\mu _2\chi ^{(1)}_{1}\chi ^{(2)}_{1}q^{-(2s+1/2)})}.\nonumber \\ \end{aligned}$$
(5.12)
The denominator of the expression is as expected, and
$$\begin{aligned} {\mathbb {P}}^0(s,1/2,f',\Phi _s')=\frac{1}{q(1-\chi ^{(2)}q^{-(2s+1)})}. \end{aligned}$$
(5.13)
We first work out the Whittaker function \(W^-_\varphi \) for a new form \(\varphi \) in \(\hat{\pi }\). It’s a classical result that such \(\varphi \) should be \(K_1(\varpi )\)-invariant, and up to a constant multiple
$$\begin{aligned} \varphi |_K=char(B(O_F)K_1(\varpi ))-q^{-1}char\left( B(O_F)\begin{pmatrix} 1 &{} \quad 0 \\ 1 &{} \quad 1 \end{pmatrix}K_1(\varpi )\right) . \end{aligned}$$
(5.14)
The corresponding \(W^-_\varphi \) is also \(K_1(\varpi )\)-invariant. According to Lemma 2.1, we just need to figure out \(W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) \) and \(W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ 1 &{} \quad 1 \end{pmatrix}\right) \). It is possible to obtain these values directly from classical theories, but we will start with a more general setting, as it will be helpful for later cases. Recall that \(W^-_\varphi \) is the Whittaker funciton associated to \(\psi ^-\). So
$$\begin{aligned} W^-_\varphi (g)=\int \limits _{m\in {\mathbb {F}}}\varphi \left( \omega \begin{pmatrix} 1 &{} \quad m \\ 0 &{} \quad 1 \end{pmatrix}g\right) \psi (m)dm. \end{aligned}$$
In general we want to write
$$\begin{aligned} \omega \begin{pmatrix} 1 &{} \quad m \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^j &{} \quad 1 \end{pmatrix}=\begin{pmatrix} \varpi ^j &{} \quad 1 \\ -\alpha -m\varpi ^j &{} \quad -m \end{pmatrix} \end{aligned}$$
in form of \(B\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}K_1(\varpi ^c)\) for \(0\le i,j \le c\). Note that if \(i=c\), then \(\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\) is absorbed into \(K_1(\varpi ^c)\). Same for j.

Lemma 5.6

  1. (1)
    Suppose that \(i=0\).
    1. (1i)

      If \(j=0\), we need \(m\notin \alpha (-1+\varpi O_F)\) for \(\begin{pmatrix} \varpi ^j &{} \quad 1 \\ -\alpha -m\varpi ^j &{} \quad -m \end{pmatrix}\in B\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}K_1(\varpi ^c)\);

       
    2. (1ii)

      If \(j>0\), we need \(v(m)\ge v(\alpha )\).

       
    Under the above conditions we can write \(\begin{pmatrix} \varpi ^j &{} \quad 1 \\ -\alpha -m\varpi ^j &{} \quad -m \end{pmatrix}\) as
    $$\begin{aligned} \begin{pmatrix} -\frac{\alpha }{\alpha +m\varpi ^j} &{} \quad \varpi ^j+\frac{\alpha }{\alpha +m\varpi ^j} \\ 0 &{} \quad -\alpha -m\varpi ^j \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ 1 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad -1+\frac{m}{\alpha +m\varpi ^j} \\ 0 &{} \quad 1 \end{pmatrix}. \end{aligned}$$
     
  2. (2)
    Suppose that \(i=c\).
    1. (2i)

      If \(j<c\), we need \(m\in \alpha \varpi ^{-j}(-1+\varpi ^{c-j}O_F)\);

       
    2. (2ii)

      If \(j=c\), we need \(v(m)\le v(\alpha )-c\).

       
    Under the above conditions, we can write \(\begin{pmatrix} \varpi ^j &{} \quad 1 \\ -\alpha -m\varpi ^j &{} \quad -m \end{pmatrix}\) as
    $$\begin{aligned} \begin{pmatrix} -\frac{\alpha }{m} &{} \quad 1 \\ 0 &{} \quad -m \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \frac{\alpha }{m}+\varpi ^j &{} \quad 1 \end{pmatrix}. \end{aligned}$$
     
  3. (3)
    Suppose that \(0<i<c\).
    1. (3i)

      If \(j<i\), we need \(m\in \alpha \varpi ^{-j}(-1+\varpi ^{i-j}O_F^*)\);

       
    2. (3ii)

      If \(j>i\), we need \(v(m)=v(\alpha )-i\);

       
    3. (3iii)

      If \(j=i\), we need \(v(m)\le v(\alpha )-i\) but \(m\notin \alpha \varpi ^{-i}(-1+\varpi O_F)\).

       
    Under the above conditions we can write \(\begin{pmatrix} \varpi ^j &{} \quad 1 \\ -\alpha -m\varpi ^j &{} \quad -m \end{pmatrix}\) as
    $$\begin{aligned} \begin{pmatrix} -\frac{\alpha \varpi ^i}{\alpha +m\varpi ^j} &{} \quad 1 \\ 0 &{} \quad -m \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\begin{pmatrix} \frac{\alpha +m\varpi ^j}{m\varpi ^i} &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}. \end{aligned}$$
     

This lemma is straightforward to check. We will leave the proof to the reader.

Corollary 5.7

Assume that \(\mu _1\) and \(\mu _2\) are unramified and \(\varphi \in \sigma (\mu _1^{-1},\mu _2^{-1})\) is given by (5.14). Let \(W^-_\varphi \) be the normalized Whittaker function associated to \(\varphi \). Then
$$\begin{aligned} W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) = {\left\{ \begin{array}{ll} \mu _1^{-v(\alpha )}q^{-v(\alpha )/2}, &{}\text { if }v(\alpha )\ge 0;\\ 0,&{}\text { if }v(\alpha )<0, \end{array}\right. } \end{aligned}$$
(5.15)
and
$$\begin{aligned} W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ 1 &{} \quad 1 \end{pmatrix}\right) = {\left\{ \begin{array}{ll} -q^{-1}\mu _1^{-v(\alpha )}q^{-v(\alpha )/2}\psi (-\alpha ), &{}\text { if }v(\alpha )\ge -1;\\ 0,&{}\text { if }v(\alpha )<-1. \end{array}\right. } \end{aligned}$$
(5.16)

Proof

Put \(c=1\), and consider \(j=1\). By formula (5.14) and (1ii) (2ii) of the above lemma:
$$\begin{aligned} W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right)&=\int \limits _{m\in {\mathbb {F}}}\varphi \left( \omega \begin{pmatrix} 1 &{} \quad m \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) \psi (m) dm \nonumber \\&=\int \limits _{v(m)\le v(\alpha )-1}\mu _1^{-1}\left( -\frac{\alpha }{m}\right) \mu _2^{-1}(-m)\left| \frac{\alpha }{m^2}\right| ^{1/2}\psi (m)dm-q^{-1}\nonumber \\&\quad \times \int \limits _{v(m)\ge v(\alpha )}\mu _1^{-1}(-1)\mu _2^{-1}(-\alpha )\left| \frac{1}{\alpha }\right| ^{1/2}\psi (m)dm\nonumber \\&={\left\{ \begin{array}{ll} (-q^{-1}-q^{-2})\mu _1^{-v(\alpha )}q^{-v(\alpha )/2}, \text { if }v(\alpha )\ge 0;\\ 0,\text { if }v(\alpha )<0. \end{array}\right. }\end{aligned}$$
(5.17)
In the last equation, one need to use Lemma 2.3 and that https://static-content.springer.com/image/art%3A10.1007%2Fs40993-016-0061-7/MediaObjects/40993_2016_61_IEq782_HTML.gif .
If we normalize \(W^-_\varphi \left( \begin{pmatrix} 1 &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) \) to be 1, then
$$\begin{aligned} W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) = {\left\{ \begin{array}{ll} \mu _1^{-v(\alpha )}q^{-v(\alpha )/2}, &{}\text { if }v(\alpha )\ge 0;\\ 0,&{}\text { if }v(\alpha )<0. \end{array}\right. } \end{aligned}$$
(5.18)
Similarly, we consider the case when \(j=0\). From (1i) and (2i):
$$\begin{aligned} W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ 1 &{} \quad 1 \end{pmatrix}\right)&=\int \limits _{m\in {\mathbb {F}}}\varphi \left( \omega \begin{pmatrix} 1 &{} \quad m \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ 1 &{} \quad 1 \end{pmatrix}\right) \psi (m) dm\nonumber \\&=\int \limits _{m\in \alpha (-1+\varpi O_F)}\mu _1^{-1}\left( -\frac{\alpha }{m}\right) \mu _2^{-1}(-m)\left| \frac{\alpha }{m^2}\right| ^{1/2}\psi (m)dm\nonumber \\&\quad -q^{-1}\int \limits _{m\notin \alpha (-1+\varpi O_F)}\mu _1^{-1}\left( -\frac{\alpha }{\alpha +m}\right) \mu _2^{-1}(-\alpha -m)\nonumber \\&\quad \times \left| \frac{\alpha }{(\alpha +m)^2}\right| ^{1/2}\psi (m)dm\nonumber \\&={\left\{ \begin{array}{ll} q^{-1}(q^{-1}+q^{-2})\mu _1^{-v(\alpha )}q^{-v(\alpha )/2}, &{}\text { if }v(\alpha )\ge 0;\\ q^{-1}(1+q^{-1})\mu _1q^{-1/2}\psi (-\alpha ),&{}\text { if }v(\alpha )=-1;\\ 0,&{}\text { if }v(\alpha )<-1. \end{array}\right. }\nonumber \\ \end{aligned}$$
(5.19)
After normalization, we get
$$\begin{aligned} W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ 1 &{} \quad 1 \end{pmatrix}\right) = {\left\{ \begin{array}{ll} -q^{-1}\mu _1^{-v(\alpha )}q^{-v(\alpha )/2}\psi (-\alpha ), &{}\text { if }v(\alpha )\ge -1;\\ 0,&{}\text { if }v(\alpha )<-1. \end{array}\right. } \end{aligned}$$
(5.20)
\(\square \)
Suppose that the chosen Schwartz function f is also \(K_1(\varpi )\)-invariant under the Weil representation \(r'\). Then by Lemma 2.2 and Corollary 5.7, we have:
$$\begin{aligned} {\mathbb {P}}(s,w,f,\Phi _s)&=\frac{1}{q+1}\int \limits _{v(\alpha )\ge 0}\mu _1^{-v(\alpha )}q^{-(w/2+1/4)v(\alpha )}\Phi _s(\alpha )^{-1}I(\alpha ,f,\Phi _s)d^*\alpha \nonumber \\&\quad +\frac{q}{q+1}\int \limits _{v(\alpha )\ge -1}-q^{-1}\mu _1^{-v(\alpha )}\psi (-\alpha ) q^{-(w/2+1/4)v(\alpha )}\Phi _s(\alpha )^{-1} \nonumber \\&\quad \times I\left( \alpha , r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ 1 &{} \quad 1 \end{pmatrix}\right) f,\Phi _s\right) d^*\alpha . \end{aligned}$$
(5.21)
Now we verify the \(K_1(\varpi )\)-invariance of f and calculate \(I(\alpha ,f,\Phi _s)\) and \(I\left( \alpha , r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ 1 &{} \quad 1 \end{pmatrix}\right) f,\Phi _s\right) \). Recall that we picked
$$\begin{aligned} f=char\left( \begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix}\begin{pmatrix} O_F &{} \quad O_F \\ \varpi O_F &{} \quad O_F \end{pmatrix}\right) \times char(O_F^*). \end{aligned}$$
It’s clearly right \(K_1(\varpi )\)-invariant. This choice of Schwartz function is motivated by Example 2.19. It can also be written as
$$\begin{aligned} f=\sum \limits _{a_0\in O_F/\varpi O_F}char\left( \begin{pmatrix} a_0+\varpi O_F &{} \quad O_F \\ a_0\sqrt{D}+\varpi O_F &{} \quad O_F \end{pmatrix}\right) \times char(O_F^*). \end{aligned}$$
(5.22)
Using Lemmas (2.10) and (2.13) one can easily check that f is \(K_1(\varpi )\)-invariant under the Weil representation, and
$$\begin{aligned} r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ 1 &{} \quad 1 \end{pmatrix}\right) f&=q^{-2}\sum \limits _{a_0\in O_F/\varpi O_F}\psi (u[(x_1-a_0)x_4-x_2(x_3-a_0\sqrt{D})])\nonumber \\ {}&\quad \times char\left( \begin{pmatrix} O_F &{} \quad \varpi ^{-1}O_F \\ O_F &{} \quad \varpi ^{-1}O_F \end{pmatrix}\right) \times char(O_F^*). \end{aligned}$$
(5.23)
This sum is also right \(K_1(\varpi )\)-invariant.

Recall that \(\Phi _s^{(2)}\) is the unique right \(K_1(\varpi )\)-invariant function supported on \(BK_1(\varpi )\), and \(\Phi _s^{(1)}\) is the standard right K-invariant function.

We will write \({\text {GL}}_2({\mathbb {F}})=\begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix}BK_1(\varpi )\cup \begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix}B\begin{pmatrix} 1 &{} \quad 0 \\ 1 &{} \quad 1 \end{pmatrix}K_1(\varpi )\). The matrix \(\begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix}\) on the left amounts to a change of variable. Then by our choice of \(\Phi _s^{(2)}\), in particular its support, we only need to integrate over \(\begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix}BK_1(\varpi )\) for \(I(\alpha ,f,\Phi _s)\) and \(I\left( \alpha ,r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ 1 &{} \quad 1 \end{pmatrix}\right) f,\Phi _s\right) \). By the right \(K_1(\varpi )\)-invariance of \(\Phi _s\), we can write
https://static-content.springer.com/image/art%3A10.1007%2Fs40993-016-0061-7/MediaObjects/40993_2016_61_Equ101_HTML.gif
(5.24)
Note that the domain and the integrand of this integral is exactly the same as (4.13) in Sect. 4.2. Denote by I the result we got in (4.17). Then
$$\begin{aligned} I(\alpha ,f,\Phi _s)=\frac{1}{q+1}I. \end{aligned}$$
(5.25)
Now we consider \(I\left( \alpha ,r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ 1 &{} \quad 1 \end{pmatrix}\right) f,\Phi _s\right) \). For the matrix \(\begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix}\begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad a_2 \end{pmatrix}=\begin{pmatrix} a_1 &{} \quad m \\ a_1\sqrt{D} &{} \quad a_2+m\sqrt{D} \end{pmatrix}\),
$$\begin{aligned} \psi (u[(x_1-a_0)x_4-x_2(x_3-a_0\sqrt{D})])=\psi \left( \alpha \left( 1-\frac{a_0}{a_1}\right) \right) . \end{aligned}$$
(5.26)
Here we have used \(u=\frac{\alpha }{\det x}\) in \(I\left( \alpha ,r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ 1 &{} \quad 1 \end{pmatrix}\right) f,\Phi _s\right) \). Then by (5.23),
https://static-content.springer.com/image/art%3A10.1007%2Fs40993-016-0061-7/MediaObjects/40993_2016_61_Equ104_HTML.gif
(5.27)
Compare the domain of each integral in this expression with the domain of (4.13), we note that we have two additional parts:
$$\begin{aligned} \{v(a_1),v(a_2)\ge 0, v(m)=-1 \}\;\;\text { and }\;\; \{v(a_1)=v(\alpha )+1,v(a_2)=-1, v(m)\ge -1 \}. \end{aligned}$$
Also note \(\psi (\alpha (1-\frac{a_0}{a_1}))=1\) if \(v(\alpha )\ge 0\) and \(v(a_1)\le v(\alpha )\). So over the common domain \(\{a_1,m,a_2\in O_F \}\), the integral gives I as in (4.17). It’s not difficult to work out the integral over the two additional parts. Then one can get
$$\begin{aligned}&I\left( \alpha ,r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ 1 &{} \quad 1 \end{pmatrix}\right) f,\Phi _s\right) \nonumber \\&\quad = {\left\{ \begin{array}{ll} \frac{q^{-1}}{q+1}\left( I+(q-1)\frac{\chi _{1,s}^{(1)}}{\chi _{2,s}^{(1)}}(\chi _{1,s}^{(1)}\chi _{2,s}^{(2)})^{v(\alpha )} \frac{1-\left( \frac{q\chi _{1,s}^{(2)}}{\chi _{2,s}^{(2)}}\right) ^{v(\alpha )+1}}{1-\frac{q\chi _{1,s}^{(2)}}{\chi _{2,s}^{(2)}}}\right) , &{}\text { if }v(\alpha )\ge 0;\\ 0,&{}\text { otherwise}. \end{array}\right. } \end{aligned}$$
(5.28)
By combining (5.25) (5.28) with (5.21), we get
$$\begin{aligned} {\mathbb {P}}(s,w,f,\Phi _s) =\frac{1-q^{-1}}{(q+1)^2}\frac{1-\frac{\chi _{1,s}^{(1)}}{\chi _{2,s}^{(1)}}}{(1-\delta \chi ^{(1)}_{2,s}\chi ^{(2)}_{1,s}\mu _2)(1-\delta \chi ^{(1)}_{1,s}\chi ^{(2)}_{2,s}\mu _2)(1-q\delta \chi ^{(1)}_{1,s}\chi ^{(2)}_{1,s}\mu _2)}.\nonumber \\ \end{aligned}$$
(5.29)
At \(w=1/2\), we have
$$\begin{aligned}&{\mathbb {P}}(s,1/2,f,\Phi _s) \nonumber \\&\quad =\frac{1}{(q+1)^2} \frac{(1-q^{-1})(1-\frac{\chi ^{(1)}_{1}}{\chi ^{(1)}_{2}}q^{-(2s+1)})}{(1{-}\mu _2\chi ^{(1)}_{2}\chi ^{(2)}_{1}q^{-1/2})(1{-}\mu _2\chi ^{(1)}_{1}\chi ^{(2)}_{2}q^{-1/2})(1{-}\mu _2\chi ^{(1)}_{1}\chi ^{(2)}_{1}q^{-(2s+1/2)})}.\nonumber \\ \end{aligned}$$
(5.30)
Now we evaluate local pairing \(\langle F_1,F \rangle \) for \(f'\) by (2.27).
Let
$$\begin{aligned} f''=char\left( \begin{pmatrix} O_F &{} \quad O_F \\ \varpi O_F &{} \quad O_F \end{pmatrix}\right) \times char(O_F^*). \end{aligned}$$
It differs from \(f'\) by conjugation, but conjugation acts trivially on the center. So
$$\begin{aligned} \langle F_1,F \rangle&=\int \limits _{NZ\backslash {\text {GL}}_{2}({\mathbb {F}}_v)}\int \limits _{{\mathbb {F}}_v^{*}} W_{\varphi }^-(h)r'(h)f'(x,x^{-2})w_{\hat{\pi }}(x)d^*x\,dh\nonumber \\&=\int \limits _{NZ\backslash {\text {GL}}_{2}({\mathbb {F}}_v)}\int \limits _{{\mathbb {F}}_v^{*}} W_{\varphi }^-(h)r'(h)f''(x,x^{-2})w_{\hat{\pi }}(x)d^*x\,dh. \end{aligned}$$
(5.31)
The latter is easier to compute, and one can easily check that
$$\begin{aligned} \langle F_1,F \rangle&=\frac{1}{q+1}\sum \limits _{n\ge 0}\mu _1^{-2n}q^{-n}(\mu _1\mu _2)^n- \frac{q^{-1}}{q+1}\sum \limits _{n\ge 0}\mu _1^{-2n}q^{-n}(\mu _1\mu _2)^n\nonumber \\&=\frac{q}{(q+1)^2}. \end{aligned}$$
(5.32)
Here we have used that \(\mu _1^{-1}\mu _2=|\cdot |\).
The expected L-factors in this case is
$$\begin{aligned}&\frac{\zeta (2)}{L(\pi ,Ad,1)}\frac{L (\Pi \otimes \Omega ,1/2)L (\pi \otimes \chi _{1}|_{{\mathbb {F}}^*},2s+1/2)}{L (\eta ,1)L (\chi ,2s+1)}\nonumber \\&\quad =\frac{(1-q^{-1})(1-\chi ^{(1)}q^{-(2s+1)})(1-\chi ^{(2)}q^{-(2s+1)})}{(1-\mu _2\chi ^{(1)}_{2}\chi ^{(2)}_{1}q^{-1/2})(1-\mu _2\chi ^{(1)}_{1}\chi ^{(2)}_{2}q^{-1/2})(1-\mu _2\chi ^{(1)}_{1}\chi ^{(2)}_{1}q^{-(2s+1/2)})}. \end{aligned}$$
(5.33)
Thus
$$\begin{aligned} {\mathbb {P}}^0(s,1/2,f',\Phi _s')=\frac{1}{q(1-\chi ^{(2)}q^{-(2s+1)})}. \end{aligned}$$
(5.34)

5.3 \(\pi \) of higher level and \({\mathbb {E}}/{\mathbb {F}}\) split

Definition 5.8

Let
$$\begin{aligned} f=char\left( \begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix}\begin{pmatrix} O_F &{} \quad O_F \\ \varpi ^cO_F &{} \quad O_F \end{pmatrix}\right) \times char(O_F^*), \end{aligned}$$
(5.35)
$$\begin{aligned} f'=char\left( \begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix}\begin{pmatrix} O_F &{} \quad O_F \\ \varpi ^cO_F &{} \quad O_F \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ -\sqrt{D} &{} \quad 1 \end{pmatrix}\right) \times char(O_F^*). \end{aligned}$$
(5.36)
Pick \(\Phi _s^{(2)}\) to be the unique right \(K_1(\varpi ^c)\)-invariant function supported on \(BK_1(\varpi ^c)\) such that \(\Phi _s^{(2)}(1)=1\), and \(\Phi _s^{(1)}\) to be the standard right K-invariant function. For \(\Phi _s=\Phi _s^{(1)}\Phi _s^{(2)}\), let
$$\begin{aligned} \Phi _s'(g)=\Phi _s\left( g\begin{pmatrix} 1 &{} \quad 0 \\ -\sqrt{D} &{} \quad 1 \end{pmatrix}\right) . \end{aligned}$$
(5.37)

In this subsection we will prove:

Proposition 5.9

Suppose that \(\pi \) is a representation of level \(c>1\) with unramified central character. Suppose that \(\chi _1\) and \(\chi _2\) are unramified and \({\mathbb {E}}/{\mathbb {F}}\) is split. Further suppose that 2 is a unit.

Then
$$\begin{aligned} {\mathbb {P}}(s,w,f',\Phi _s')={\mathbb {P}}(s,w,f,\Phi _s)=\frac{1-q^{-1}}{(q+1)^2q^{2c-2}}\left( 1-\frac{\chi _{1}^{(1)}}{\chi _{2}^{(1)}}q^{-(2s+1)}\right) . \end{aligned}$$
(5.38)
Apart from the constant term, the denominator is the same as \(L_v (\Pi \otimes \Omega ,1/2)L _v(\pi \otimes \chi _{1}|_{{\mathbb {F}}^*},2s+1/2)\), which is 1 in this case. Then
$$\begin{aligned} {\mathbb {P}}^0(s,1/2,f',\Phi _s')=\frac{1}{q^c}\frac{L(\pi , Ad,1)}{1-\chi ^{(2)}q^{-(2s+1)}}. \end{aligned}$$
(5.39)

5.3.1 Supercuspidal representations

We first assume that \(\pi \) and \(\hat{\pi }\) are supercuspidal representations. Then it is easier to describe the group actions using the Kirillov model.

For basic properties of the Kirillov model, one can read [15]. For the level and the new form of the Kirillov model of a supercuspidal representation, we mainly follow [5]. Here we just recount part of the facts necessary for our computations.

For a fixed additive character \(\psi ^-\), the Kirillov model of \(\hat{\pi }\) is a unique realization on the space of Schwartz functions \(S({\mathbb {F}}^*)\) such that
$$\begin{aligned} \hat{\pi }\left( \begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad a_2 \end{pmatrix}\right) \varphi (x)=w_{\hat{\pi }}(a_2)\psi (-ma_2^{-1}x)\varphi (a_1a_2^{-1}x), \end{aligned}$$
(5.40)
where \(w_{\hat{\pi }}\) is the central character for \(\hat{\pi }\). By Bruhat decomposition, one just has to know the action of \(\omega =\begin{pmatrix} 0 &{} \quad 1 \\ -1 &{} \quad 0 \end{pmatrix}\) to understand the whole group action.
Define
$$\begin{aligned} \mathbf 1 _{\nu ,n}(x)={\left\{ \begin{array}{ll} \nu (u), &{}\text {if } x=u\varpi ^n\text { for } u\in O_F^*;\\ 0,&{}\text {otherwise}. \end{array}\right. } \end{aligned}$$
Roughly speaking, it’s the character \(\nu \) supported at \(v(x)=n\). Such functions provide a basis for \(S({\mathbb {F}}^*)\). We can then describe the action of \(\omega =\begin{pmatrix} 0 &{} \quad 1 \\ -1 &{} \quad 0 \end{pmatrix}\) on \(\mathbf 1 _{\nu ,n}\) explicitly according to [15]:
$$\begin{aligned} \hat{\pi }(\omega )\mathbf 1 _{\nu ,n}=C_{\nu w_0^{-1}}z_0^{-n}\mathbf 1 _{\nu ^{-1}w_0,-n+n_{\nu ^{-1}}}. \end{aligned}$$
(5.41)
Here \(z_0=w_{\hat{\pi }}(\varpi )\) and \(w_0=w_{\hat{\pi }}|_{O_F^*}\). \(n_\nu \) is an integer. \(C_\nu \) and \(n_\nu \) are decided by the representation \(\hat{\pi }\) and the character \(\nu \) (and independent of n).
The relation \(\omega ^2=-\begin{pmatrix} 1 &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\) implies that
$$\begin{aligned} n_{\nu }=n_{\nu ^{-1}w_0^{-1}}, C_\nu C_{\nu ^{-1}w_0^{-1}}=w_0(-1)z_0^{n_\nu }. \end{aligned}$$
(5.42)
It’s well-known that
$$\begin{aligned} n_{\nu }\le -2 \end{aligned}$$
for any \(\nu \). According to Proposition A.1, when the supercuspidal representation is fixed with the central character unramified, \(n_\nu \) only depends on the level of \(\nu \). When we pick \(\nu \) to be the trivial character, the number \(-n_1\) is actually the level of this supercuspidal representation, that is
$$\begin{aligned} c=-n_1. \end{aligned}$$
The argument in [5] with slight modification can show that there is a unique up to constant element \(\varphi \) in the supercupidal representation which is invariant under \(K_1(\varpi ^{c})\). One can easily check that \(\varphi =\mathbf 1 _{1,0}\) is such an element.
From now on, we assume that the central character \(w_{\hat{\pi }}\) is unramified, so \(w_0=w_{\hat{\pi }}|_{O_F^*}=1\). For the newform \(\varphi =\mathbf 1 _{1,0}\), its associated Whittaker function \(W^-_\varphi \) is also right \(K_1(\varpi ^c)\)-invariant. We can calculate \(W_\varphi ^- \) according to the relation between the Kirillov model and the Whittaker model:
$$\begin{aligned} W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) =\hat{\pi }\left( \begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) \varphi (\alpha ). \end{aligned}$$
(5.43)
It’s difficult to describe \(W^-_\varphi \) explicitly. But it will suffice to know only some specific integrals for \(W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) \).

Lemma 5.10

Suppose that \(\hat{\pi }\) is a supercuspidal representation with unramified central character. Let \(W^-_\varphi \) be the Whittaker function associated to \(\varphi =\mathbf 1 _{1,0}\in \hat{\pi }\).
  1. (1)

    \(W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) =\mathbf 1 _{1,0}(\alpha )\). For \(0\le i<c\), \(W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) \) is supported only at \(v(\alpha )=\min \{0, 2i-c\}\).

     
  2. (2)

    \(\displaystyle \int \limits _{v(\alpha )=\min \{0, 2i-c\}}W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) d^*\alpha = {\left\{ \begin{array}{ll} 1, &{}\text { if }i\ge c;\\ -\frac{1}{q-1}, &{}\text { if }i=c-1;\\ 0, &{}\text { otherwise}. \end{array}\right. }\)

     
  3. (3)

    \(\displaystyle \int \limits _{v(\alpha )=\min \{0, 2i-c\}}W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) \psi (\varpi ^{-i}\alpha ) d^*\alpha ={\left\{ \begin{array}{ll} C_1, &{}\text { if }i=0;\\ -\frac{1}{q-1}C_1w_{\hat{\pi }}, &{}\text { if }i=1;\\ 0, &{}\text { otherwise}. \end{array}\right. }\)

     

Proof

The first statement of (1) is clear. Now let \(0\le i<c\). According to Proposition A.1, if \(\nu \) is a character of level i, then \(n_\nu =\min \{n_1,-2i\}\). Note that
$$\begin{aligned}&\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}=-\omega \begin{pmatrix} 1 &{} \quad -\varpi ^i \\ 0 &{} \quad 1 \end{pmatrix}\omega ,\\&\hat{\pi }(\omega )\mathbf 1 _{1,0}=C_{w_0^{-1}}\mathbf 1 _{w_0,n_1}=C_1\mathbf 1 _{1,n_1}. \end{aligned}$$
The action of \(\begin{pmatrix} 1 &{} \quad -\varpi ^i \\ 0 &{} \quad 1 \end{pmatrix}\) for \(i< c\) will give a non-trivial factor \(\psi (\varpi ^i x)\) at \(v(x)=n_1\). By the classical result about Gauss sum, \(\hat{\pi }\left( \begin{pmatrix} 1 &{} \quad -\varpi ^i \\ 0 &{} \quad 1 \end{pmatrix}\omega \right) \mathbf 1 _{1,0}\) is a linear combination of all characters of level \(-n_1-i=c-i\), supported at \(v(x)=n_1\). (It should be understood that if \(c-i=1\), then this is a linear combination of all characters of level 1 and 0.) After another action of \(\omega \) their levels will not be changed, but supported at
$$\begin{aligned} v(x)=-n_1+\min \{n_1, -2(c-i)\}=\min \{0, 2i-c\}. \end{aligned}$$
This finishes the proof of part (1).
When we integrate \(W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) \) in \(\alpha \), we are just finding the level 0 component of it. By the discussion above, this is only possible when \(i=c\) or \(c-1\). The integral in the case \(i=c\) is obvious. When \(i=c-1\), one can compute that the level 0 component of \(\hat{\pi }\left( \begin{pmatrix} 1 &{} \quad -\varpi ^i \\ 0 &{} \quad 1 \end{pmatrix}\omega \right) \mathbf 1 _{1,0}\) is
$$\begin{aligned} -\frac{1}{q-1}C_1\mathbf 1 _{1,n_1}, \end{aligned}$$
as \(\int \limits _{x\in \varpi ^{-1}O_F^*}\psi (x)d^*x=-\frac{1}{q-1}\). Then the action of \(\omega \) will map it to \(-\frac{1}{q-1}\mathbf 1 _{1,0},\) using (5.42). Thus
$$\begin{aligned} \int \limits _{v(\alpha )=0}W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^{c-1} &{} \quad 1 \end{pmatrix}\right) d^*\alpha =-\frac{1}{q-1}. \end{aligned}$$
(5.44)
Part (2) is proved.
Now to integrate \(W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) \) against \(\psi (\varpi ^{-i}\alpha )\), the idea is to interpret \(\psi (\varpi ^{-i}\alpha )\) as a factor one can get by the group action in the Kirillov model. More specifically,
$$\begin{aligned} W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) \psi (\varpi ^{-i}\alpha )&=\hat{\pi }\left( \begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) \varphi (\alpha )\psi (\varpi ^{-i}\alpha ) \nonumber \\ {}&=\hat{\pi }\left( \begin{pmatrix} 1 &{} \quad -\varpi ^{-i} \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) \varphi (\alpha )\\ \nonumber&=\hat{\pi }\left( -\omega \begin{pmatrix} \varpi ^i &{} \quad 1 \\ 0 &{} \quad \varpi ^{-i} \end{pmatrix}\right) \varphi (\alpha ). \end{aligned}$$
The integral of \(W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) \psi (\varpi ^{-i}\alpha )\) is then the same as to find level 0 component of the expression above. One can do this similarly as in the proof of part (2). We will leave the rest to the reader. \(\square \)
Now we will give the formula for \({\mathbb {P}}(s,w,f,\Phi _s)\). We will basically follow the technique used for the unramified special representation case in the last subsection, so we will skip some details and give results directly. Recall that we have chosen
$$\begin{aligned} f=char\left( \begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix}\begin{pmatrix} O_F &{} \quad O_F \\ \varpi ^cO_F &{} \quad O_F \end{pmatrix}\right) \times char(O_F^*). \end{aligned}$$
It is \(K_1(\varpi ^c)\)-invariant under the right action and the Weil representation. One can calculate for \(0\le i\le c\) that
$$\begin{aligned} r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) f&=q^{2(i-c)}\sum \limits _{a_0\in O_F/\varpi ^{c}O_F}\psi (u\varpi ^{-i}[(x_1-a_0)x_4-x_2(x_3-a_0\sqrt{D})]) \nonumber \\ {}&\quad \times char\left( \begin{pmatrix} a_0+\varpi ^{i}O_F &{} \quad \varpi ^{i-c}O_F \\ a_0\sqrt{D}+\varpi ^iO_F &{} \quad \varpi ^{i-c}O_F \end{pmatrix}\right) \times char(O_F^*). \end{aligned}$$
(5.45)
The sum is right \(K_1(\varpi ^c)\)-invariant for any i.
The local integral can be written as
$$\begin{aligned} {\mathbb {P}}(s,w,f,\Phi _s)= & {} \sum \limits _{0\le i\le c}A_i\int \limits _{v(\alpha )=\min \{0, 2i-c\}}W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) |\alpha |^{\frac{w}{2}-\frac{1}{4}}\Phi _s(\alpha )^{-1} \nonumber \\&\quad \times I(\alpha , r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) f,\Phi _s)d^*\alpha , \end{aligned}$$
(5.46)
where \(A_i\)’s were given in Lemma 2.2.
Recall that \(\Phi _s=\Phi _s^{(1)}\Phi _s^{(2)}\), and \(\gamma _0\) should be understood as \(\begin{pmatrix} 1 &{} \quad 0 \\ (\sqrt{D},-\sqrt{D}) &{} \quad 1 \end{pmatrix}\). \(\Phi _s^{(2)}\) is the unique right \(K_1(\varpi ^c)\)-invariant function supported on \(BK_1(\varpi ^c)\) such that \(\Phi _s^{(2)}(1)=1\), and \(\Phi _s^{(1)}\) is the standard right K-invariant function. Then
$$\begin{aligned}&I\left( \alpha ,r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) f,\Phi _s\right) \nonumber \\&\quad =A_c\int \Phi _s^{(1)}\left( \begin{pmatrix} 1 &{} \quad 0 \\ 2\sqrt{D} &{} \quad 1 \end{pmatrix}\begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad a_2 \end{pmatrix}\right) \Phi _s^{(2)}\left( \begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad a_2 \end{pmatrix}\right) \nonumber \\&\qquad \times r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) f\left( \begin{pmatrix} a_1 &{} \quad m \\ a_1\sqrt{D} &{} \quad a_2+m\sqrt{D} \end{pmatrix},\frac{\alpha }{a_1a_2}\right) dm\,d^*a_2|a_1|^{-1}\,d^*a_1,\nonumber \\ \end{aligned}$$
(5.47)
where
$$\begin{aligned}&r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) f\left( \begin{pmatrix} a_1 &{} \quad m \\ a_1\sqrt{D} &{} \quad a_2+m\sqrt{D} \end{pmatrix},\frac{\alpha }{a_1a_2}\right) \nonumber \\&=q^{2(i-c)}\sum \limits _{a_0\in O_F/\varpi ^{c}O_F}\psi \left( \varpi ^{-i}\alpha \left( 1-\frac{a_0}{a_1}\right) \right) char\left( \begin{pmatrix} a_0+\varpi ^{i}O_F &{} \quad \varpi ^{i-c}O_F \\ a_0\sqrt{D}+\varpi ^iO_F &{} \quad \varpi ^{i-c}O_F \end{pmatrix}\right) \nonumber \\&\quad \times \left( \begin{pmatrix} a_1 &{} \quad m \\ a_1\sqrt{D} &{} \quad a_2+m\sqrt{D} \end{pmatrix}\right) . \end{aligned}$$
(5.48)
For each \(a_0\), the corresponding term in the above expression is not zero if and only if
$$\begin{aligned} a_1\equiv a_0 \mod (\varpi ^iO_F), m,a_2\in \varpi ^{i-c}O_F. \end{aligned}$$
(5.49)

Lemma 5.11

For any fixed \(0\le i\le c\) and fixed \(v(\alpha )\), \(I\left( \alpha ,r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) f,\Phi _s\right) \) as a function of \(\alpha \) is a linear combination of the constant function independent of \(\alpha \) and \(\psi (\varpi ^{-i}\alpha )\).

Proof

For fixed \(a_0\), the corresponding term in \(I\left( \alpha ,r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) f,\Phi _s\right) \) is
https://static-content.springer.com/image/art%3A10.1007%2Fs40993-016-0061-7/MediaObjects/40993_2016_61_Equ127_HTML.gif
(5.50)
There are two cases. If \(a_0\in \varpi ^iO_F\), the domain for \(a_1\) is \(\varpi ^iO_F\). We fix \(a_2\) and m and integrate in \(a_1\) for fixed \(v(a_1)\) first. Note that \(\Phi _s^{(1)}\left( \begin{pmatrix} a_1 &{} \quad m \\ 2a_1\sqrt{D} &{} \quad a_2+2m\sqrt{D} \end{pmatrix}\right) \) only depends on \(v(a_1)\) instead of the specific value of \(a_1\), as one can see from Lemma 4.2. So we are essentially integrating
$$\begin{aligned} \psi \left( \varpi ^{-i}\alpha \left( 1-\frac{a_0}{a_1}\right) \right) =\psi (\varpi ^{-i}\alpha )\psi \left( -\varpi ^{-i}\alpha \frac{a_0}{a_1}\right) . \end{aligned}$$
Then we get either 0 or a multiple of \(\psi (\varpi ^{-i}\alpha )\).
If \(a_0\notin \varpi ^i O_F\), we consider the sum in \(a_0\) for fixed \(v(a_0)<i\). Note that \(v(a_1)=v(a_0)\) would also be fixed. As the value of \(\Phi _s\) and the domains for the integrals in m and \(a_2\) are actually independent of \(a_0\), we can change the order of the integral in \(a_2\), m and the summation in \(a_0\). Then the sum in \(a_0\) is essentially
$$\begin{aligned}&\sum \limits _{a_0}\int \limits _{a_1\equiv a_0\mod (\varpi ^i)}\psi \left( \varpi ^{-i}\alpha \frac{a_1-a_0}{a_1}\right) d^*a_1\nonumber \\&\quad =\int \limits _{a_1}\sum \limits _{a_0\equiv a_1\mod (\varpi ^i)}\psi \left( \varpi ^{-i}\alpha \frac{a_1-a_0}{a_1}\right) d^*a_1. \end{aligned}$$
(5.51)
One can now easily see that the inner sum is either 0 or a constant independent of \(\alpha \). \(\square \)

As a result of this Lemma and Lemma 5.10, we only have to care about the constant part when \(i=c,c-1\) and the \(\psi (\varpi ^{-i}\alpha )\) part when \(i=0,1\).

Lemma 5.12

Suppose that \(v(\alpha )=\min \{0,2i-c\}\).
  1. (1)

    \(I(\alpha ,f,\Phi _s)=A_c\). The constant part of \(I\left( \alpha ,r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^{c-1} &{} \quad 1 \end{pmatrix}\right) f,\Phi _s\right) \) is \(A_c\left[ q^{-1}+(1-q^{-1})\frac{\chi _{1,s}^{(1)}}{\chi _{2,s}^{(1)}}\right] \).

     
  2. (2)

    \(I\left( \alpha ,r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ 1 &{} \quad 1 \end{pmatrix}\right) f,\Phi _s\right) =0 \). The \(\psi (\varpi ^{-1}\alpha )\) part of \(I\left( \alpha ,r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ \varpi &{} \quad 1 \end{pmatrix}\right) f,\Phi _s\right) \) is 0.

     

We will however not provide proof here. Basically one can use (5.50) to do the calculations, and at some steps switch the order of the integral and the summation in \(a_0\) as in the proof of the previous lemma. It’s complicated, but not difficult.

Now we combine Lemmas 5.10 and 5.12 to compute (5.46). Note that only \(i=0,1\) terms are non-zero, and \(v(\alpha )=0\) for these terms. Then
$$\begin{aligned} {\mathbb {P}}(s,w,f,\Phi _s)&=A_c\int \limits _{v(\alpha )=0}W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) I(\alpha , f,\Phi _s)d^*\alpha \nonumber \\ {}&\quad +A_{c-1}\int \limits _{v(\alpha )=0}W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^{c-1} &{} \quad 1 \end{pmatrix}\right) \nonumber \\ {}&\quad \times I\left( \alpha , r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^{c-1} &{} \quad 1 \end{pmatrix}\right) f,\Phi _s\right) d^*\alpha \nonumber \\&=A_c\cdot A_c +A_{c-1}\left( -\frac{1}{q-1}\right) A_c\left[ q^{-1}+(1-q^{-1})\frac{\chi _{1,s}^{(1)}}{\chi _{2,s}^{(1)}}\right] \nonumber \\&=\frac{1-q^{-1}}{(q+1)^2q^{2c-2}}\left( 1-\frac{\chi _{1}^{(1)}}{\chi _{2}^{(1)}}q^{-(2s+1)}\right) . \end{aligned}$$
(5.52)
Note that this result is independent of w.

5.3.2 Highly ramified principal series

Now we consider the case when \(\pi \simeq \pi (\mu _1,\mu _2)\) is highly ramified. In the case when \(\pi \) is a highly ramified special representation, we will get the same result as we can choose the same new form.

We still assume that \(\chi _1\) and \(\chi _2\) are unramified and \({\mathbb {E}}/{\mathbb {F}}\) is split. This implies that \(\mu _1\) and \(\mu _2\) should be ramified of the same level k. Let \(c=2k\ge 2\) be the level of \(\pi \) and \(\hat{\pi }\). It’s a classical result (refer to [5]) that the \(K_1(\varpi ^c)\)-invariant new form \(\varphi \) is supported on \(B\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^{k} &{} \quad 1 \end{pmatrix}K_1(\varpi ^c)\).

The idea is to compare the Whittaker function for \(\varphi \) with Lemma 5.10. If we can get similar properties, then we can choose the same f and \(\Phi _s\) as in the previous situation and get the results directly.

Lemma 5.13

  1. (1)
    If \(k<i\le c\), then \(W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) \) is not zero only when \(v(\alpha )=0\). In that case, its integral against \(\psi (\varpi ^{-i}\alpha )\) is always 0.
    $$\begin{aligned} \int \limits _{v(\alpha )=0}W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) d^*\alpha = {\left\{ \begin{array}{ll} 1, &{}\text { if }i=c;\\ -\frac{1}{q-1},&{}\text { if }i=c-1>k;\\ 0,&{}\text { otherwise }. \end{array}\right. } \end{aligned}$$
    (5.53)
     
  2. (2)
    If \(i<k\), then \(W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) \) is not zero only when \(v(\alpha )=2i-c\). In that case, its integral against 1 is always 0.
    $$\begin{aligned}&\int \limits _{v(\alpha )=2i-c}W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) \psi (\varpi ^{-i}\alpha )d^*\alpha \nonumber \\&\quad = w_{\pi }(\varpi ^{k-i})\mu _1^{-1}(-1) {\left\{ \begin{array}{ll} 1, &{}\text { if }i=0;\\ -\frac{1}{q-1}, &{}\text { if }i=1<k;\\ 0, &{}\text { otherwise }. \end{array}\right. } \end{aligned}$$
    (5.54)
     
  3. (3)

    If \(i=k\), the integral of \(W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^k &{} \quad 1 \end{pmatrix}\right) \) against 1 or \(\psi (\varpi ^{-k}\alpha )\) is always zero if either \(k>1\) or \(v(\alpha )\ne 0\). When \(k=1\) and \(v(\alpha )=0\), its integral against 1 is the same as expected from (1) as the limit case, and its integral against \(\psi (\varpi ^{-k}\alpha )\) is the same as expected from (2).

     

Proof

According to part (3) of Lemma 5.6, we can always write
$$\begin{aligned} W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right)= & {} \int \mu _1^{-1}\left( -\frac{\alpha \varpi ^k}{\alpha +m\varpi ^i}\right) \mu _2^{-1}(-m) \nonumber \\&\times \left| \frac{\alpha \varpi ^k}{m(\alpha +m\varpi ^i)}\right| ^{1/2}\psi (m)dm. \end{aligned}$$
(5.55)
The difference is the domain for m, which was given in Lemma 5.6. For the sake of conciseness, we will only prove part (1) here.
When \(k<i\le c\), the domain for m is \(v(m)=v(\alpha )-k\). Write \(m=\varpi ^{-k}\alpha u\) for \(u\in O_F^*\). The integral becomes
$$\begin{aligned} W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right)= & {} \int \limits _{u\in O_F^*} \mu _1^{-1}\left( -\frac{\varpi ^k}{1+u\varpi ^{i-k}}\right) \mu _2^{-1}(-\varpi ^{-k}\alpha u) \nonumber \\&\quad \times q^{-v(\alpha )/2}\psi (\varpi ^{-k}\alpha u)du. \end{aligned}$$
(5.56)
As functions in u, \(\mu _1^{-1}(-\frac{\varpi ^k}{1+u\varpi ^{i-k}})\) is at most of level \(2k-i<k\), \(\mu _2^{-1}(-\varpi ^{-k}\alpha u)\) is multiplicative of level k and \(\psi (-\varpi ^{-k}\alpha u)\) is additive of level \(k-v(\alpha )\). So if \(v(\alpha )\ne 0\), the integral will be zero for level reason.
When \(v(\alpha )=0\) and \(i=c\), \(\mu _1^{-1}(-\frac{\varpi ^k}{1+u\varpi ^{i-k}})=\mu _1^{-1}(-\varpi ^k)\) as \(\mu _1\) is level k. Then
$$\begin{aligned} W^-_\varphi \left( \begin{pmatrix} 1 &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right)&=\mu _1^{-1}(-\varpi ^k)\int \limits _{u\in O_F^*} \mu _2^{-1}(-\varpi ^{-k}\alpha u)\psi (\varpi ^{-k}\alpha u)du \nonumber \\&=\mu _1^{-1}(\varpi ^k)\int \limits _{u\in O_F^*} \mu _2^{-1}(\varpi ^{-k} u)\psi (\varpi ^{-k} u)d^*u\cdot (1-q^{-1}). \end{aligned}$$
(5.57)
We have used \(\mu _1\mu _2(-1)=1\) here.
We first consider the integral
$$\begin{aligned} \int \limits _{v(\alpha )=0}W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) d^*\alpha . \end{aligned}$$
One can switch the order and integrate in \(\alpha \) first. Note that
$$\begin{aligned} \int \limits _{v(\alpha )}\psi (\varpi ^{-k}\alpha u))\mu _2^{-1}(-\varpi ^{-k}\alpha u) d^*\alpha =\int \limits _{v(\alpha )}\psi (\varpi ^{-k}\alpha ))\mu _2^{-1}(-\varpi ^{-k}\alpha ) d^*\alpha \end{aligned}$$
(5.58)
is independent of u. \(q^{-v(\alpha )}=1\). So
$$\begin{aligned}&\int \limits _{v(\alpha )=0}W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) d^*\alpha \nonumber \\&\quad =\int \limits _{u\in O_F^*} \mu _1^{-1}\left( -\frac{\varpi ^k}{1+u\varpi ^{i-k}}\right) du \int \limits _{v(\alpha )}\psi (\varpi ^{-k}\alpha ))\mu _2^{-1}(-\varpi ^{-k}\alpha ) d^*\alpha \nonumber \\&\quad =\mu _1^{-1}(\varpi ^k) \int \limits _{v(\alpha )}\psi (\varpi ^{-k}\alpha ))\mu _2^{-1}(\varpi ^{-k}\alpha ) d^*\alpha \cdot {\left\{ \begin{array}{ll} 1-q^{-1}, &{}\text { if }i=c;\\ -q^{-1},&{}\text { if }i=c-1>k;\\ 0,&{}\text { otherwise}. \end{array}\right. } \end{aligned}$$
(5.59)
In the last equation we have used Lemma 2.4.
Now we integrate \(W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) \) against \(\psi (\varpi ^{-i}\alpha )\) when \(v(\alpha )=0\) and \(k<i\le c\) . Again we shall switch the order of the integrals and get
$$\begin{aligned}&\int \limits _{v(\alpha )=0}W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) \psi (\varpi ^{-i}\alpha )d^*\alpha \nonumber \\&\quad =\int \limits _{u\in O_F^*} \mu _1^{-1}\left( -\frac{\varpi ^k}{1+u\varpi ^{i-k}}\right) \int \limits _{v(\alpha )=0}\mu _2^{-1}(-\varpi ^{-k}\alpha u)\psi (\varpi ^{-k}\alpha u)\psi (\varpi ^{-i}\alpha )d^*\alpha du.\nonumber \\ \end{aligned}$$
(5.60)
As functions in \(\alpha \), \(\mu _2^{-1}(-\varpi ^{-k}\alpha u)\) is of level k, and \(\psi (\varpi ^{-i}\alpha (1+u\varpi ^{i-k}))\) is of level \(i>k\). Thus the integral in \(\alpha \) would be zero.

Lastly when we normalize \(W^-_\varphi \left( \begin{pmatrix} 1 &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) \) to be 1, we will get the formulae as claimed. \(\square \)

5.3.3 Normalization

If we use \(f'\) to do Theta lifting, we will get Gross–Prasad test vectors for both \(F_1\) and F. Let
$$\begin{aligned} f''=char\left( \begin{pmatrix} O_F &{} \quad O_F \\ \varpi ^cO_F &{} \quad O_F \end{pmatrix}\right) \times char(O_F^*) \end{aligned}$$
We can compute \(\langle F_1,F \rangle \) using \(f''\) just as in (5.31):
$$\begin{aligned} \langle F_1,F \rangle= & {} \sum \limits _{0\le i\le c}A_i\int \int W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) \nonumber \\&\times \, f''(\alpha x, \alpha ^{-1}x^{-2})w_{\hat{\pi }}(x)d^*x\,d^*\alpha \end{aligned}$$
(5.61)
Only \(i=c\) or \(c-1\) has non-zero contribution because of Lemmas 5.10, 5.13 and the fact that \(f''\) is invariant under the action of center. Then one can easily compute that
$$\begin{aligned} \langle F_1,F \rangle =A_c+A_{c-1}\left( -\frac{1}{q-1}\right) q^{-1}=\frac{q-1}{q+1}\frac{1}{q^c}. \end{aligned}$$
(5.62)
The expected L-factors in this case is
$$\begin{aligned}&\frac{\zeta (2)}{L(\pi ,Ad,1)}\frac{L (\Pi \otimes \Omega ,1/2)L (\pi \otimes \chi _{1}|_{{\mathbb {F}}^*},2s+1/2)}{L (\eta ,1)L (\chi ,2s+1)}\nonumber \\&\quad =\frac{1}{(1-q^{-2})L(\pi ,Ad,1)}(1-q^{-1})(1-\chi ^{(1)}q^{-(2s+1)})(1-\chi ^{(2)}q^{-(2s+1)}). \end{aligned}$$
(5.63)
Then
$$\begin{aligned} {\mathbb {P}}^0(s,1/2,f',\Phi _s')=\frac{1}{q^c}\frac{L(\pi , Ad,1)}{1-\chi ^{(2)}q^{-(2s+1)}}. \end{aligned}$$
(5.64)

5.4 Ramification in \(\Phi _s\)

In this subsection, we consider the case when \(\Phi _s\) is ramified while \(\pi \) is unramified. To make things simple, we assume that \(\chi _{1}\) is ramified of level c and \(\chi _{2}\) is unramified. Note \(\mu _1\mu _2(\chi _1\chi _2)|_{{\mathbb {F}}^*}=1\) implies that \(\chi _{1}|_{{\mathbb {F}}^*}\) is still unramified.

Proposition 5.14

Suppose that \(\pi \) is unramified and \({\mathbb {E}}/{\mathbb {F}}\) is inert. Suppose that \(\chi _{1}\) is ramified of level c, but \(\chi _{1}|_{{\mathbb {F}}^*}\) and \(\chi _2\) are unramified. Pick
$$\begin{aligned} f=char\left( \begin{pmatrix} O_F &{} \quad \varpi ^cO_F \\ \varpi ^{-c}O_F &{} \quad O_F \end{pmatrix}\right) \times char(O_F^*). \end{aligned}$$
Pick \(\Phi _s\) to be the new form, that is, the \(K_1(\varpi ^c)\)-invariant function supported on \(B\begin{pmatrix} 1 &{} \quad 0 \\ 1 &{} \quad 1 \end{pmatrix}K_1(\varpi ^c)\). Then
$$\begin{aligned} {\mathbb {P}}(s,w,f,\Phi _s)=\frac{{\mathbb {P}}'(w)}{(1-\delta \mu _1\chi _{1,s}(\varpi )q)(1-\delta \mu _2\chi _{1,s}(\varpi )q)}, \end{aligned}$$
(5.65)
where \({\mathbb {P}}'(w)\) denotes the expression
$$\begin{aligned}&\frac{\left( \frac{\chi _{2,s}}{q\chi _{1,s}}\right) ^{-c}}{\mu _2-\mu _1}\left[ \mu _2\frac{(1-(\delta \mu _2\chi _{2,s})^{c+1})-\frac{\chi _{2,s}}{q^2\chi _{1,s}}(1-(\delta \mu _2\chi _{2,s})^c)}{1-\delta \mu _2\chi _{2,s}}(1-q\delta \mu _1\chi _{1,s})\right. \nonumber \\&\quad \left. -\mu _1\frac{(1-(\delta \mu _1\chi _{2,s})^{c+1})-\frac{\chi _{2,s}}{q^2\chi _{1,s}}(1-(\delta \mu _1\chi _{2,s})^c)}{1-\delta \mu _1\chi _{2,s}}(1-q\delta \mu _2\chi _{1,s}) \right] . \end{aligned}$$
(5.66)
Here \(\delta =q^{-(w/2+1/4)}\). When \(w=1/2\),
$$\begin{aligned} {\mathbb {P}}^0(s,1/2,f,\Phi _s)=\frac{{\mathbb {P}}'(1/2)}{1+q^{-1}}. \end{aligned}$$
(5.67)
Recall the local integral
$$\begin{aligned} {\mathbb {P}}(s,w,f,\Phi _s)=\int \limits _{ZN\backslash {\text {GL}}_{2}({\mathbb {F}})}W^-_\varphi (h)\Delta (h)^{w-1/2}\int \limits _{{\text {GL}}_{2}({\mathbb {F}})}r'(h)f(g,\det (g)^{-1})\Phi _{s}(\gamma _0 g) dg\,dh.\nonumber \\ \end{aligned}$$
(5.68)
We’ve already known well the K-invariant Whittaker function \(W^-_\varphi \). The choice
$$\begin{aligned} f=char\left( \begin{pmatrix} O_F &{} \quad \varpi ^cO_F \\ \varpi ^{-c}O_F &{} \quad O_F \end{pmatrix}\right) \times char(O_F^*) \end{aligned}$$
(5.69)
is motivated by Example 2.21. This Schwartz function is K-invariant under the Weil representation. Thus as in the unramified case,
$$\begin{aligned} {\mathbb {P}}(s,w,f,\Phi _s)=\int \limits _{{\mathbb {F}}^{*}}W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) |\alpha |^{w/2-1/4} \Phi _{s}(\alpha )^{-1} I(\alpha ,f,\Phi _s)d^*\alpha . \end{aligned}$$
(5.70)
The key point in this subsection is to work out
$$\begin{aligned} I(\alpha , f, \Phi _s)=\int \limits _{g\in {\text {GL}}_2,v(\det g)=v(\alpha )}f\left( g,\frac{\alpha }{\det g}\right) \Phi _s(\gamma _0g)dg. \end{aligned}$$
(5.71)
Note that when the extension \({\mathbb {E}}/{\mathbb {F}}\) is inert,
$$\begin{aligned} {\text {GL}}_2=O_E^*\cdot B. \end{aligned}$$
(5.72)
Recall that \(\Phi _s(\gamma _0 t g)=\chi _{1,s}(\bar{t})\chi _{2,s}(t)\Phi _s(\gamma _0g)\) for \(t=a+b\sqrt{D}\). By our assumption on \(\chi _{1,s}\) and \(\chi _{2,s}\), \(\Phi _s(\gamma _0g)\) as a function of g is left invariant under \(O_F^*+\varpi ^cO_E\). f is also left invariant under \(\left\{ \begin{pmatrix} a &{} \quad b \\ bD &{} \quad a \end{pmatrix}|a\in O_F^*, b\in \varpi ^cO_F\right\} \simeq O_F^*+\varpi ^c O_E\). As a result,
https://static-content.springer.com/image/art%3A10.1007%2Fs40993-016-0061-7/MediaObjects/40993_2016_61_Equ150_HTML.gif
(5.73)
We have used that the right Haar measure for the Borel subgroup is
$$\begin{aligned} |a_2|^{-1}dmd^*a_1d^*a_2. \end{aligned}$$
The coset representatives \((O_F^*+\varpi ^c O_E)\backslash O_E^*\) can be chosen as
$$\begin{aligned} \{1+b_1\sqrt{D}|b_1\in O_F/\varpi ^cO_F \}\cup \{b_2+\sqrt{D}|b_2\in \varpi O_F/\varpi ^c O_F \}. \end{aligned}$$
One can easily see that this set has \((q+1)q^{c-1}\) elements. That’s why we have \(\frac{1}{(q+1)q^{c-1}}\) in front of the integral above.

Lemma 5.15

We need \(v(\frac{a_1\sqrt{D}}{a_2+m\sqrt{D}})\le 0\) for
$$\begin{aligned} \begin{pmatrix} a_1 &{} \quad m \\ a_1\sqrt{D} &{} \quad a_2+m\sqrt{D} \end{pmatrix}\in B\begin{pmatrix} 1 &{} \quad 0 \\ 1 &{} \quad 1 \end{pmatrix}K_1(\varpi ^c). \end{aligned}$$
Under the above condition we can write \(\begin{pmatrix} a_1 &{} \quad m \\ a_1\sqrt{D} &{} \quad a_2+m\sqrt{D} \end{pmatrix}\) as
$$\begin{aligned} \begin{pmatrix} \frac{a_2}{\sqrt{D}} &{} \quad a_1-\frac{a_2}{\sqrt{D}} \\ 0 &{} \quad a_1\sqrt{D} \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ 1 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad -1+\frac{a_2+m\sqrt{D}}{a_1\sqrt{D}} \\ 0 &{} \quad 1 \end{pmatrix}. \end{aligned}$$

Corollary 5.16

Assume that \(\chi _{2,s}\) is unramified and \(\chi _{1,s}\) is ramified of level c, such that \(\chi _{1,s}|_{{\mathbb {F}}^*}\) is still unramified. Suppose that \(\Phi _s\) is the unique \(K_1(\varpi ^c)\)-invariant function supported on \(B\begin{pmatrix} 1 &{} \quad 0 \\ 1 &{} \quad 1 \end{pmatrix}K_1(\varpi ^c)\). Then \(\Phi _s\left( \begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix}\begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad a_2 \end{pmatrix}\right) \) is non-zero when \(v(a_1)\le v(a_2+m\sqrt{D})\). In that case, it’s equal to
$$\begin{aligned} \chi _{1,s}\left( \frac{a_2}{\sqrt{D}}\right) \chi _{2,s}\left( a_1\sqrt{D}\right) =\frac{\chi _{1,s}^{v(a_2)} \chi _{2,s}^{v(a_1)}}{\chi _{1,s}(\sqrt{D})}. \end{aligned}$$
For each representative \(t\in (O_F^*+\varpi ^c O_E)\backslash O_E^*\), we decide now the domain of the integral which is given by the condition
$$\begin{aligned} t\begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad a_2 \end{pmatrix}\in \begin{pmatrix} O_F &{} \quad \varpi ^cO_F \\ \varpi ^{-c}O_F &{} \quad O_F \end{pmatrix}. \end{aligned}$$
If \(t=1+b_1\sqrt{D}\), then by
$$\begin{aligned} \begin{pmatrix} 1 &{} \quad b_1 \\ b_1D &{} \quad 1 \end{pmatrix}\begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad a_2 \end{pmatrix} \in \begin{pmatrix} O_F &{} \quad \varpi ^cO_F \\ \varpi ^{-c}O_F &{} \quad O_F \end{pmatrix} \end{aligned}$$
we get \(v(a_1),v(a_2)\ge 0\), \(m\equiv -b_1a_2\mod (\varpi ^c)\). Similarly for \(t=\sqrt{D}+b_2\),
$$\begin{aligned} \begin{pmatrix} b_2 &{} \quad 1 \\ D &{} \quad b_2 \end{pmatrix}\begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad a_2 \end{pmatrix} \in \begin{pmatrix} O_F &{} \quad \varpi ^cO_F \\ \varpi ^{-c}O_F &{} \quad O_F \end{pmatrix}. \end{aligned}$$
The domain will be \(v(a_1)\ge -v(b_2),v(m)\ge 0, a_2\equiv -mb_2\mod (\varpi ^c)\).

The key observation here is that although the domain depends on the specific choice of \(b_1\) or \(b_2\), the integral of \(\Phi _s\) over the domain only depends on \(v(b_1)\) and \(v(b_2)\). Indeed in Corollary 5.16, the requirement that \(v(a_1)\le v(a_2+m\sqrt{D})\) and the value of \(\Phi _s\) both depend only on the valuations of \(a_1,a_2\) and m. The domains differ slightly but the different parts have the same volume.

More specifically for fixed \(v(b_1)\) or \(v(b_2)\), let \(t_0\) be a fixed representative for \(1+b_1\sqrt{D}\) or \(b_2+\sqrt{D}\). Then we have
$$\begin{aligned} \int \limits _{B}\chi _{1,s}(\bar{t})\Phi _s(\gamma _0b)f\left( tb,\frac{\alpha }{\det tb}\right) db=\chi _{1,s}(\bar{t})\int \limits _{B}\Phi _s(\gamma _0b)f\left( t_0b,\frac{\alpha }{\det t_0b}\right) db . \end{aligned}$$
So when we sum over t for fixed \(v(b_1)\) or \(v(b_2)\), we are essentially just summing \(\chi _{1,s}(\bar{t})\). Then we need a lemma similar to Lemma 2.4:

Lemma 5.17

Let \(\chi \) be a character of level c on \({\mathbb {E}}^*\) which is unramified when restricted to \({\mathbb {F}}^*\).
  1. (1)
    If \(c\ge 2\), we have
    $$\begin{aligned} \sum \limits _{b_1 \in O_F/\varpi ^cO_F, v(b_1)=i}\chi (1+b_1\sqrt{D})= {\left\{ \begin{array}{ll} 1, \text { if }i=c;\\ -1, \text { if }i=c-1;\\ 0, \text { otherwise}. \end{array}\right. } \end{aligned}$$
    (5.74)
    $$\begin{aligned} \sum \limits _{b_2 \in O_F/\varpi ^cO_F, v(b_2)=i}\chi (\sqrt{D}+b_2)=\chi (\sqrt{D}) {\left\{ \begin{array}{ll} 1, \text { if }i=c;\\ -1, \text { if }i=c-1;\\ 0, \text { otherwise}. \end{array}\right. } \end{aligned}$$
    (5.75)
     
  2. (2)
    If \(c=1\), we have
    $$\begin{aligned} \sum \limits _{b_1\in O_F/\varpi O_F}\chi (1+b_1\sqrt{D})+\chi (\sqrt{D})=0. \end{aligned}$$
    (5.76)
     
This lemma enable us to greatly simplify (5.73) as
$$\begin{aligned} I(\alpha , f, \Phi _s)&=\frac{1}{(q+1)q^{c-1}}\left[ \left( \int \limits _{B} \Phi _s(\gamma _0g)f\left( g,\frac{\alpha }{\det g}\right) dg \right. \right. \nonumber \\&\quad \left. -\int \limits _{B}\Phi _s(\gamma _0g)f\left( \begin{pmatrix} 1 &{} \quad \varpi ^{c-1} \\ \varpi ^{c-1}D &{} \quad 1 \end{pmatrix}g,\frac{\alpha }{\det g}\right) dg\right) \nonumber \\&\quad + \chi _{1,s}(\sqrt{D})\left( \int \limits _{B} \Phi _s(\gamma _0g)f(\begin{pmatrix} 0 &{} \quad 1 \\ D &{} \quad 0 \end{pmatrix}g,\frac{\alpha }{\det g}\right) dg \nonumber \\&\quad \left. \left. -\int \limits _{B} \Phi _s(\gamma _0g)f\left( \begin{pmatrix} \varpi ^{c-1} &{} \quad 1 \\ D &{} \quad \varpi ^{c-1} \end{pmatrix}g,\frac{\alpha }{\det g}\right) dg\right) \right] . \end{aligned}$$
(5.77)
Here we have chosen \(\begin{pmatrix} 1 &{} \quad \varpi ^{c-1} \\ \varpi ^{c-1}D &{} \quad 1 \end{pmatrix}\) as a representative for \(1+b_1\sqrt{D}\) with \(v(b_1)=c-1\), and \(\begin{pmatrix} \varpi ^{c-1} &{} \quad 1 \\ D &{} \quad \varpi ^{c-1} \end{pmatrix}\) as a representative for \(\sqrt{D}+b_2\) with \(v(b_2)=c-1\). This formula is true even if \(c=1\).

To integrate (5.77), it’s easier to compare the domains of the integrals, as the common part can be cancelled when we do subtraction. We will only give the result here:

Lemma 5.18

$$\begin{aligned} I(\alpha ,f,\Phi _s)= & {} \frac{1}{(q+1)q^{c-1}}\\&\times {\left\{ \begin{array}{ll} (q\chi _{1,s})^{v(\alpha )}((\frac{\chi _{2,s}}{q\chi _{1,s}})^{-c}-(\frac{\chi _{2,s}}{q\chi _{1,s}})^{v(\alpha )-c+1})\frac{1-\frac{\chi _{2,s}}{q^2\chi _{1,s}}}{1-\frac{\chi _{2,s}}{q\chi _{1,s}}},&{}\text { if }0\le v(\alpha )\le c-1;\\ (q\chi _{1,s})^{v(\alpha )}(\frac{\chi _{2,s}}{q\chi _{1,s}})^{-c}\frac{1-\frac{\chi _{2,s}}{q^2\chi _{1,s}}}{1-\frac{\chi _{2,s}}{q\chi _{1,s}}}-(1-q^{-1})(q\chi _{1,s})^{v(\alpha )}\frac{\frac{\chi _{2,s}}{q\chi _{1,s}}}{1-\frac{\chi _{2,s}}{q\chi _{1,s}}},&{}\text { if }v(\alpha )\ge c. \end{array}\right. } \end{aligned}$$
From this one can get
$$\begin{aligned} {\mathbb {P}}(s,w,f,\Phi _s) =\frac{{\mathbb {P}}'(w)}{(1-\delta \mu _1\chi _{1,s}(\varpi )q)(1-\delta \mu _2\chi _{1,s}(\varpi )q)}, \end{aligned}$$
(5.78)
where \({\mathbb {P}}'\) denotes the expression
$$\begin{aligned}&\frac{\left( \frac{\chi _{2,s}}{q\chi _{1,s}}\right) ^{-c}}{\mu _2-\mu _1}\left[ \mu _2\frac{(1-(\delta \mu _2\chi _{2,s})^{c+1})-\frac{\chi _{2,s}}{q^2\chi _{1,s}}(1-(\delta \mu _2\chi _{2,s})^c)}{1-\delta \mu _2\chi _{2,s}}(1-q\delta \mu _1\chi _{1,s})\right. \nonumber \\&\quad \quad \left. -\mu _1\frac{(1-(\delta \mu _1\chi _{2,s})^{c+1})-\frac{\chi _{2,s}}{q^2\chi _{1,s}}(1-(\delta \mu _1\chi _{2,s})^c)}{1-\delta \mu _1\chi _{2,s}}(1-q\delta \mu _2\chi _{1,s}) \right] . \end{aligned}$$
(5.79)
Now we compute \( \langle F_1,F \rangle \). Since the Schwartz function is K-invariant under the Weil representation, we have
$$\begin{aligned} \langle F_1,F \rangle =\int \int W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) f(\alpha x, \alpha ^{-1}x^{-2})(\mu _1\mu _2)^{-1}(x)d^*xd^*\alpha . \end{aligned}$$
(5.80)
This turns out to be the same as in the unramified case, that is,
$$\begin{aligned} \langle F_1,F \rangle =\frac{L(\pi ,Ad,1)}{\zeta (2)}. \end{aligned}$$
(5.81)
The expected L-factors in this case is
$$\begin{aligned}&\frac{\zeta (2)}{L(\pi ,Ad,1)}\frac{L (\Pi \otimes \Omega ,1/2)L (\pi \otimes \chi _{1}|_{{\mathbb {F}}^*},2s+1/2)}{L (\eta ,1)L (\chi ,2s+1)}\nonumber \\&\quad =\frac{\zeta (2)}{L(\pi ,Ad,1)}\frac{1+q^{-1}}{(1-\mu _1\chi _{1}(\varpi )q^{-(2s+1/2)})(1-\mu _2\chi _{1}(\varpi )q^{-(2s+1/2)})}. \end{aligned}$$
(5.82)
Thus
$$\begin{aligned} {\mathbb {P}}^0(s,1/2,f,\Phi _s)=\frac{{\mathbb {P}}'(1/2)}{1+q^{-1}}. \end{aligned}$$
(5.83)

5.5 Joint ramification

In this section we consider a very special case of the local integral when both \(\pi (\mu _1,\mu _2)\) and \(\Phi _s\) are ramified.

Proposition 5.19

Suppose that \(\mu _1\) is unramified and \(\mu _2\) is of level \(c>0\) for the principal series \(\pi (\mu _1,\mu _2)\). Suppose that \(\chi _{2}\) is unramified and \(\chi _{1}\), \(\chi _{1}|_{{\mathbb {F}}^*}\) are both ramified of level c. Assume that \({\mathbb {E}}/{\mathbb {F}}\) is inert. Pick
$$\begin{aligned} f= & {} char\left( \begin{pmatrix} 1+\varpi ^c O_F &{} \quad O_F \\ \varpi ^c O_F &{} \quad O_F \end{pmatrix}\right) \times char(1+\varpi ^c O_F),\\ f'= & {} {\sum \limits _{a\in (O_F/\varpi ^cO_F)^*}}'char\left( \begin{pmatrix} a^{-1}+\varpi ^c O_F &{} \quad O_F \\ \varpi ^c O_F &{} \quad O_F \end{pmatrix}\right) \times char(a+\varpi ^c O_F) \nonumber \\= & {} {\sum \limits _{a\in (O_F/\varpi ^cO_F)^*}}'r'\left( \begin{pmatrix} a &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) f, \end{aligned}$$
where \(\sum '\) means the average. Pick \(\Phi _s\) to be the unique up to constant \(K_1(\varpi ^c)\)-invariant function supported on \(B\begin{pmatrix} 1 &{} \quad 0 \\ 1 &{} \quad 1 \end{pmatrix} K_1(\varpi ^c)\) such that \(\Phi _s\left( \left( \begin{array}{ll}1&{}0\\ 1&{}1\end{array}\right) \right) =1\). Then
$$\begin{aligned} {\mathbb {P}}(s,w,f,\Phi _s)={\mathbb {P}}(s,w,f',\Phi _s)=\frac{1}{(q^2 - 1)^2q^{4c-4}\chi _{1,s}(\sqrt{D})}\frac{1}{1-q\delta \mu _2\chi _{1,s}}. \end{aligned}$$
(5.84)
Here \(\delta =q^{-(w/2+1/4)}\). When \(w=1/2\), we have
$$\begin{aligned} {\mathbb {P}}(s,1/2,f',\Phi _s)=\frac{1}{(q^2 - 1)^2q^{4c-4}\chi _{1}(\sqrt{D})}\frac{1}{1-\mu _2\chi _{1}q^{-(2s+1/2)}}. \end{aligned}$$
(5.85)
The denominator is as expected since
$$\begin{aligned} L_v (\Pi \otimes \Omega ,1/2)L _v(\pi \otimes \chi _{1}|_{{\mathbb {F}}^*},2s+1/2)=\frac{1}{1-\mu _2\chi _{1}q^{-(2s+1/2)}} \end{aligned}$$
in this case, and
$$\begin{aligned} {\mathbb {P}}^0(s,1/2,f',\Phi _s)=\frac{1}{(q+1){q^{c-1}}\chi _{1}(\sqrt{D})}. \end{aligned}$$
(5.86)

First note that our choice for f is only \(K_1^1(\varpi ^c)\)-invariant under the Weil representation, where \(K_1^1(\varpi ^c)\) is the subgroup of K whose elements are congruent to \(\begin{pmatrix} 1 &{} \quad * \\ 0 &{} \quad 1 \end{pmatrix} \mod (\varpi ^c)\). On the other hand, \(f'\) is \(K_1(\varpi ^c)\)-invariant under left, right, and Weil representation. For the following we will only compute \({\mathbb {P}}(s,w,f,\Phi _s)\). But all pieces in \(f'\) can be computed very similarly, and the result doesn’t depend on a. Thus an average will give the same result.

Lemma 5.20

$$\begin{aligned} {\text {GL}}_2=\coprod \limits _{0\le i\le c, \beta \in (O_F/\varpi ^{\min \{i,c-i\}} O_F)^*}B\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\begin{pmatrix} \beta &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}K_1^1(\varpi ^c). \end{aligned}$$
(5.87)

Proof

First of all,
$$\begin{aligned} K_1(\varpi ^c)=\coprod \limits _{\beta \in (O_F/\varpi ^c O_F)^*}\begin{pmatrix} \beta &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}K_1^1(\varpi ^c). \end{aligned}$$
We know by Lemma (2.1)
$$\begin{aligned} {\text {GL}}_2=\coprod \limits _{0\le i\le c}B\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}K_1(\varpi ^c). \end{aligned}$$
We just need to check when \(B\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\begin{pmatrix} \beta &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}K_1^1(\varpi ^c)=B\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\begin{pmatrix} \beta ' &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}K_1^1(\varpi ^c)\).
This is equvalent to that when modulo \(\varpi ^c\),
$$\begin{aligned} \begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\begin{pmatrix} \beta /\beta ' &{} \quad * \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ -\varpi ^i &{} \quad 1 \end{pmatrix} \end{aligned}$$
is upper triangular. That is
$$\begin{aligned} \varpi ^i\beta /\beta '-\varpi ^i-\varpi ^{2i}*\equiv 0 \mod (\varpi ^c). \end{aligned}$$
Then the conclusion is clear. \(\square \)
We pick \(\varphi \in \pi (\mu _1^{-1},\mu _2^{-1})\) to be the unique \(K_1(\varpi ^c)\)-invariant function supported on \(BK_1(\varpi ^c)\). Then by the above lemma, the local integral \({\mathbb {P}}(s,w,f,\Phi _s)\) should be
$$\begin{aligned}&\sum \limits _{0\le i\le c,\beta }A_{i,\beta }\int W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) |\alpha |^{\frac{w}{2}-\frac{1}{4}}\Phi _s(\alpha )^{-1}\nonumber \\&\quad \times I\left( \alpha , r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\begin{pmatrix} \beta &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) f,\Phi _s\right) d^*\alpha , \end{aligned}$$
(5.88)
where
$$\begin{aligned} A_{i,\beta }=\frac{A_i}{\sharp (O_F/\varpi ^{\min \{i,c-i\}} O_F)^*}. \end{aligned}$$
We shall work out the integral \(I\left( \alpha , r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\begin{pmatrix} \beta &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) f,\Phi _s\right) \) first. By (2.13),
$$\begin{aligned} r'\left( \begin{pmatrix} \beta &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) f=char\left( \begin{pmatrix} \beta ^{-1}+\varpi ^c O_F &{} \quad O_F \\ \varpi ^c O_F &{} \quad O_F \end{pmatrix}\right) \times char(\beta +\varpi ^c O_F). \end{aligned}$$
(5.89)
Then by Lemma 2.10 and the remark after it,
$$\begin{aligned} r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\begin{pmatrix} \beta &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) f= & {} q^{2(i-c)}\psi (u\varpi ^{-i}[(x_1-\beta ^{-1})x_4-x_2x_3]) \nonumber \\&\quad \times char\left( \begin{pmatrix} \beta ^{-1}+\varpi ^i O_F &{} \quad \varpi ^{i-c}O_F \\ \varpi ^i O_F &{} \quad \varpi ^{i-c}O_F \end{pmatrix}\right) \nonumber \\&\quad \times char(\beta +\varpi ^c O_F). \end{aligned}$$
(5.90)

Lemma 5.21

$$\begin{aligned} \int \Phi _s(\gamma _0g)f(g,\frac{\alpha }{\det g})dg= {\left\{ \begin{array}{ll} \frac{1}{(q-1)(q^2-1)q^{3c-3}}\chi _{1,s}(\frac{\alpha }{\sqrt{D}})q^{v(\alpha )}, &{}\text { if }v(\alpha )\ge 0,\\ 0,&{}\text { otherwise.} \end{array}\right. } \end{aligned}$$
(5.91)
$$\begin{aligned} \int \Phi _s(\gamma _0g)r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\begin{pmatrix} \beta &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) f\left( g,\frac{\alpha }{\det g}\right) dg\equiv 0 \quad \text {for any } i<c \;{and}\; \beta .\nonumber \\ \end{aligned}$$
(5.92)

Proof

Recall that we can write \({\text {GL}}_2=O_E^*B\). \(\Phi _s(\gamma _0g)\) and \(r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\begin{pmatrix} \beta &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) f\) are both left invariant under \(1+\varpi ^c O_E\). Note that \(O_E^*/1+\varpi ^cO_E\simeq (O_E/\varpi ^cO_E)^*\) is of cardinality \((q^2-1)q^{2c-2}\). Then
$$\begin{aligned}&I\left( \alpha , r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\begin{pmatrix} \beta &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) f,\Phi _s\right) \nonumber \\&\quad =\frac{1}{(q^2-1)q^{2c-2}} \sum \limits _{t\in (O_E/\varpi ^cO_E)^*}\int \Omega (t)\Phi _s\left( \gamma _0 \begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad a_2 \end{pmatrix}\right) r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\begin{pmatrix} \beta &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) \nonumber \\&\quad \quad \times f\left( t\begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad a_2 \end{pmatrix}, \frac{\alpha }{N(t)a_1a_2}\right) |a_2|^{-1}dmd^*a_1d^*a_2. \end{aligned}$$
(5.93)
If we write \(t=b_1+b_2\sqrt{D}=\begin{pmatrix} b_1 &{} \quad b_2 \\ b_2D &{} \quad b_1 \end{pmatrix}\), then \(N(t)=b_1^2-b_2^2D\), and
$$\begin{aligned} \Omega (t)=\chi _{1}(\bar{t})\chi _{2}(t)=\chi _{1}(b_1-b_2\sqrt{D}) \end{aligned}$$
as \(\chi _{2}\) is unramified.
First let \(i=c\) and \(\beta =1\). To satisfy
$$\begin{aligned} t\begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad a_2 \end{pmatrix} \in \begin{pmatrix} 1+\varpi ^c O_F &{} \quad O_F \\ \varpi ^c O_F &{} \quad O_F \end{pmatrix}, \end{aligned}$$
we need \(a_1b_1\in 1+\varpi ^cO_F\), \(a_1b_2D\in \varpi ^cO_F\), and \(a_2,m\in O_F\). If \(b_2\notin \varpi ^cO_F\), then it’s impossible for \(a_1\) to satisfy the first two conditions. Thus we only need to consider those t with \(b_2\in \varpi ^cO_F, b_1\in (O_F/\varpi ^cO_F)^*\). Then the domain for the integral is
$$\begin{aligned} a_1\equiv b_1^{-1}+\varpi ^cO_F, m\in O_F\text { and }a_2\in \frac{\alpha }{b_1}(1+\varpi ^cO_F), \end{aligned}$$
(5.94)
as we also need \(\frac{\alpha }{N(t)a_1a_2}\in 1+\varpi ^cO_F\).
By Lemma 5.15,
$$\begin{aligned} \Phi _s\left( \begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix}\begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad a_2 \end{pmatrix}\right) =\chi _{1,s}\left( \frac{a_2}{\sqrt{D}}\right) \chi _{2,s}(a_1\sqrt{D}), \end{aligned}$$
when the condition \(v(a_1)\le v(a_2+m\sqrt{D})\) is satisfied. In particular over the domain in (5.94),
$$\begin{aligned} \Phi _s={\left\{ \begin{array}{ll} \chi _{1,s}\left( \frac{\alpha }{b_1\sqrt{D}}\right) ,&{}\text { if }v(\alpha )\ge 0;\\ 0,&{}\text { otherwise}, \end{array}\right. } \end{aligned}$$
as \(\chi _{2,s}\) is unramified and \(v(a_1\sqrt{D})=0\). Also \(\Omega (t)=\chi _{1,s}(b_1-b_2\sqrt{D})=\chi _{1,s}(b_1)\). Then for \(v(\alpha )\ge 0\) the integral is easily computed to be
https://static-content.springer.com/image/art%3A10.1007%2Fs40993-016-0061-7/MediaObjects/40993_2016_61_Equ172_HTML.gif
(5.95)
Now suppose \(0<i<c\). For any fixed \(t=b_1+b_2\sqrt{D}\) and \(\beta \), we can do the integral similarly. In particular one would need \(v(a_1)\le v(a_2+m\sqrt{D}),\) \(b_1\in O_F^*\) and \(b_2\in \varpi ^iO_F\) for the integral to be nonzero. And when that’s the case, the domain of the integral is
$$\begin{aligned} v(m)\ge 0,\quad a_2\in \frac{b_1\alpha }{(b_1^2-b_2^2D)}(1+\varpi ^iO_F)\;\; \text { and }\;\;a_1\in \frac{\beta ^{-1}\alpha }{(b_1^2-b_2^2D)a_2}(1+\varpi ^cO_F). \end{aligned}$$
Over this domain, we have
$$\begin{aligned} \psi (u\varpi ^{-i}[(x_1-\beta ^{-1})x_4-x_2x_3])= & {} \psi (\varpi ^{-i}\alpha )\psi \left( -\frac{\beta ^{-1}\alpha \varpi ^{-i}}{(b_1^2-b_2^2D)a_1a_2}(a_2b_1+mb_2D)\right) \nonumber \\= & {} \psi (\varpi ^{-i}\alpha )\psi \left( -\frac{b_1^2\alpha \varpi ^{-i}}{b_1^2-b_2^2D}\right) , \end{aligned}$$
(5.96)
which turns out to be constant over the domain.
Now we do the integral of \(\chi _{1,s}(\frac{a_2}{\sqrt{D}})\chi _{2,s}(a_1\sqrt{D})\) over the above domain. When integrating in \(a_1\) first, we are essentially integrating a constant as \(\chi _{2,s}\) is unramified. Then the integral in \(a_2\) is essentially
$$\begin{aligned} \int \limits _{a_2\in \frac{b_1\alpha }{(b_1^2-b_2^2D)}(1+\varpi ^iO_F)}\chi _{1,s}(a_2)d^*a_2, \end{aligned}$$
(5.97)
which is zero according to Lemma 2.4.

When \(i=0\), the proof is similar. We will leave this case to the reader. \(\square \)

According to this Lemma, we only need to compute one integral for (5.88) and we only care about \(W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) \).

Lemma 5.22

Assume that \(\mu _1\) is unramified and \(\mu _2\) is of level \(c>0\). Suppose that \(\varphi \in \pi (\mu _1^{-1},\mu _2^{-1})\) is the unique \(K_1(\varpi ^c)-\) invariant function supported on \(BK_1(\varpi ^c)\). Then
$$\begin{aligned} W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) = {\left\{ \begin{array}{ll} q^{-v(\alpha )/2}\mu _1^{-v(\alpha )},&{}\text { if }v(\alpha )\ge 0;\\ 0,&{}\text { if }v(\alpha )<0. \end{array}\right. } \end{aligned}$$
(5.98)

Proof

By Lemma 5.6, in particular by part (2ii), we have
$$\begin{aligned} \varphi \left( \omega \begin{pmatrix} 1 &{} \quad m \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) =\mu _1^{-1}\left( -\frac{\alpha }{m}\right) \mu _2^{-1}(-m)\left| \frac{\alpha }{m^2}\right| ^{1/2}, \end{aligned}$$
when \(v(m)\le v(\alpha )-c\). Recall that \(W^-_\varphi \) is the Whittaker function for \(\varphi \) associated to \(\psi ^- \). Then
$$\begin{aligned} W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right)&=\int \limits _{v(m)\le v(\alpha )-c}\mu _1^{-1}\left( -\frac{\alpha }{m}\right) \mu _2^{-1}(-m)\left| \frac{\alpha }{m^2}\right| ^{1/2}\psi (m)dm\nonumber \\&={\left\{ \begin{array}{ll} C'q^{-v(\alpha )/2}\mu _1^{-v(\alpha )},&{}\text { if }v(\alpha )\ge 0;\\ 0,&{}\text { if }v(\alpha )<0. \end{array}\right. } \end{aligned}$$
(5.99)
where
$$\begin{aligned} C'=\frac{1}{q^c\mu _1^c}\int \limits _{v(m)=-c}\mu _2^{-1}(-m)\psi (m)dm \end{aligned}$$
is a non-zero constant and will be cancelled after normalization. \(\square \)
Now we combine Lemma 5.21 and 5.22 into (5.88). One can easily see that
$$\begin{aligned} {\mathbb {P}}(s,w,f,\Phi _s)&=A_c\int \limits _{v(\alpha )\ge 0} q^{-v(\alpha )/2}\mu _1^{-v(\alpha )}|\alpha |^{\frac{w}{2}-\frac{1}{4}}\Phi _s(\alpha )^{-1}\nonumber \\ {}&\quad \times \frac{1}{(q-1)(q^2-1)q^{3c-3}}\chi _{1,s}\left( \frac{\alpha }{\sqrt{D}}\right) q^{v(\alpha )}d^*\alpha \nonumber \\&=\frac{1}{(q^2 - 1)^2q^{4c-4}\chi _{1,s}(\sqrt{D})}\frac{1}{1-q\delta \mu _2\chi _{1,s}}. \end{aligned}$$
(5.100)
Here \(\delta =q^{-(w/2+1/4)}\). We have used that \(\mu _1\mu _2\chi _{1,s}\chi _{2,s}=1\) and \(\mu _2\chi _{1,s}\) is unramified. When \(w=1/2\), we have
$$\begin{aligned} {\mathbb {P}}(s,1/2,f,\Phi _s)=\frac{1}{(q^2 - 1)^2q^{4c-4}\chi _{1,s}(\sqrt{D})}\frac{1}{1-\mu _2\chi _{1}q^{-(2s+1/2)}}. \end{aligned}$$
(5.101)
To compute \(\langle F_1,F \rangle \), we shall compute the corresponding integral using \(f'\).
$$\begin{aligned} \langle F_1,F \rangle= & {} \sum \limits _{0\le i\le c, \beta }A_{i}\int \int W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) \nonumber \\&\times \, f'(\alpha x, \alpha ^{-1}x^{-2})(\mu _1\mu _2)^{-1}(x)d^*xd^*\alpha . \end{aligned}$$
(5.102)
By Formula (5.90), we have for any i,
$$\begin{aligned} r'\left( \begin{pmatrix} 1 &{} \quad 0 \\ \varpi ^i &{} \quad 1 \end{pmatrix}\right) f'= & {} {\sum \limits _{a\in (O_F/\varpi ^cO_F)^{*}}}'q^{2(i-c)}\psi (u\varpi ^{-i}[(x_1-a^{-1})x_4-x_2x_3])\nonumber \\&\times char\left( \begin{pmatrix} a^{-1}+\varpi ^i O_F &{} \quad \varpi ^{i-c}O_F \\ \varpi ^i O_F &{} \quad \varpi ^{i-c}O_F \end{pmatrix}\right) \times char(a+\varpi ^c O_F)\nonumber \\ \end{aligned}$$
(5.103)
\((\alpha x, \alpha ^{-1}x^{-2})\) is in the support if and only if \(x\in 1+\varpi ^iO_F\). On the support, it’s a constant function in x. Then integral against \((\mu _1\mu _2)(x)\) is nonzero if and only if \(i=c\). Thus
$$\begin{aligned} \langle F_1,F \rangle&=A_c\int \int W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) f'(\alpha x, \alpha ^{-1}x^{-2})(\mu _1\mu _2)^{-1}(x)d^*x\,d^*\alpha \nonumber \\&=A_c\int \limits _{v(\alpha )=0} W^-_\varphi \left( \begin{pmatrix} \alpha &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\right) \frac{1}{(q-1^2)(q^{2c-2})} d^*\alpha \nonumber \\&=\frac{1}{(q-1)(q^{2}-1)(q^{3c-3})}. \end{aligned}$$
(5.104)
The expected L-factors in this case is
$$\begin{aligned} \frac{\zeta (2)}{L(\pi ,Ad,1)}\frac{L (\Pi \otimes \Omega ,1/2)L (\pi \otimes \chi _{1}|_{{\mathbb {F}}^*},2s+1/2)}{L (\eta ,1)L (\chi ,2s+1)}=\frac{1}{1-\mu _2\chi _{1}q^{-(2s+1/2)}}. \end{aligned}$$
(5.105)
Thus
$$\begin{aligned} {\mathbb {P}}^0(s,1/2,f,\Phi _s)= \frac{1}{(q+1){q^{c-1}}\chi _{1}(\sqrt{D})}. \end{aligned}$$
(5.106)

5.6 The last case when only \(\pi \) is ramified

In this section, we consider a finite place where \(\pi \) is highly ramified, \(\chi _1\) and \(\chi _2\) are unramified and \({\mathbb {E}}/{\mathbb {F}}\) is inert. As mentioned in the end of Section 3, the local integral of our problem can also be formulated in terms of matrix coefficients:
$$\begin{aligned} \int \limits _{{\mathbb {F}}^*\backslash {\text {GL}}_2({\mathbb {F}})}\Phi _s(\gamma _0 g) \langle F_{1},\pi (g) F \rangle dg, \end{aligned}$$
(5.107)
where \(F_1\in \hat{\pi }\), \(F\in \pi \) and \( \langle \cdot ,\cdot \rangle \) is a bilinear and \({\text {GL}}_2({\mathbb {F}})-\) invariant pairing between \(\hat{\pi }\) and \(\pi \).

If the level c of the representation \(\pi \) is odd, then the local integral is automatically zero according to Theorem 2.14 and Example 2.15. Assume from now on that \(c=2k\).

Define \(\tilde{K}\) to be the subgroup of \({\text {GL}}_2(O_E)\) whose elements are congruent to \(\begin{pmatrix} 1 &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\mod (\varpi ^k O_E)\). Define \(\Phi _s\) to be the unique up to constant function from the induced representation such that it’s right \(\tilde{K}\)-invariant and supported on \(B\begin{pmatrix} 0 &{} \quad 1 \\ -1 &{} \quad -\frac{\sqrt{D}}{D} \end{pmatrix}\tilde{K}. \) It will be normalized such that
$$\begin{aligned} \Phi _s\left( \begin{pmatrix} 0 &{} \quad 1 \\ -1 &{} \quad -\frac{\sqrt{D}}{D} \end{pmatrix}\right) =1. \end{aligned}$$
As motivated by Example 2.18, we will pick \(F_{1}\) and F to be the unique up to constant elements from respective representations which are invariant under
$$\begin{aligned} \left\{ \begin{pmatrix} a+\varpi ^k O_F &{} \quad b+\varpi ^k O_F \\ bD+\varpi ^k O_F &{} \quad a+\varpi ^k O_F \end{pmatrix}|a+b\sqrt{D}\in O_E^* \right\} . \end{aligned}$$
Let \(F_{1}\) and F be so normalized that
$$\begin{aligned} \langle F_{1},F \rangle =1. \end{aligned}$$

Proposition 5.23

Suppose that \(\pi \) is highly ramified of level \(c=2k\), \(\chi _1\) and \(\chi _2\) are both unramified, and \({\mathbb {E}}/{\mathbb {F}}\) is inert. Then for the choice of \(F_{1}\in \hat{\pi }\), \(F\in \pi \) and \(\Phi _s\) as given above, we have
$$\begin{aligned} \int \limits _{{\mathbb {F}}^*\backslash {\text {GL}}_2({\mathbb {F}})}\Phi _s(\gamma _0 g)\langle F_{1},\pi (g) F \rangle dg=\frac{1}{(q-1)q^{c-1}}, \end{aligned}$$
(5.108)
and
$$\begin{aligned} {\mathbb {P}}^0=\frac{L(\pi ,Ad,1)}{q^c(1-\chi q^{-(4s+2)})}. \end{aligned}$$
(5.109)

We first give the property for \(\Phi _s\).

Lemma 5.24

Let \(\Phi _s\) be the unique normalized element from the induced representation which is supported on \(B\begin{pmatrix} 0 &{} \quad 1 \\ -1 &{} \quad -\frac{\sqrt{D}}{D} \end{pmatrix}\tilde{K}\). Then
$$\begin{aligned} \Phi _s\left( \gamma _0\begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad 1 \end{pmatrix}\right) = {\left\{ \begin{array}{ll} 1,&{}\text { if } v(m)\ge k\text { and }a_1\equiv 1\mod (\varpi ^kO_F);\\ 0, &{}\text { otherwise }. \end{array}\right. } \end{aligned}$$
(5.110)

Proof

Let’s consider when the matrix \(\begin{pmatrix} 1 &{} \quad 0 \\ \sqrt{D} &{} \quad 1 \end{pmatrix}\begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad 1 \end{pmatrix}\) can be in the support \(B\begin{pmatrix} 0 &{} \quad 1 \\ -1 &{} \quad -\frac{\sqrt{D}}{D} \end{pmatrix}\tilde{K}\). This is equivalent to say if there exists \(k\in \tilde{K}\) such that
$$\begin{aligned} \begin{pmatrix} a_1 &{} \quad m \\ a_1\sqrt{D} &{} \quad 1+m\sqrt{D} \end{pmatrix}k\begin{pmatrix} -\frac{\sqrt{D}}{D} &{} \quad -1 \\ 1 &{} \quad 0 \end{pmatrix} \end{aligned}$$
is upper triangular. This in turn is equivalent to that
$$\begin{aligned} -a_1+1+m\sqrt{D}\equiv 0\mod (\varpi ^kO_E). \end{aligned}$$
Thus one get the conditions for \(\Phi _s\left( \gamma _0\begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad 1 \end{pmatrix}\right) \) to be non-zero as in the lemma. When these conditions are satisfied, the rest are easy to check. \(\square \)
Now we can prove Proposition 5.23 easily. As \(\chi _1\) and \(\chi _2\) are unramified, \(\Phi _s(\gamma _0tg)=\Phi _s(\gamma _0g)\) for \(t\in O_E^*\). Note that \({\mathbb {F}}^*\backslash {\text {GL}}_2=O_E^*\left\{ \begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad 1 \end{pmatrix}\right\} \). Then the local integral (5.107) becomes
$$\begin{aligned}&\int \limits _{{\mathbb {F}}^*\backslash {\text {GL}}_2({\mathbb {F}})}\Phi _s(\gamma _0 g) \langle F_{1},\pi (g) F\rangle dg \nonumber \\&\quad =\int \limits _{t\in O_E^*}\int \limits _{a_1,m}\Omega (t)\Phi _s\left( \gamma _0\begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad 1 \end{pmatrix}\right) \left\langle \hat{\pi }(t^{-1})F_{1},\pi \left( \begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad 1 \end{pmatrix}\right) F \right\rangle d^*a_1\,dm\,d^*t \nonumber \\&\quad =\int \limits _{a_1,m}\Phi _s\left( \gamma _0\begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad 1 \end{pmatrix}\right) \left\langle F_{1},\pi \left( \begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad 1 \end{pmatrix}\right) F \right\rangle d^*a_1\,dm \nonumber \\&\quad =\int \limits _{v(m)\ge k\text { and }a_1\equiv 1\text { mod}{(\varpi ^k)}}\left\langle F_{1},\pi \left( \begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad 1 \end{pmatrix}\right) F \right\rangle d^*a_1\,dm. \end{aligned}$$
(5.111)
Here we have used the fact that \(F_1\) is invariant under \(O_E^*\) for the second equality, and Lemma 5.24 for the last equality.
Note that when \(v(m)\ge k\text { and }a_1\equiv 1\text { mod}{(\varpi ^k)}\),
$$\begin{aligned} \begin{pmatrix} a_1 &{} \quad m \\ 0 &{} \quad 1 \end{pmatrix}\in \left\{ \begin{pmatrix} a+\varpi ^k O_F &{} \quad b+\varpi ^k O_F \\ bD+\varpi ^k O_F &{} \quad a+\varpi ^k O_F \end{pmatrix}|a+b\sqrt{D}\in O_E^*\right\} , \end{aligned}$$
under the action of which F is invariant.
Thus
$$\begin{aligned}&\int \limits _{{\mathbb {F}}^*\backslash {\text {GL}}_2({\mathbb {F}})}\Phi _s(\gamma _0 g)\langle F_{1},\pi (g) F \rangle dg \nonumber \\&\quad =\int \limits _{v(m)\ge k\text { and }a_1\equiv 1\text {mod}{(\varpi ^k)}} \langle F_{1},F \rangle d^*a_1dm=\frac{1}{(q-1)q^{c-1}}. \end{aligned}$$
(5.112)
The expected L-factors in this case is
$$\begin{aligned}&\frac{\zeta (2)}{L(\pi ,Ad,1)}\frac{L (\Pi \otimes \Omega ,1/2)L (\pi \otimes \chi _{1}|_{{\mathbb {F}}^*},2s+1/2)}{L (\eta ,1)L (\chi ,2s+1)}\nonumber \\&\quad =\frac{1}{(1-q^{-2})L(\pi ,Ad,1)}(1+q^{-1})(1-\chi q^{-(4s+2)}). \end{aligned}$$
(5.113)
So
$$\begin{aligned} {\mathbb {P}}^0(s,1/2,f,\Phi _s)=\frac{L(\pi ,Ad,1)}{q^c(1-\chi q^{-(4s+2)})}. \end{aligned}$$
(5.114)

6 Archimedean places

The local integral at archimedean places in general can be very complicated to compute. In this section we shall restrict ourselves to the following setting. Suppose that \({\mathbb {E}}\) is totally real. In particular all infinity places of \({\mathbb {F}}\) are real and they split in \({\mathbb {E}}\). Suppose that the cusp form F is anti-holomorphic and is of parallel weight \(-2n\), and the Eisenstein series is holomorphic and is of parallel weight n.

In terms of local components, we assume that \(\pi \) is of form \(\sigma (|\cdot |^{\frac{2n-1}{2}},|\cdot |^{-\frac{2n-1}{2}})\). Pick \(F_1\) and F to be weight \(-2n\) elements. It is well-known that in general the matrix coefficient associated to holomorphic weight \(2n>0\) elements is
$$\begin{aligned} \varphi \left( \begin{pmatrix} a &{} \quad m \\ 0 &{} \quad 1 \end{pmatrix}\right) =\frac{a^{n}(2i)^2n}{(m+(a+1)i)^2n}. \end{aligned}$$
(6.1)
Note that it is normalized so that \(\varphi (1)=1\). Then the matrix coefficient associated to weight \(-2n\) elements is
$$\begin{aligned} \varphi \left( \begin{pmatrix} a &{} \quad m \\ 0 &{} \quad 1 \end{pmatrix}\right) =\frac{a^{n}(2i)^2n}{(-m+(a+1)i)^2n}. \end{aligned}$$
(6.2)
Here we have conjugated the formula by \(\begin{pmatrix} -1 &{} \quad 0 \\ 0 &{} \quad 1 \end{pmatrix}\) to get the matrix coefficient for weight \(-2n\) elements.
Since this real place splits in \({\mathbb {E}}\), we shall write \(\Phi _s=\Phi _s^{(1)}\Phi _s^{(2)}\), where we pick \(\Phi _s^{(i)}\) to be the weight n element from \(Ind_{B}^{{\text {GL}}_2}(sgn^\delta (\cdot )|\cdot |^{\frac{n-1}{2}}, |\cdot |^{-\frac{n-1}{2}})\). Here \(\delta \) is 0 if n is even and 1 if n is odd. We shall think \(\sqrt{D}=(\sqrt{D},-\sqrt{D})\) in this case. Recall that the local integral is
$$\begin{aligned} {\mathbb {P}}=\int \limits _{{\mathbb {F}}^*\backslash {\text {GL}}_2({\mathbb {F}})}\langle F_{1},\pi (g)F \rangle \Phi _s(\gamma _0g)dg=\int \limits _{{\mathbb {F}}^*\backslash {\text {GL}}_2({\mathbb {F}})}\varphi (g)\Phi _s(\gamma _0 g)dg. \end{aligned}$$
(6.3)
Here we shall use the decomposition
$$\begin{aligned} {\mathbb {F}}^*\backslash {\text {GL}}_2({\mathbb {F}})=\left\{ \begin{pmatrix} a &{} \quad m \\ 0 &{} \quad 1 \end{pmatrix}|a\in {\mathbb {R}}^*,m\in {\mathbb {R}}\right\} K \end{aligned}$$
for \(K=SO(2)\), and the Haar measure is
$$\begin{aligned} \frac{1}{2\pi }\frac{da \,dm \,d\theta }{a^2}. \end{aligned}$$
(6.4)
Note that
$$\begin{aligned} \begin{pmatrix} a &{} \quad m \\ \pm a\sqrt{D} &{} \quad 1\pm m\sqrt{D} \end{pmatrix}= & {} \begin{pmatrix} \frac{a}{\sqrt{a^2D+(1\pm m\sqrt{D})^2}} &{} \quad \frac{\pm a^2\sqrt{D}+m\pm m^2\sqrt{D}}{\sqrt{a^2D+(1\pm m\sqrt{D})^2}} \\ 0 &{} \quad \sqrt{a^2D+(1\pm m\sqrt{D})^2} \end{pmatrix} \nonumber \\&\times \begin{pmatrix} \frac{1\pm m\sqrt{D}}{\sqrt{a^2D+(1\pm m\sqrt{D})^2}} &{} \quad \frac{\mp a\sqrt{D}}{\sqrt{a^2D+(1\pm m\sqrt{D})^2}} \\ \frac{\pm a\sqrt{D}}{\sqrt{a^2D+(1\pm m\sqrt{D})^2}} &{} \quad \frac{1\pm m\sqrt{D}}{\sqrt{a^2D+(1\pm m\sqrt{D})^2}} \end{pmatrix} \end{aligned}$$
(6.5)
So by definition,
$$\begin{aligned} {\mathbb {P}}&=\int \limits _{a>0}\int \limits _{m}\frac{a^n2^{2n}}{(-m+(a+1)i)^{2n}}\left( \frac{a}{a^2D+(1+ m\sqrt{D})^2}\right) ^{\frac{n}{2}}\left( \frac{1+ m\sqrt{D}-ia\sqrt{D}}{\sqrt{a^2D+(1+ m\sqrt{D})^2}}\right) ^{n} \nonumber \\&\quad \times \left( \frac{a}{a^2D+(1- m\sqrt{D})^2}\right) ^{\frac{n}{2}}\left( \frac{1- m\sqrt{D}+ia\sqrt{D}}{\sqrt{a^2D+(1- m\sqrt{D})^2}} \right) ^{n}a^{-2}da\,dm. \end{aligned}$$
(6.6)
We have used that \(\varphi \) is only supported on \(\det (g)>0\). It can be easily simplified as
$$\begin{aligned} {\mathbb {P}}=\frac{2^{2n}(-1)^n}{D^n}\int \limits _{a>0}\int \limits _{m}\frac{a^{2n-2}}{(m-(a+1)i)^{2n}\left( m+\frac{1}{\sqrt{D}}+ia\right) ^n \left( m-\frac{1}{\sqrt{D}}+ia \right) ^n} da\,dm.\nonumber \\ \end{aligned}$$
(6.7)
As an analytic function in m, the integrand has a pole of order 2n at \(m=(a+1)i\) in the upper half plane, and two poles of order n at \(m=\pm \frac{1}{\sqrt{D}}-ia\) in the lower half plane. To fully make use of the symmetry, we shall shift the contour integral upward and use basic complex analysis.
$$\begin{aligned}&{\mathbb {P}}=\frac{2\pi i}{(2n-1)!}\frac{2^{2n}(-1)^n}{D^n}\int \limits _{a>0}a^{2n-2} \left. \left( \frac{d}{dm}\right) ^{2n-1}\left( \frac{1}{\left( (m+ia)^2-\frac{1}{D}\right) ^n}\right) \right| _{m=(a+1)i} da.\qquad \end{aligned}$$
(6.8)
Now we make use of the linear relation between m and a in the expression, and do the following change of variable:
$$\begin{aligned} u=\frac{m+ia-i}{2i}. \end{aligned}$$
(6.9)
Then
$$\begin{aligned}&\left( \frac{d}{dm}\right) ^{2n-1}\left( \frac{1}{\left( (m+ia)^2-\frac{1}{D}\right) ^n}\right) |_{m=(a+1)i}\nonumber \\&\quad =\left( \frac{1}{2i}\right) ^{2n-1}\left( \frac{d}{du}\right) ^{2n-1}\left( \frac{1}{\left( (2u+1)^2+\frac{1}{D}\right) ^n(-1)^n}\right) |_{u=a}.\nonumber \\ \end{aligned}$$
(6.10)
So
$$\begin{aligned} {\mathbb {P}}=-\frac{4\pi }{(2n-1)!D^n}\int \limits _{a>0}a^{2n-2} \left( \frac{d}{du}\right) ^{2n-1}\left. \left( \frac{1}{\left( (2u+1)^2+\frac{1}{D}\right) ^n}\right) \right| _{u=a} da. \end{aligned}$$
(6.11)
Now we can use integration by parts multiple times to get
$$\begin{aligned} {\mathbb {P}}&= -\frac{4\pi }{(2n-1)!D^n}\left[ {a^{2n - 2}}{\left( {\frac{d}{{da}}} \right) ^{2n - 2}}\left. {\left( {\frac{1}{{{{\left( {{{(2a + 1)}^2} + \frac{1}{D}} \right) }^n}}}} \right) } \right| _{a = 0}^\infty \right. \nonumber \\&\quad - (2n - 2){a^{2n - 3}}\left. {{{\left( {\frac{d}{{da}}} \right) }^{2n - 3}}\left( {\frac{1}{{{{\left( {{{(2a + 1)}^2} + \frac{1}{D}} \right) }^n}}}} \right) } \right| _{a = 0}^\infty + \cdots \nonumber \\&\quad \left. {\left. { + (2n - 2)!\left( {\frac{1}{{{{({{(2a + 1)}^2} + \frac{1}{D})}^n}}}} \right) } \right| _{a = 0}^\infty } \phantom {\int \limits ^{ssdf}} \right] \nonumber \\&= \frac{{4\pi }}{{{(2n-1)!D^n}}}(2n - 2)!\frac{1}{{{{(1 + \frac{1}{D})}^n}}} = \frac{4\pi }{2n-1} \frac{1}{(1+D)^n}. \end{aligned}$$
(6.12)
Here we have used that \(a^i (\frac{d}{da})^{i}(\frac{1}{((2a+1)^2+\frac{1}{D})^n})|_{a=0}^{\infty } =0\) for \(0<i\le 2n-2\). We will not normalize the local integral by L-factors for archimedean places.

Declarations

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Acknowledgements

I would like to thank my advisor, Prof. Tonghai Yang. He suggested this problem to me and has given me guidance throughout this paper. I would like to thank Robert Harron, Dipendra Prasad, Michael Woodbury and Lei Zhang for helpful discussions. I would also like to thank referee for suggestions to improve the paper. This paper is partially supported by Graduate School Grant and NSF Grant of Prof. Tonghai Yang.

Authors’ Affiliations

(1)
ETH Zürich

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