On the Blasius-Deligne conjecture for the standard $L$-functions of symplectic type for $\textrm{GL}_{2n}$

TL;DR

This paper provides an unconditional proof of the Blasius-Deligne conjecture concerning the critical values of standard L-functions for symplectic type on GL_{2n}, completing a significant research project.

Contribution

It offers the first unconditional proof of the conjecture for all n ≥ 1, advancing understanding of L-functions of symplectic type.

Findings

Unconditional proof of the Blasius-Deligne conjecture for GL_{2n} L-functions

Complete the project initiated in JST19

Establish critical value formulas for symplectic type L-functions

Abstract

In this paper we give an unconditional proof of the Blasius-Deligne conjecture for the critical values of the $\textrm{GL}_{2n}$-standard $L$-functions of symplectic type with $n\geq 1$ and complete the project started in [JST19].

Equations

$${\mathrm{k}}_{\infty }:=\mathrm{k}{\otimes }_{\mathbb{Q}}\mathbb{R}=\prod _{v\mid \infty }{\mathrm{k}}_v\hookrightarrow \mathrm{k}{\otimes }_{\mathbb{Q}}\mathbb{C}=\prod _{\iota \in {\mathcal{E}}_{\mathrm{k}}}\mathbb{C},$$

$$H_{\mathrm{ct}}^{\ast }({\mathbb{R}}_{+}^{\times }\backslash {\mathrm{GL}}_{2n}({\mathrm{k}}_{\infty }{)}^0;{\Pi }_{\infty }\otimes F_{\mu }^{\vee })\ne \{0\},$$

$${\mu }_1^{\iota }+{\mu }_{2n}^{\iota }={\mu }_2^{\iota }+{\mu }_{2n-1}^{\iota }=\cdots ={\mu }_n^{\iota }+{\mu }_{n-1}^{\iota }=w_{\iota }.$$

$${\chi }_{\infty }={\chi }_{♮}{|}_{{\mathrm{k}}_{\infty }^{\times }}\cdot {\chi }^{♮}\text{for a unique quadratic character χ♮ of k∞×},$$

$${\xi }_{\mu ,{\chi }_{♮}}:={\otimes }_{\iota \in {\mathcal{E}}_{\mathrm{k}}}({\det }^{d{\chi }_{\iota }}⊠{\det }^{-d{\chi }_{\iota }-w_{\iota }})$$

$${\mathrm{Hom}}_{H(\mathrm{k}{\otimes }_{\mathbb{Q}}\mathbb{C})}(F_{\mu }^{\vee }\otimes {\xi }_{\mu ,{\chi }_{♮}}^{\vee },\mathbb{C})\ne \{0\}.$$

$${\Omega }_{\mu ,{\chi }_{♮}\cdot {\otimes }_{\iota \in {\mathcal{E}}_{\mathrm{k}}}{\iota }^j}={\mathrm{i}}^{jn[\mathrm{k}:\mathbb{Q}]}\cdot {\Omega }_{\mu ,{\chi }_{♮}}.$$

$$L(\frac{1}{2},{}^{{\sigma }^{\prime }}\Pi \otimes {}^{{\sigma }^{\prime }}{\chi }^{\prime })\ne 0.$$

$$\sigma \left(\frac{L(\frac{1}{2},\Pi \otimes \chi )}{{\Omega }_{\mu ,{\chi }_{♮}}\cdot \mathcal{G}(\chi {)}^n\cdot {\Omega }_{\epsilon }(\Pi ,\mathbf{\eta })}\right)=\frac{L(\frac{1}{2},{}^{\sigma }\Pi \otimes {}^{\sigma }\chi )}{{\Omega }_{\mu ,{\chi }_{♮}}\cdot \mathcal{G}({}^{\sigma }\chi {)}^n\cdot {\Omega }_{\epsilon }({}^{\sigma }\Pi ,{}^{\sigma }\mathbf{\eta })}$$

$$\frac{L(\frac{1}{2},\Pi \otimes \chi )}{{\Omega }_{\mu ,{\chi }_{♮}}\cdot \mathcal{G}(\chi {)}^n\cdot {\Omega }_{\epsilon }(\Pi ,\mathbf{\eta })}\in \mathbb{Q}(\Pi ,\mathbf{\eta },\chi ),$$

$${\pi }_{\lambda }={\mathrm{Ind}}_P^{G_m}({\tau }_{\lambda })={\mathrm{Ind}}_P^{G_m}({\tau }_1|\cdot {|}_{\text{mathbbm}k}^{{\lambda }_1}\hat{\otimes }{\tau }_2|\cdot {|}_{\text{mathbbm}k}^{{\lambda }_2}\hat{\otimes }\cdots \hat{\otimes }{\tau }_r|\cdot {|}_{\text{mathbbm}k}^{{\lambda }_r}),$$

$$M\cong G_{n_1}\times G_{n_2}\times \cdots \times G_{n_r},n_1+n_2+\cdots +n_r=m,$$

$$\min \Re (\lambda ):=\underset{i=1,2,\dots ,r}{\min }\Re ({\lambda }_i),\max \Re (\lambda ):=\underset{i=1,2,\dots ,r}{\max }\Re ({\lambda }_i).$$

$$L(s,\pi ,{\wedge }^2\otimes {\eta }^{-1})=L(s,{\wedge }^2{\varphi }_{\pi }\otimes {\eta }^{-1}),$$

$$\epsilon (s,\pi ,{\wedge }^2\otimes {\eta }^{-1},\psi )=\epsilon (s,{\wedge }^2{\varphi }_{\pi }\otimes {\eta }^{-1},\psi ),$$

$$\gamma (s,\pi ,{\wedge }^2\otimes {\eta }^{-1},\psi )=\epsilon (s,\pi ,{\wedge }^2\otimes {\eta }^{-1},\psi )\cdot \frac{L(1-s,{\pi }^{\vee },{\wedge }^2\otimes \eta )}{L(s,\pi ,{\wedge }^2\otimes {\eta }^{-1})},$$

$$L(s,{\pi }_{\lambda },{\wedge }^2\otimes {\eta }^{-1})=$$

$$\cdot \prod _{1\le j<k\le r}L(s+{\lambda }_j+{\lambda }_k,{\varphi }_{{\tau }_j}\otimes {\varphi }_{{\tau }_k}\otimes {\eta }^{-1}),$$

$$L(s,{\pi }_{\lambda },{\wedge }^2\otimes {\eta }^{-1})=L(s,{\pi }_{\lambda }^0,{\wedge }^2\otimes {\eta }^{-1}),\epsilon (s,{\pi }_{\lambda },{\wedge }^2\otimes {\eta }^{-1},\psi )=\epsilon (s,{\pi }_{\lambda }^0,{\wedge }^2\otimes {\eta }^{-1},\psi )$$

$${\mathcal{F}}_{{\psi }^{\prime }}(\varphi )(x)={\int }_{\text{mathbbm}{k}^n}\varphi (y){\psi }^{\prime }(y{}^{t}x)dy,x\in \text{mathbbm}{k}^n.$$

$$S_{m}:=\begin{cases}\Set{\begin{bmatrix}g&Xg\\ 0&g\end{bmatrix}}{g\in G_{n},X\in M_{n}},&\textrm{if }m=2n,\\ \Set{\begin{bmatrix}g&Xg&y\\ 0&g&0\\ 0&xg&1\end{bmatrix}}{\begin{array}[]{l}g\in G_{n},X\in M_{n},\\ y\in\mathbbm{k}^{n\times 1},x\in\mathbbm{k}^{1\times n}\end{array}},&\textrm{if }m=2n+1,\end{cases}$$

$${\phi }_{2n}:S_{2n}\to {\mathbb{C}}^{\times },\left[\begin{array}{cc}g& Xg\\ & g\end{array}\right]\mapsto \eta (g)\psi (\mathrm{tr}X).$$

$$h=\left[\begin{array}{cc}g& Xg\\ & g\end{array}\right]:\text{mathbbm}{k}^n\to \text{mathbbm}{k}^n,v\mapsto vg.$$

$$R_{{\phi }_{2n}}(h)\varphi (v):={\phi }_{2n}(h)\varphi (v.h)={\phi }_{2n}(h)\varphi (vg),\varphi \in \mathcal{S}(\text{mathbbm}{k}^n),$$

$${\phi }_{2n+1}:S_{2n+1}\cap P_{2n+1}\to {\mathbb{C}}^{\times },\left[\begin{array}{ccc}g& Xg& y\\ & g& 0\\ & & 1\end{array}\right]\mapsto \eta (g)\psi (\mathrm{tr}X),$$

$${\sigma }_m:=\{\begin{array}{ll}\left(\begin{array}{cccccccc}1& 2& \cdots & n& n+1& n+2& \cdots & 2n\\ 1& 3& \cdots & 2n-1& 2& 4& \cdots & 2n\end{array}\right),& \text{if }m=2n,\\ \left(\begin{array}{ccccccccc}1& 2& \cdots & n& n+1& n+2& \cdots & 2n& 2n+1\\ 1& 3& \cdots & 2n-1& 2& 4& \cdots & 2n& 2n+1\end{array}\right),& \text{if }m=2n+1.\end{array}$$

$$Z_{\mathrm{JS}}(s,W,\varphi ,{\phi }_m^{-1}):=\{\begin{array}{ll}{\int }_{{\overline{S}}_m}W({\sigma }_{m}h)R_{{\phi }_m^{-1}}(h)\varphi (e_n)|h{|}_{\text{mathbbm}k}^{\frac{s}{2}}dh,& \text{if }m=2n,\\ {\int }_{{\overline{S}}_m}W({\sigma }_{m}h)R_{{\phi }_m^{-1}}(h)\varphi (0)|h{|}_{\text{mathbbm}k}^{\frac{s}{2}}dh,& \text{if }m=2n+1,\end{array}$$

$${\tau }_m:=\left[\begin{array}{cc}0& 1_n\\ 1_n& 0\end{array}\right]resp.\left[\begin{array}{ccc}0& 1_n& \\ 1_n& 0& \\ & & 1\end{array}\right],\text{if }m=2nresp.2n+1.$$

$$\frac{Z_{\mathrm{JS}}(1-s,{\tau }_m.\tilde{W},\hat{\varphi },{\phi }_m)}{L(1-s,{\pi }_{\lambda }^{\vee },{\wedge }^2\otimes \eta )}=\eta (-1{)}^{mn}\epsilon (s,{\pi }_{\lambda },{\wedge }^2\otimes {\eta }^{-1},\psi )\frac{Z_{\mathrm{JS}}(s,W,\varphi ,{\phi }_m^{-1})}{L(s,{\pi }_{\lambda },{\wedge }^2\otimes {\eta }^{-1})},$$

$$\hat{\varphi }:=\{\begin{array}{ll}{\mathcal{F}}_{\psi }(\varphi ),& \text{if m is even},\\ {\mathcal{F}}_{\bar{\psi }}(\varphi ),& \text{if m is odd.}\end{array}$$

$$s\mapsto Z_{\mathrm{JS}}^{\circ }(s,W,\varphi ,{\phi }_m^{-1}):=\frac{Z_{\mathrm{JS}}(s,W,\varphi ,{\phi }_m^{-1})}{L(s,{\pi }_{\lambda },{\wedge }^2\otimes {\eta }^{-1})}$$

$$\epsilon (s,\delta ,\bar{\psi })=\det (\delta )(-1)\epsilon (s,\delta ,\psi )$$

Full text

1. 1 Introduction

2. 2 Main Local Results

3. 2.1 Jacquet-Shalika integrals and modifying factors

4. 2.1.1 Representations and exterior square local factors

5. 2.1.2 Jacquet-Shalika integrals

6. 2.1.3 Open orbit integrals and modifying factors

7. 2.1.4 The ideas of the proof

8. 2.2 Friedberg-Jacquet integrals and modifying factors

9. 2.3 Archimedean period relations

10. 3 Basic Properties of Jacquet-Shalika Integrals

11. 3.1 Preliminaries on Whittaker functions

12. 3.2 Jacquet-Shalika integrals revisited

13. 3.3 Convergence and continuity

14. 3.4 A non-vanishing result

15. 4 Reductions of $({\rm FE}_{m})$

16. 4.1 Reductions of inducing data

17. 4.1.1 Reduction of spectral parameters

18. 4.1.2 Reduction to principal series representations

19. 4.2 $({\rm MF}_{m})+({\rm FE}^{\prime}_{m})\Rightarrow({\rm FE}_{m})$

20. 5 Proof of $({\rm FE}_{m}^{\prime})$

21. 5.1 Convergence and continuation

22. 5.2 The even case

23. 5.3 The odd case

24. 6 $({\rm MF}_{m})+({\rm FE}_{m})\Rightarrow({\rm MF}_{m+1})$

25. 6.1 Godement sections

26. 6.2 The case $G_{2n}\to G_{2n+1}$

27. 6.2.1 $\operatorname{Z}_{\rm JS}$-side

28. 6.2.2 $\Lambda_{\rm JS}$-side

29. 6.3 The case $G_{2n-1}\to G_{2n}$

30. 6.3.1 $\operatorname{Z}_{\rm JS}$-side

31. 6.3.2 $\Lambda_{\rm JS}$-side

32. 7 Friedberg-Jacquet integrals and modifying factors

33. 7.1 Proof of Theorem 2.11

34. 7.2 Proof of Proposition 2.13

35. 7.3 Proof of Proposition 2.14

36. 7.4 Proof of Theorem 2.15

37. 8 Proof of Archimdedean period relations

38. 9 Cohomology groups and modular symbols

39. 9.1 Preliminaries on cohomology groups

40. 9.2 Modular symbols and a commutative diagram

41. 9.2.1 Global modular symbol

42. 9.2.2 Archimedean modular symbol

43. 9.2.3 Non-Archimedean modular symbol

44. 9.2.4 A commutative diagram

45. 10 Shalika periods and the Blasius-Deligne conjecture

46. 10.1 The kernels of modular symbols

47. 10.2 Shalika periods and the end of proof

1. Introduction

The Blasius-Deligne conjecture ([D79, B97]) for automorphic $L$-functions is about the period relations and the algebraicity of critical $L$-values. In the paper, we give an unconditional proof of the Blasius-Deligne conjecture for the ${\mathrm{GL}}_{2n}$-standard $L$-functions of symplectic type with $n\geq 1$ and completes the project started in [JST19]. We refer to the introduction of [JST19, LLS24] for historical comments on earlier work of lower rank cases and relevant work for higher rank cases.

Let ${\mathrm{k}}$ be a number field with adele ring ${\mathbb{A}}$. Let ${\mathrm{k}}_{v}$ be the local field at a local place $v$ of ${\mathrm{k}}$, and write ${\mathbb{A}}={\mathbb{A}}_{f}\times{\mathrm{k}}_{\infty}$ with ${\mathbb{A}}_{f}=\bigotimes^{\prime}_{v\nmid\infty}{\mathrm{k}}_{v}$ being the finite part of ${\mathbb{A}}$ and ${\mathrm{k}}_{\infty}$ being the so-called $\infty$-part of ${\mathbb{A}}$, which has the following realization:

[TABLE]

where ${\mathcal{E}}_{\mathrm{k}}$ is the set of field embeddings $\iota:{\mathrm{k}}\hookrightarrow{\mathbb{C}}$.

Let $\Pi=\Pi_{f}\otimes\Pi_{\infty}$ be a regular algebraic irreducible cuspidal automorphic representation of ${\mathrm{GL}}_{2n}({\mathbb{A}})$ ($n\geq 1$) in the sense of [Cl90]. Then up to isomorphism there is a unique irreducible algebraic representation $F_{\mu}$ of ${\mathrm{GL}}_{2n}({\mathrm{k}}\otimes_{\mathbb{Q}}{\mathbb{C}})$, say of highest weight $\mu=\{\mu^{\iota}\}_{\iota\in{\mathcal{E}}_{\mathrm{k}}}\in({\mathbb{Z}}^{2n})^{{\mathcal{E}}_{\mathrm{k}}}$, such that the total continuous cohomology

[TABLE]

where ${\mathbb{R}}^{\times}_{+}$ is the diagonal central torus. Here and henceforth, a superscript ∨ indicates the contragradient representation, and $X^{0}$ denotes the identity component of a topological group $X$. The representation $F_{\mu}$ is called the coefficient system of $\Pi$. For $\sigma\in{\mathrm{Aut}}({\mathbb{C}})$, denote by ${}^{\sigma}\Pi$ the $\sigma$-twist of $\Pi$ in the sense of [Cl90], which is also a regular algebraic irreducible cuspidal automorphic representation of ${\mathrm{GL}}_{2n}({\mathbb{A}})$. Similarly denote by ${}^{\sigma}F_{\mu}$ the coefficient system of ${}^{\sigma}\Pi$.

Assume that $\Pi$ is of symplectic type, which is equivalent to that there is a character $\boldsymbol{\eta}:{\mathrm{k}}^{\times}\backslash{\mathbb{A}}^{\times}\to{\mathbb{C}}^{\times}$ such that the complete twisted exterior square $L$-function $\operatorname{L}(s,\Pi,\wedge^{2}\otimes\boldsymbol{\eta}^{-1})$ has a pole at $s=1$ ([JST19, Definition 2.3]). For each $\iota\in{\mathcal{E}}_{\mathrm{k}}$ write $\mu^{\iota}=(\mu^{\iota}_{1},\mu^{\iota}_{2},\dots,\mu^{\iota}_{2n})\in{\mathbb{Z}}^{2n}$. Then there exists $w_{\iota}\in{\mathbb{Z}}$ such that

[TABLE]

For an arbitrary algebraic Hecke character $\chi=\chi_{f}\otimes\chi_{\infty}:{\mathrm{k}}^{\times}\backslash{\mathbb{A}}^{\times}\to{\mathbb{C}}^{\times}$, there exists a unique family $\{\operatorname{d}\!\chi_{\iota}\in{\mathbb{Z}}\}_{\iota\in{\mathcal{E}}_{\mathrm{k}}}$ of integers such that

[TABLE]

where $\chi_{\natural}:=\otimes_{\iota\in{\mathcal{E}}_{\mathrm{k}}}\iota^{\operatorname{d}\!\chi_{\iota}}$ is a character of $({\mathrm{k}}\otimes_{\mathbb{Q}}{\mathbb{C}})^{\times}$. That is, $\chi_{\natural}$ is the coefficient system of $\chi$. Note that the formal sum $\sum_{\iota\in{\mathcal{E}}_{\mathrm{k}}}\operatorname{d}\!\chi_{\iota}\cdot\iota\in{\mathbb{Z}}[{\mathcal{E}}_{\mathrm{k}}]$ is referred as the infinite type of $\chi$ in the literature. View $H:={\mathrm{GL}}_{n}\times{\mathrm{GL}}_{n}$ as a standard Levi subgroup of ${\mathrm{GL}}_{2n}$. Define a character

[TABLE]

of $H({\mathrm{k}}\otimes_{\mathbb{Q}}{\mathbb{C}})$.

Definition 1.1**.**

With the above notation, we say that $\chi_{\natural}$ is $F_{\mu}$-balanced if

[TABLE]

Remark 1.2**.**

Some remarks are in order.

1. (1)

If $\chi_{\natural}$ is $F_{\mu}$-balanced, then the integers $j$ such that $\chi_{\natural}\cdot\otimes_{\iota\in{\mathcal{E}}_{\mathrm{k}}}\iota^{j}$ is $F_{\mu}$-balanced are in bijection with the critical places $\frac{1}{2}+j$ of $\operatorname{L}(s,\Pi\otimes\chi)$. This can be proved in the same way as that of **[JST19, Proposition 2.20]**. 2. (2)

Set $\Omega_{\mu,\chi_{\natural}}:={\rm i}^{\sum_{\iota\in{\mathcal{E}}_{\mathrm{k}}}\sum^{n}_{i=1}(\mu^{\iota}_{i}+\operatorname{d}\!\chi_{\iota})}$ with ${\rm i}=\sqrt{-1}$. Then we must have that

[TABLE] 3. (3)

If ${\mathrm{k}}$ contains no CM field, then

• •

the integer $\operatorname{d}\!\chi_{\iota}$ is independent of $\iota\in{\mathcal{E}}_{\mathrm{k}}$;

• •

$\chi_{\natural}$* is $F_{\mu}$-balanced if and only if $\frac{1}{2}$ is a critical place of $\operatorname{L}(s,\Pi\otimes\chi)$;*

• •

$\frac{1}{2}$* is a critical place of $\operatorname{L}(s,\Pi\otimes\chi)$ for some algebraic Hecke characters $\chi:{\mathrm{k}}^{\times}\backslash\mathbb{A}^{\times}\rightarrow\mathbb{C}^{\times}$.*

See **[JST19, Remark 2.23]**.

We identify the set of quadratic characters of ${\mathrm{k}}_{\infty}^{\times}$ with the set of characters $\widehat{\pi_{0}({\mathrm{k}}_{\infty}^{\times})}$ of the component group $\pi_{0}({\mathrm{k}}_{\infty}^{\times})$, so that $\chi^{\natural}\in\widehat{\pi_{0}({\mathrm{k}}_{\infty}^{\times})}$. Let $\varepsilon\in\widehat{\pi_{0}({\mathrm{k}}_{\infty}^{\times})}$. We introduce the following assumption for the pair $(\Pi,\varepsilon)$.

Assumption 1.3**.**

There exist $\sigma^{\prime}\in{\mathrm{Aut}}({\mathbb{C}})$ and an algebraic Hecke character $\chi^{\prime}$ of ${\mathrm{k}}^{\times}\backslash{\mathbb{A}}^{\times}$ such that $\chi^{\prime}_{\natural}$ is $F_{\mu}$-balanced, $\chi^{\prime\natural}=\varepsilon$ and

[TABLE]

Let us explain the meaning of Assumption 1.3. Note that the Blasius-Deligne conjecture is about the algebraicity of the critical values of $\operatorname{L}(s,\Pi\otimes\chi)$ and its reciprocity law. One may only consider that of the central value $\operatorname{L}(\frac{1}{2},\Pi\otimes\chi)$ because of the generality of the algebraic Hecke character $\chi$. If Assumption 1.3 fails, then $\operatorname{L}(\frac{1}{2},{}^{\sigma}\Pi\otimes{}^{\sigma}\chi)=0$ for all $\sigma\in{\mathrm{Aut}}({\mathbb{C}})$ and all algebraic Hecke characters $\chi$ such that $\chi_{\natural}$ is $F_{\mu}$-balanced and $\chi^{\natural}=\varepsilon$. Hence, at least when ${\mathrm{k}}$ contains no CM field, there is nothing to prove if Assumption 1.3 fails. Under Assumption 1.3, we are able to define a canonical family of Shalika periods as in Definition 10.3, which is the key step towards the formulation and the proof of Theorem 1.4 below, which is the Blasius-Deligne conjecture for this case. It may be important to point out that without Assumption 1.3, the definition of a canonical family of Shalika periods as in Definition 10.3 is currently unavailable when the underlying number field ${\mathrm{k}}$ has a complex local place, due to the appearance of multi-dimensional cohomology groups in the modular symbols. The main result of this paper is the following theorem.

Theorem 1.4** (Blasius-Deligne conjecture).**

*Let $\Pi$ be a regular algebraic irreducible cuspidal automorphic representation of ${\mathrm{GL}}_{2n}({\mathbb{A}})$ that is of symplectic type. For a given $\varepsilon\in\widehat{\pi_{0}({\mathrm{k}}_{\infty}^{\times})}$, the following reciprocity identity *

[TABLE]

holds for every $\sigma\in{\mathrm{Aut}}({\mathbb{C}})$ and every algebraic Hecke character $\chi$ of ${\mathrm{k}}^{\times}\backslash{\mathbb{A}}^{\times}$ such that $\chi_{\natural}$ is $F_{\mu}$-balanced and $\chi^{\natural}=\varepsilon,$ where

• •

$\Omega_{\mu,\chi_{\natural}}={\rm i}^{\sum_{\iota\in{\mathcal{E}}_{\mathrm{k}}}\sum^{n}_{i=1}(\mu^{\iota}_{i}+\operatorname{d}\!\chi_{\iota})}$* with ${\rm i}=\sqrt{-1}$;*

• •

${\mathcal{G}}(\chi)={\mathcal{G}}(\chi_{f})$* is the Gauss sum of $\chi$;*

• •

$\{\Omega_{\varepsilon}({}^{\sigma}\Pi,{}^{\sigma}\boldsymbol{\eta})\}_{\sigma\in{\mathrm{Aut}}(\mathbb{C})}$* is the family of Shalika periods in Definition 10.3.*

In particular,

[TABLE]

where ${\mathbb{Q}}(\Pi,\boldsymbol{\eta},\chi)$ is the composition of the rationality fields of $\Pi,\boldsymbol{\eta}$ and $\chi$.

The theorem has the following important consequence, the general conjecture of which is attributed to P. Deligne and some relevant progress on which can be found in [CK23].

Corollary 1.5**.**

With the notation and assumption as in Theorem 1.4, if $\operatorname{L}(\frac{1}{2},\Pi\otimes\chi)\neq 0$, then $\operatorname{L}(\frac{1}{2},{}^{\sigma}\Pi\otimes{}^{\sigma}\chi)\neq 0$ for all $\sigma\in{\mathrm{Aut}}({\mathbb{C}})$.

Here are some more detailed remarks regarding Theorem 1.4, which give an outline of the strategy and byproducts of its proof. The main result of [JST19] is the algebracity (1.6) when $\chi$ is of finite order. Theorem 1.4 is the first time to consider the Blasius-Deligne conjecture with general algebraic Hecke characters.

Among others, there are two technical key results needed for the formulation and the proof of Theorem 1.4: the nonvanishing of the Archimedean modular symbols and the Archimedean period relations. The methods in [JST19] and the current paper are quite different. In [JST19], both the nonvanishing of the Archimedean modular symbols and the Archimedean period relations are proved based on the explicit calculations of uniform cohomological test vectors in [CJLT20, LT20]. For the reciprocity law considered in Theorem 1.4, the nonvanishing of the Archimedean modular symbols can be deduced from the proofs in [JST19]. However, the results on the uniform cohomological test vectors in [CJLT20, LT20] are not enough to establish the refined Archimedean period relations (Theorem 2.16), which are needed for the reciprocity law in Theorem 1.4, by means of the arguments in [JST19].

In this paper we prove the refined Archimedean period relations (Theorem 2.16) via a robust application of Zuckerman translation functors and the method of modifying factors. This approach has been used in [LLS24] for the Rankin-Selberg case. The arguments in this paper combined with those in [LLS24] represent a new and more effective approach to the reciprocity law in the Blasius-Deligne conjecture for automorphic $L$-functions.

As proved in [JST19], the periods for this case considered in this paper (and in [JST19]) are defined in terms of the Friedberg-Jacquet local zeta integrals ([FJ93]). The definition of such integrals needs a local Shalika functional. In order to establish refined Archimedean period relations (Theorem 2.16), we need an explicitly normalized local Shalika functional to define explicit Friedberg-Jacquet local zeta integrals. We follow the approach by means of open-orbit integrals, as used in [LLS24], to construct such explicitly normalized local Shalika functionals by means of the Jacquet-Shalika local zeta integrals ([JS90]). Hence the first local result of this paper is to establish the Archimedean theory of Jacquet-Shalika integrals almost completely for ${\mathrm{GL}}_{m}$ with $m\geq 1$, which treats principal series representations of ${\mathrm{GL}}_{m}$ for all local fields (Theorem 2.2). Then we compare the local zeta integrals for the principal series representations as in Theorem 2.2 with the local integrals defined over the open-orbits when the relevant spherical subgroups acting on the flag variety.

This general open-orbit comparison method yields substantial arithmetic applications. In the Jacquet-Shalika case, it leads to the modifying factors in the sense of J. Coates and B. Perrin-Rion for exterior square $L$-functions (Theorem 2.6) compatible with the prediction for $p$-adic $L$-functions in [CPR89, C89]. Meanwhile, we also use the local Rankin-Selberg zeta integrals ([JPSS83]) and the local Godement-Jacquet zeta integrals ([GJ72]) to construct the different kind Shalika functionals, with which the open-orbit comparison method for the Friedberg-Jacquet local zeta integrals leads to the modifying factors for standard $L$-functions of symplectic type via Friedberg-Jacquet integrals (Theorem 2.15). The local theory of Jacquet-Shalika integrals in the even case gives an explicit realization of Shalika functionals (Theorem 2.11). As an application of modifying factors, we prove the Archimedean period relations for Friedberg-Jacquet integrals (Theorem 2.16) in terms of translation functors between regular algebraic representations. It is important to mention that those local results have interesting applications to arithmetic problems, including the theory of $p$-adic $L$-functions for higher rank groups and the methods to prove those local results could be extended to treat the arithmetic problems for more general automorphic $L$-functions.

This paper is organized as follows. In Section 2 we give a summary of the above local results with more detailed discussions. A large portion (Section 3–Section 6) is devoted to the local theory of Jacquet-Shalika integrals and the corresponding modifying factors, which is the most technical part of the paper. In brief, the novelty of our approach is to prove Theorem 2.2 and Theorem 2.6 together inductively, using Godement sections. In Section 7 we establish the modifying factors for Friedberg-Jacquet integrals, and we prove the Archimedean period relations in Section 8. We turn to the global setting in Section 9, where we introduce certain cohomology groups and the global and local modular symbols for Friedberg-Jacquet integrals. Finally in Section 10 we define the family of Shalika periods and prove the Blasius-Deligne conjecture (Theorem 1.4).

2. Main Local Results

In this section, we develop the local theory for relevant local zeta integrals, which form the main local results of this paper and the main ingredients to establish the refined Archimedean period relations for Friedberg-Jacquet integrals (Theorem 2.16). They will be established through Section 3 to Section 8.

2.1. Jacquet-Shalika integrals and modifying factors

We discuss the theory of local Jacquet-Shalika zeta integrals ([JS90]) and the associated local integrals from the open-orbit method. The goal is to construct refined explicit local Shalika functionals.

2.1.1. Representations and exterior square local factors

Assume that $\mathbbm{k}$ is an arbitrary local field, with normalized absolute value $|\cdot|_{\mathbbm{k}}$. For a connected reductive group $G$ over $\mathbbm{k}$, denote by $\mathrm{Irr}(G)$ the set of isomorphism classes of irreducible admissible representations of $G$, which are assumed to be Casselman-Wallach if $\mathbbm{k}$ is Archimedean. Let $\Pi_{2}(G)$ be the subset of square-integrable classes in $\mathrm{Irr}(G)$. More precisely, $\pi\in\mathrm{Irr}(G)$ is square-integrable if its central character is unitary and the absolute values of its matrix coefficients are functions in $L^{2}(G/Z)$, with $Z$ the center of $G$.

For a positive integer $m$, write $G_{m}:={\mathrm{GL}}_{m}(\mathbbm{k})$ and let $N_{m}$ be the upper triangular maximal unipotent subgroup of $G_{m}$. Fix a nontrivial unitary character $\psi$ of $\mathbbm{k}$, and define a character $\psi_{m}:N_{m}\to{\mathbb{C}}$ with $[x_{i,j}]_{m\times m}\mapsto\psi\left(\sum^{m-1}_{i=1}x_{i,i+1}\right)$. To shorten the notation, in this paper we write $\omega(g)=\omega(\det g)$ and $|g|_{\mathbbm{k}}=|\det g|_{\mathbbm{k}}$ for a character $\omega$ of $\mathbbm{k}^{\times}$ and $g\in G_{m}$.

We consider a representation of $G_{m}$ given by the normalized smooth parabolic induction

[TABLE]

where

• •

$P$ is a parabolic subgroup of $G_{m}$ with Levi subgroup

[TABLE]

• •

$\tau=\tau_{1}\,\widehat{\otimes}\,\tau_{2}\,\widehat{\otimes}\cdots\widehat{\otimes}\,\tau_{r}\in\Pi_{2}(M)$ and

• •

$\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{r})\in X^{*}(M)\otimes{\mathbb{C}}\cong{\mathbb{C}}^{r}$, where $X^{*}(M)$ is the character lattice of $M$.

Note that if $\mathbbm{k}$ is Archimedean, then in (2.1) one has that $n_{i}=1$ or $2$, $i=1,2,\dots,r$. The following facts are well-known:

• •

$\dim{\mathrm{Hom}}_{N_{m}}(\pi_{\lambda},\psi_{m})=1$.

• •

For fixed $\tau\in\Pi_{2}(M)$, $\pi_{\lambda}$ is irreducible for $\lambda$ outside a measure zero subset of ${\mathbb{C}}^{r}$.

• •

Any $\pi\in\mathrm{Irr}_{\rm gen}(G_{m})$, the subset of generic classes in $\mathrm{Irr}(G_{m})$, is isomorphic to an induced representation $\pi_{\lambda}$ of the form (2.1).

We will use the following notation: for $\lambda=(\lambda_{1},\lambda_{2},\dots\lambda_{r})\in{\mathbb{C}}^{r}$, write

[TABLE]

Following [BP21], $\pi_{\lambda}$ in (2.1) is called nearly tempered if $|\Re(\lambda_{i})|<1/4$ for all $i=1,2,\ldots,r$. It is known that nearly tempered representations $\pi_{\lambda}$ are irreducible.

For $\pi\in\mathrm{Irr}(G_{m})$, denote by $\phi_{\pi}$ the Langlands parameter of $\pi$ under the local Langlands correspondence, which is an $m$-dimensional admissible representation of the Weil-Deligne group $W_{\mathbbm{k}}^{\prime}$ of $\mathbbm{k}$. Fix a character $\eta$ of $\mathbbm{k}^{\times}$. We have the twisted exterior square local factors (see [CST17, Sh24])

[TABLE]

where the right hand sides are as in [T79]. For the parabolic induction $\pi_{\lambda}$ in (2.1), we have

[TABLE]

and $\varepsilon(s,\pi_{\lambda},\wedge^{2}\otimes\eta^{-1},\psi)$ and $\gamma(s,\pi_{\lambda},\wedge^{2}\otimes\eta^{-1})$ are similar.

By the compatibility of local Langlands correspondence with parabolic induction and unramified twists, if $\pi_{\lambda}^{0}$ denotes the unique Langlands subquotient of $\pi_{\lambda}$, then

[TABLE]

where the right hand sides are given by (2.3). In particular, $\eqref{exL}$ and $\eqref{exL2nt}$ coincide when $\pi_{\lambda}$ is irreducible.

2.1.2. Jacquet-Shalika integrals

We follow from [JS90]. Fix the self-dual Haar measure on $\mathbbm{k}$ with respect to $\psi$. For integers $n,n^{\prime}\geq 0$, denote by $\mathbbm{k}^{n\times n^{\prime}}$ the space of $n\times n^{\prime}$ matrices over $\mathbbm{k}$, and write $M_{n}:=\mathbbm{k}^{n\times n}$. We endow $\mathbbm{k}^{n\times n^{\prime}}$ with the product measure, and fix the Haar measure on $G_{n}$ to be $\operatorname{d}\!g=|g|_{\mathbbm{k}}^{-n}\cdot\prod_{i,j=1,2,\dots,n}\operatorname{d}\!g_{i,j}$ for $g=[g_{i,j}]_{n\times n}\in G_{n}$. For $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$, the space of Schwartz functions on $\mathbbm{k}^{n}:=\mathbbm{k}^{1\times n}$, define its Fourier transform with respect to a nontrivial unitary character $\psi^{\prime}$ of $\mathbbm{k}$ by

[TABLE]

Here and thereafter, ${}^{t}(\cdot)$ indicates the transpose of a matrix.

Assume that $m=2n$ or $2n+1$. The Shalika subgroup $S_{m}$ of $G_{m}$ is defined by

[TABLE]

which is a unimodular group. In the following we introduce a representation $R_{\varphi_{m}}$ of $S_{m}$, where $\varphi_{m}$ is a certain character determined by $\eta$ and $\psi$. Similarly, one can define a representation $R_{\varphi_{m}^{-1}}$, which will be omitted.

If $m=2n$ is even, we first define a character

[TABLE]

Let $S_{2n}$ act on $\mathbbm{k}^{n}$ from the right by

[TABLE]

Then we define a representation $R_{\varphi_{2n}}$ of $S_{2n}$ on ${\mathcal{S}}(\mathbbm{k}^{n})$ by

[TABLE]

where $h\in S_{2n}$ acts on $\mathbbm{k}^{n}$ as in (2.6).

If $m=2n+1$ is odd, we first define a character

[TABLE]

where $P_{m}$ denotes the mirabolic subgroup of $G_{m}$, i.e., the subgroup of matrices with last row $e_{m}:=(0,0,\dots,0,1)\in\mathbbm{k}^{m}.$ Then we define $R_{\varphi_{2n+1}}:={\rm ind}^{S_{2n+1}}_{S_{2n+1}\cap P_{2n}}\varphi_{2n+1}$ (the Schwartz induction), which is also realized on the space ${\mathcal{S}}(\mathbbm{k}^{n})$ (see Section 3.2 for details).

We identify the symmetric group $\mathfrak{S}_{m}$ with the group of permutation matrices in $G_{m}$, and introduce the following element of $\mathfrak{S}_{m}$,

[TABLE]

Assume that $\pi_{\lambda}$ is an induced representation of $G_{m}$ as in (2.1). Denote by ${\mathcal{W}}(\pi_{\lambda},\psi)$ the Whittaker model of $\pi_{\lambda}$ with respect to $(N_{m},\psi_{m})$. For $W\in{\mathcal{W}}(\pi_{\lambda},\psi)$, $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$ with $n=\lfloor m/2\rfloor$ and $s\in{\mathbb{C}}$, the Jacquet-Shalika integral introduced in [JS90] can be uniformly reformulated as

[TABLE]

where $e_{n}=(0,0,\ldots,0,1)\in\mathbbm{k}^{n}$ as above and $\overline{S}_{m}:=\sigma_{m}^{-1}N_{m}\sigma_{m}\cap S_{m}\backslash S_{m}.$ Here and thereafter, the Haar measures on $S_{m}$ and $N_{m}$ etc. are induced from the fixed Haar measures on $G_{n}$ and $\mathbbm{k}$, and $\overline{S}_{m}$ is equipped with the right invariant quotient measure. In general, we always take right invariant measures (when such measures exist) on locally compact topological groups and homogeneous spaces under the right actions of such groups in this paper.

Remark 2.1**.**

The integral (2.9) converges absolutely when $\Re(s)$ is sufficiently large, and its meromorphic continuation and functional equation were only proven for $\mathbbm{k}$ non-Archimedean and $\eta$ trivial (see [KR12, M14, CM15, Jo20]). However, it is not known whether the local exterior square $\varepsilon$-factors in the functional equation obtained in the non-Archimedean case are the same as the Artin local factors in (2.3) (see [CST17, Sh24]). Moreover, much less was known for the Archimedean case. We will establish the Archimedean theory of Jacquet-Shalika integrals almost completely, and our treatment of principal series representations is uniform for all local fields. In particular we will obtain the expected Artin local factors, which in general are crucial for arithmetic applications.

Let $w_{m}$ be the longest element of $\mathfrak{S}_{m}$, i.e., the $m\times m$ anti-diagonal permutation matrix. For $W\in{\mathcal{W}}(\pi_{\lambda},\psi)$, define $\widetilde{W}(h):=W(w_{m}{}^{t}h^{-1})$ for $h\in G_{m}$. Introduce the following element of $\mathfrak{S}_{m}$:

[TABLE]

Here and thereafter, $1_{n}$ denotes the $n\times n$ identity matrix. Denote by $\widehat{\mathbbm{k}^{\times}}$ the set of characters of $\mathbbm{k}^{\times}$, and for any $\omega\in\widehat{\mathbbm{k}^{\times}}$ let $\Re(\omega)$ be the real number (which is denoted by ${\rm ex}(\omega)$ in [LLSS23]) such that $|\omega(a)|=|a|_{\mathbbm{k}}^{\Re(\omega)}$ for $a\in\mathbbm{k}^{\times}$. Our first main result on the local theory of Jacquet-Shalika integrals is as follows.

Theorem 2.2** (${\rm FE}_{m}$).**

Assume that $\pi_{\lambda}={\mathrm{Ind}}^{G_{m}}_{P}(\tau_{\lambda})$ is an induced representation of $G_{m}$ as in (2.1), where $P$ is assumed to be a Borel subgroup if $\mathbbm{k}$ is non-archimedean. Let $W\in{\mathcal{W}}(\pi_{\lambda},\psi)$ and $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$ with $n=\lfloor m/2\rfloor$. Then the following hold.

1. (1)

$\operatorname{Z}_{\rm JS}(s,W,\phi,\varphi_{m}^{-1})$* converges absolutely when $\Re(s)>\Re(\eta)-2\min\Re(\lambda)$, and extends to a meromorphic function on ${\mathbb{C}}$.* 2. (2)

It holds the functional equation

[TABLE]

where

[TABLE] 3. (3)

The function

[TABLE]

has a holomorphic continuation to ${\mathbb{C}}$ which is of finite order in vertical strips (in the sense of **[BP21, 2.8]**). 4. (4)

If $\max\Re(\lambda)<\min\Re(\lambda)+1/2$, then for every $s_{0}\in{\mathbb{C}}$ there exist $W\in{\mathcal{W}}(\pi_{\lambda},\psi)$ and $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$ such that $\operatorname{Z}_{\rm JS}^{\circ}(s_{0},W,\phi,\varphi_{m}^{-1})\neq 0.$

In particular, we have the following:

• •

Theorem 2.2 holds for any $\pi\in\mathrm{Irr}_{\rm gen}(G_{m})$ when $\mathbbm{k}$ is Archimedean.

• •

If $\pi_{\lambda}\otimes|\eta|^{-\frac{1}{2}}$ is nearly tempered, where $|\eta|^{-\frac{1}{2}}$ indicates the character $|\eta(\det(\cdot))|^{\frac{1}{2}}$ of $G_{m}$, then the condition in Theorem 2.2 (4) clearly holds.

Remark 2.3**.**

In view of ${\mathcal{F}}_{\bar{\psi}}(\phi)(x)={\mathcal{F}}_{\psi}(\phi)(-x)$ and that

[TABLE]

for an admissible representation $\delta$ of the Weil-Deligne group $W_{\mathbbm{k}}^{\prime}$, it is easy to show that the functional equation (2.11) in Theorem 2.2 can be equivalently written as

[TABLE]

where $\omega_{\pi_{\lambda}}$ is the central character of $\pi_{\lambda}$. It seems that different conventions for the local $\varepsilon$-factors have been used in the literature. In this paper we stick to the convention in Tate’s classical treatments [T50, T79], which in the abelian case is given by (2.19).

2.1.3. Open orbit integrals and modifying factors

Our proof of Theorem 2.2 is purely local and uses the idea from [LLSS23] which studies the modifying factors for the Rankin-Selberg convolution case. The strategy is to compare the Jacquet-Shalika integrals of principal series representations with the integrals over the open orbit of the Shalika subgroup $S_{m}$ acting on a certain variety. Note that $S_{m}$ is a spherical subgroup of $G_{m}$.

Such a comparison in turn produces certain modifying factors, which are compatible in the non-Archimedean case with the conjecture for $p$-adic $L$-functions given by Coates and Perrin-Riou in [CPR89, C89]. This kind of phenomena has been observed for several families of periods (see [LSS21, LLSS23, LS25]). In particular, the Friedberg-Jacquet case has been established in [LS25], which leads to the construction of nearly ordinary standard $p$-adic $L$-functions of symplectic type. It will be established in a different setting later in this paper, the Archimedean case of which is crucial for our proof of the Archimedean period relations for Friedberg-Jacquet integrals (Theorem 2.16) and of the Blasius-Deligne conjecture for standard $L$-functions of symplectic type (Theorem 1.4).

The comparison in the Jacquet-Shalika case is carried out inductively via the theory of Godement sections. Thus we have labeled Theorem 2.2 as $({\rm FE}_{m})$ for the purpose of induction. To explain the details, we introduce an $S_{m}$-variety ${\mathcal{X}}_{m}$ as follows. Let $\overline{B}_{m}$ be the lower triangular Borel subgroup of $G_{m}$, and let ${\mathcal{B}}_{m}:=\overline{B}_{m}\backslash G_{m}$ be the flag variety on which $G_{m}$ acts from the right. Define ${\mathcal{X}}_{m}:={\mathcal{B}}_{m}\times\mathbbm{k}^{n}$ with $n=\lfloor m/2\rfloor$. We have specified a right action of $S_{m}$ on $\mathbbm{k}^{n}$ when $m$ is even in (2.6). If $m=2n+1$, then we have a right action of $S_{m}$ on $\mathbbm{k}^{n}$ given by

[TABLE]

The diagonal action of $S_{m}$ on ${\mathcal{X}}_{m}$ has a unique Zariski-open orbit, with a base point

[TABLE]

where

[TABLE]

Moreover, the stabilizer of $x_{m}$ in $S_{m}$ is trivial.

View an element $\xi=(\xi_{1},\xi_{2},\dots,\xi_{m})\in(\widehat{\mathbbm{k}^{\times}})^{m}$ as a character of $\overline{B}_{m}$ in the obvious way and put $I(\xi):={\rm Ind}^{G_{m}}_{\overline{B}_{m}}(\xi).$ For $f\in I(\xi)$, $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$ and $s\in{\mathbb{C}}$, formally define an integral

[TABLE]

where $v_{n}$ is given by (2.14). Denote by $W_{f}\in{\mathcal{W}}(I(\xi),\psi)$ the Whittaker function associated to $f$ and $\psi$ via the Jacquet integral

[TABLE]

in the sense of holomorphic continuation (see [W92, Theorem 15.4.1] for detailed explanation).

Define

[TABLE]

and for $\xi\in(\widehat{\mathbbm{k}^{\times}})^{m}$ define $\Omega_{\xi,\eta}:=\set{s\in{\mathbb{C}}}{(s,\xi)\in\Omega_{\eta}^{m}}$. Note that $\Omega_{\xi,\eta}$ may be empty. Put $\tilde{\xi}:=(\xi_{m}^{-1},\dots,\xi_{1}^{-1})$ and for $f\in I(\xi)$ define $\tilde{f}(h):=f(w_{m}{}^{t}h^{-1})$ for $h\in G_{m}.$ Note that $\tilde{f}\in I(\tilde{\xi})$ and $\widetilde{W_{f}}=W_{\tilde{f}}\in{\mathcal{W}}(I(\tilde{\xi}),\bar{\psi}).$ Here and thereafter, by abuse of notation we write $W_{\tilde{f}}$ for the Whittaker function associated to $\tilde{f}$ and $\bar{\psi}$, which should not cause any confusion.

The connected component ${\mathcal{M}}$ of $(\widehat{\mathbbm{k}^{\times}})^{m}$ containing $\xi$ is the set of all the unramified twists of $\xi$, which is a complex affine space of dimension $m$. A standard section on ${\mathcal{M}}$ is a map $\xi^{\prime}\mapsto f_{\xi^{\prime}}\in I(\xi^{\prime})$, $\xi^{\prime}\in{\mathcal{M}}$ such that $f_{\xi^{\prime}}|_{K_{m}}$ does not depend on $\xi^{\prime}$, where $K_{m}$ is the standard maximal compact subgroup of $G_{m}$. For any $f\in I(\xi)$, there is a unique standard section $\xi^{\prime}\mapsto f_{\xi^{\prime}}$ such that $f_{\xi}=f$.

The relevant analytic properties of $\Lambda_{\rm JS}(s,f,\phi,\varphi_{m}^{-1})$ are established in the following theorem.

Theorem 2.4** (${\rm FE}^{\prime}_{m}$).**

Let $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$ with $n=\lfloor m/2\rfloor$.

1. (1)

For $(s,\xi)\in\Omega_{\eta}^{m}$ and $f\in I(\xi)$, the integral $\Lambda_{\rm JS}(s,f,\phi,\varphi_{m}^{-1})$ in (2.15) converges absolutely, and it holds that

[TABLE]

where

[TABLE] 2. (2)

Let $\xi\mapsto f_{\xi}$ be a standard section on a connected component ${\mathcal{M}}$ of $(\widehat{\mathbbm{k}^{\times}})^{m}$. Then the function

[TABLE]

has a meromorphic continuation to ${\mathbb{C}}\times{\mathcal{M}}^{\circ}$, where

[TABLE]

In view of Theorem 2.2 and Theorem 2.6 below, the meromorphic continuation in Theorem 2.4 (2) in fact holds over ${\mathbb{C}}\times{\mathcal{M}}$. However we first need this weaker version, in order to prove Theorem 2.6.

For any subset $I$ of ${\mathbb{R}}$, write

[TABLE]

Remark 2.5**.**

It is easy to see that

1. (1)

$\Omega_{\tilde{\xi},\eta^{-1}}=\Set{1-s}{s\in\Omega_{\xi,\eta}}$. Thus the first assertion in Theorem 2.4 implies that the defining integral of $\Lambda_{\rm JS}(1-s,\tau_{m}.\tilde{f},\hat{\phi},\varphi_{m})$ also converges absolutely when $(s,\xi)\in\Omega_{\eta}^{m}$. 2. (2)

If $I(\xi)\otimes|\eta|^{-\frac{1}{2}}$ is nearly tempered and $\xi\in{\mathcal{M}}^{\circ}$, then there exists $\epsilon>0$ such that $\Omega_{\xi,\eta}\supset{\mathcal{H}}_{(\frac{1}{2}-\epsilon,\frac{1}{2}+\epsilon)}.$

For completeness, we recall the gamma factor

[TABLE]

for $\omega\in\widehat{\mathbbm{k}^{\times}}$ defined as in Tate’s thesis ([T50, K03]), which is holomorphic and non-vanishing when $-\Re(\omega)<\Re(s)<1-\Re(\omega)$. More precisely, the Tate integral

[TABLE]

where $\phi\in{\mathcal{S}}(\mathbbm{k})$ and $\operatorname{d}\!^{\times}a=|a|_{\mathbbm{k}}^{-1}\operatorname{d}\!a$, converges absolutely for $\Re(s)>-\Re(\omega)$. It has a meromorphic continuation to $s\in{\mathbb{C}}$ and satisfies a functional equation

[TABLE]

where both sides are holomorphic. We have the following basic facts:

• •

$\varepsilon(s,\omega,\bar{\psi})=\omega(-1)\varepsilon(s,\omega,\psi)$,

• •

$\gamma(1-s,\omega^{-1},\bar{\psi})\gamma(s,\omega,\psi)=\varepsilon(1-s,\omega^{-1},\bar{\psi})\varepsilon(s,\omega,\psi)=1$.

The Jacquet-Shalika integral $\operatorname{Z}_{\rm JS}(s,W_{f},\phi,\varphi_{m}^{-1})$ and the open orbit integral $\Lambda_{\rm JS}(s,f,\phi,\varphi_{m}^{-1})$ are related as follows.

Theorem 2.6** (${\rm MF}_{m}$).**

For $(s,\xi)\in\Omega^{m}_{\eta}$, $f\in I(\xi)$ and $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$ with $n=\lfloor m/2\rfloor$, it holds that

[TABLE]

2.1.4. The ideas of the proof

We will prove Theorem 2.4 (${\rm FE}^{\prime}_{m}$) in Section 5 using [LLSS23] and Tate’s thesis. Theorem 2.2 $({\rm FE}_{m})$ and Theorem 2.6 $({\rm MF}_{m})$ will be proved together inductively. Let us outline the strategy of the proof.

We first establish the basic analytic properties of Jacquet-Shalika integrals in Section 3, and reduce Theorem 2.2 to the case of principal series representations in the convergence range in Section 4, a large portion of which is parallel to the work [BP21] on the local zeta integrals for the local Asai $L$-functions. More precisely, we make a reduction to Theorem 4.2, which amounts to the functional equation (2.11) for $I(\xi)$ when $(s,\xi)\in\Omega^{m}_{\eta}$. In this case, on both sides of (2.11) the integrals are absolutely convergent and the $L$-functions are holomorphic. Theorem 4.2 will be also referred as (${\rm FE}_{m}$), and at this point it is clear that

[TABLE]

Applying the theory of Godement sections (see [J09]), we finish the main induction step

[TABLE]

in Section 6, which together with Section 5 forms the most essential and technical part of the proof.

As the starting point of the induction, we give the following low rank examples.

Example 2.7**.**

1. (1)

For $m=1$, all three theorems $({\rm FE}_{1})$, $({\rm FE}^{\prime}_{1})$ and $({\rm MF}_{1})$ are obviously trivial. 2. (2)

For $m=2$, we have $S_{2}=Z_{2}N_{2}$ where $Z_{2}$ is the center of $G_{2}$, and the elements $\sigma_{2}=z_{2}=1_{2}$ and $\tau_{2}=w_{2}$. In this case both $({\rm FE}_{2})$ and $({\rm FE}^{\prime}_{2})$ follow from Tate’s thesis for the character $\xi_{1}\xi_{2}\eta^{-1}$, while $({\rm MF}_{2})$ amounts to the Jacquet integral

[TABLE]

which converges absolutely when $\Re(\xi_{1})<\Re(\xi_{2})$.

Remark 2.8**.**

The work [BP21] on the Archimedean theory of the local zeta integrals for the local Asai $L$-functions uses global method, by choosing an auxiliary split place (for a quadratic extension of number fields) and reducing to the known Rankin-Selberg case $($[JPSS83, J09]$)$. This trick is unavailable for the Jacquet-Shalika case. The global method also relies on the comparison between the Langlands-Shahidi local factors and the Artin local factors. On the other hand, our approach is purely local, and the result on modifying factors has important arithmetic applications towards automorphic and $p$-adic $L$-functions.

2.2. Friedberg-Jacquet integrals and modifying factors

We now give the applications of Theorems 2.2, 2.4 and 2.6 towards twisted Shalika models and Friedberg-Jacquet integrals.

Definition 2.9**.**

Let $\xi=(\xi_{1},\xi_{2},\dots,\xi_{m})\in(\widehat{\mathbbm{k}^{\times}})^{m}$. We say that

1. (1)

$\xi$* is of Whittaker type if $I(\xi)$ has a unique irreducible generic quotient $\pi(\xi)$;* 2. (2)

$\xi$* is $\eta$-symmetric if $m=2n$ is even and $\xi_{1}\xi_{2n}=\xi_{2}\xi_{2n-1}=\cdots=\xi_{n}\xi_{n+1}=\eta.$*

Remark 2.10**.**

We have the following remarks regarding Definition 2.9.

1. (1)

If $\Re(\xi_{1})\geq\Re(\xi_{2})\geq\cdots\geq\Re(\xi_{m})$, then $\xi$ is of Whittaker type by $(1)$ and **[J09, Lemma 2.5]**, since we use the opposite Borel subgroup $\overline{B}_{m}$. 2. (2)

If $\xi$ is of Whittaker type, then $\tilde{\xi}$ is of Whittaker type as well and $\pi(\tilde{\xi})\cong\pi(\xi)^{\vee}$ by the properties of MVW involution $($[MVW87]$)$. 3. (3)

If $\xi\in(\widehat{\mathbbm{k}^{\times}})^{2n}$ is of Whittaker type, then

[TABLE]

are both of Whittaker type by the exactness of parabolic induction functor. If moreover $\xi$ is $\eta$-symmetric, then by $(3)$ it holds that $\pi(\xi^{2})\cong\pi(\xi^{1})^{\vee}\otimes\eta$.

Note that there is an $S_{2n}$-equivariant quotient map $\pi\,\widehat{\otimes}\,{\mathcal{S}}(\mathbbm{k}^{n})\twoheadrightarrow\pi$ induced by

[TABLE]

Our main result on twisted Shalika models is as follows.

Theorem 2.11**.**

Assume that $\xi\in(\widehat{\mathbbm{k}^{\times}})^{2n}$ is $\eta$-symmetric, and $I(\xi)$ has an irreducible generic quotient $\pi(\xi)$ such that $\pi(\xi)\otimes|\eta|^{-\frac{1}{2}}$ is nearly tempered. Then

1. (1)

$\operatorname{Z}^{\circ}_{\rm JS}(0,W,\phi,\varphi_{2n}^{-1})=0$* for all $W\in{\mathcal{W}}(\pi(\xi),\psi)$ and $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$ with $\phi(0)=0$;* 2. (2)

${\mathrm{Hom}}_{S_{2n}}(\pi(\xi),\varphi_{2n})\neq\{0\}$* and is spanned by the functional*

[TABLE]

where $\phi$ is an arbitrary element of ${\mathcal{S}}(\mathbbm{k}^{n})$ such that $\phi(0)=1$.

In the following we reinterpret the generator of ${\mathrm{Hom}}_{S_{2n}}(\pi(\xi),\varphi_{2n})$, which will be crucial for the study of modifying factors and the proof of Archimedean period relations for standard $L$-functions of symplectic type (Theorem 2.16) via the Friedberg-Jacquet local zeta integrals.

In view of Theorem 2.6, for $\xi\in(\widehat{\mathbbm{k}^{\times}})^{m}$ define the modified exterior square $L$-function

[TABLE]

Remark 2.12**.**

In the $p$-adic case, under certain slope conditions (nearly ordinary or non-critical slope) ${\mathcal{L}}(s,I(\xi),\wedge^{2}\otimes\eta^{-1})$ is expected to be the factor at $p$ of certain exterior square $p$-adic $L$-function, which justifies the notion of modifying factors.

Assume that $\xi\in(\widehat{\mathbbm{k}^{\times}})^{2n}$ and ${\mathcal{M}}$ is the connected component of $(\widehat{\mathbbm{k}^{\times}})^{2n}$ containing $\xi$. By Theorem 2.2 and Theorem 2.6, for any standard section $\xi^{\prime}\mapsto f_{\xi^{\prime}}$ on ${\mathcal{M}}$ and $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$, the function on ${\mathbb{C}}\times{\mathcal{M}}$ given by

[TABLE]

is holomorphic and coincides with

[TABLE]

However, the last function might vanish at $s=0$ and $\xi^{\prime}=\xi$. To remedy this issue, we introduce

[TABLE]

and denote by $d_{\xi}$ the order of $\Gamma(s,I(\xi),\wedge^{2}\otimes\eta^{-1},\psi)$ at $s=0$.

Proposition 2.13**.**

Keep the assumptions of Theorem 2.11. Let $\lambda_{\pi(\xi)}\in{\mathrm{Hom}}_{S_{2n}}(\pi(\xi),\varphi_{2n})$ be a generator. Then the following hold.

1. (1)

The functional

[TABLE]

is holomorphic and non-vanishing at $s=0$, and its value at $s=0$ factors through the quotient $I(\xi)\,\widehat{\otimes}\,{\mathcal{S}}(\mathbbm{k}^{n})\twoheadrightarrow I(\xi)$. 2. (2)

There is a unique $p_{\xi}\in{\mathrm{Hom}}_{G_{2n}}(I(\xi),\pi(\xi))$ such that $\lambda_{\pi(\xi)}\circ p_{\xi}=\lambda_{I(\xi)}$, where $\lambda_{I(\xi)}\in{\mathrm{Hom}}_{S_{2n}}(I(\xi),\varphi_{2n})$ is given by

[TABLE]

for an arbitrary element $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$ such that $\phi(0)=1$.

Using the twisted Shalika functional $\lambda_{\pi(\xi)}$ in the last proposition, we proceed to the Friedberg-Jacquet integrals introduced in [FJ93]. Let $\chi\in\widehat{\mathbbm{k}^{\times}}$. The Friedberg-Jacquet integral for $\pi(\xi)$ and $\chi$ is defined by

[TABLE]

It converges absolutely for $\Re(s)$ sufficiently large and extends to a holomorphic multiple of $\operatorname{L}(s,\pi(\xi)\otimes\chi)$ on the complex plane. By definition, if $f\in I(\xi)$ has image $v\in\pi(\xi)$, then

[TABLE]

Note that in this expression of the local Friedberg-Jacquet zeta integrals, the local Shalika functional $\lambda_{I(\xi)}$ is defined in Part (2) of Proposition 2.13, in terms of the local integral defined by the open-orbit method.

We now introduce another type of integrals, whose comparison with the Friedberg-Jacquet integral yields the modifying factors for standard $L$-functions of symplectic type. To this end, we first introduce certain Rankin-Selberg period. For a standard section $\xi^{\prime}\mapsto f_{\xi^{\prime}}$ on ${\mathcal{M}}$ and $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$, it follows easily from [LLSS23] that the function

[TABLE]

is holomorphic on $\Omega^{2n}_{\eta}\cap({\mathbb{C}}\times{\mathcal{M}})$ and has a meromorphic continuation to ${\mathbb{C}}\times{\mathcal{M}}$. As in Remark 2.10 (4), for $\xi=(\xi_{1},\xi_{2},\ldots,\xi_{2n})\in(\widehat{\mathbbm{k}^{\times}})^{2n}$ write $\xi^{1}=(\xi_{1},\xi_{2},\dots,\xi_{n})$.

Proposition 2.14**.**

Assume that $\xi\in(\widehat{\mathbbm{k}^{\times}})^{2n}$ is of Whittaker type and $\eta$-symmetric. Then the functional

[TABLE]

is holomorphic and non-vanishing at $s=0$, and its value at $s=0$ factors through the quotient $I(\xi)\,\widehat{\otimes}\,{\mathcal{S}}(\mathbbm{k}^{n})\twoheadrightarrow I(\xi)$.

Under the assumptions of Proposition 2.14, we have a nonzero functional $\lambda_{I(\xi)}^{\prime}$ in the space ${\mathrm{Hom}}_{G_{n}}(I(\xi),\eta)$ (viewing $G_{n}$ as a subgroup of $S_{2n}$) given by

[TABLE]

where $\phi$ is an arbitrary element of ${\mathcal{S}}(\mathbbm{k}^{n})$ such that $\phi(0)=1$. Let

[TABLE]

which is a spherical subgroup of $G_{2n}$. Let $\overline{Q}_{n}$ be the lower triangular maximal parabolic subgroup of $G_{2n}$ with Levi subgroup $H_{n}$. Then the right action of $H_{n}$ on the Grassmannian $\overline{Q}_{n}\backslash G_{2n}$ has a unique open orbit with a base point $\overline{Q}_{n}\gamma_{n}$, where

[TABLE]

and the stabilizer of $\overline{Q}_{n}\gamma_{n}$ in $H_{n}$ is $S_{2n}\cap H_{n}$, i.e., the diagonal $G_{n}$.

Consider the following space

[TABLE]

and for $f\in I(\xi)^{\sharp}$ introduce the integral

[TABLE]

The following is our main result on Friedberg-Jacquet integrals and the corresponding modifying factors.

Theorem 2.15**.**

Assume that $\xi\in(\widehat{\mathbbm{k}^{\times}})^{2n}$ is of Whittaker type and $\eta$-symmetric.

1. (1)

For $f\in I(\xi)^{\sharp}$, the integral $\Lambda_{\rm FJ}(s,f,\chi)$ converges absolutely and defines a holomorphic function of $s\in{\mathbb{C}}$. 2. (2)

For any $s_{0}\in{\mathbb{C}}$, there exists $f\in I(\xi)^{\sharp}$ such that $\Lambda_{\rm FJ}(s_{0},f,\chi)\neq 0$. 3. (3)

If moreover $\pi(\xi)\otimes|\eta|^{-\frac{1}{2}}$ is nearly tempered, then for $f\in I(\xi)^{\sharp}$ it holds that

[TABLE]

It is worth pointing out that the proof of Theorem 2.11, Propositions 2.13, 2.14 and Theorem 2.15, which will be given in Section 7, utilizes the strength of many ingredients such as the following:

• •

theory of Jacquet-Shalika integrals (Theorem 2.2) and the corresponding modifying factors (Theorem 2.6);

• •

theory of Rankin-Selberg integrals for ${\mathrm{GL}}_{n}\times{\mathrm{GL}}_{n}$ ([JPSS83, J09]) and the corresponding modifying factors ([LLSS23]);

• •

uniqueness of Rankin-Selberg periods ([SZ12, S12]);

• •

theory of Godement-Jacquet integrals ([GJ72]).

The key idea for the proof of Theorem 2.15 is to relate the Godement-Jacquet integrals for $G_{n}$ and the Friedberg-Jacquet integrals for $G_{2n}$. Such a relation has been used in [LS25] to evaluate the modifying factors for nearly ordinary standard $p$-adic $L$-functions of symplectic type as we mentioned earlier.

2.3. Archimedean period relations

Finally we give the application of Theorem 2.15 towards the Archimedean period relations for standard $L$-functions of symplectic type.

We set up some notation and refer to [JST19, LLS24] for more details. Assume that $\mathbbm{k}$ is Archimedean, and denote by ${\mathcal{E}}_{\mathbbm{k}}$ the set of continuous field embeddings $\iota:\mathbbm{k}\hookrightarrow{\mathbb{C}}$. For a subgroup $H$ of $G_{2n}$ defined over $\mathbb{R}$, denote $H_{\mathbb{C}}\subset G_{2n,{\mathbb{C}}}={\mathrm{GL}}_{2n}(\mathbbm{k}\otimes_{\mathbb{R}}{\mathbb{C}})$ its complexification.

Let $\mu=(\mu^{\iota})_{\iota\in{\mathcal{E}}_{\mathbbm{k}}}\in({\mathbb{Z}}^{2n})^{{\mathcal{E}}_{\mathbbm{k}}}$ be a pure weight in the sense of [Cl90], where $\mu^{\iota}=(\mu^{\iota}_{1},\mu^{\iota}_{2},\ldots,\mu^{\iota}_{2n})\in{\mathbb{Z}}^{2n}$. Then we have an irreducible algebraic representation $F_{\mu}$ of $G_{2n,{\mathbb{C}}}$ with highest weight $\mu$, and a unique irreducible generic essentially unitarizable Casselman-Wallach representation $\pi_{\mu}$ of $G_{2n}$, such that the total continuous cohomology

[TABLE]

where ${\mathbb{R}}^{\times}_{+}$ is the split component of the center of $G_{2n}$.

Assume that $\pi_{\mu}$ is of symplectic type, which is equivalent to that for each $\iota\in{\mathcal{E}}_{\mathbbm{k}}$, there exists $w_{\iota}\in{\mathbb{Z}}$ such that

[TABLE]

Put $\eta_{\mu}:=\otimes_{\iota\in{\mathcal{E}}_{\mathbbm{k}}}\iota^{w_{\iota}}$, which is a character of $(\mathbbm{k}\otimes_{\mathbb{R}}{\mathbb{C}})^{\times}$. By abuse of notation, also write $\eta_{\mu}$ for its restriction to $\mathbbm{k}^{\times}$. As is well-known, $\pi_{\mu}\otimes|\eta_{\mu}|^{-\frac{1}{2}}$ is tempered.

Fix $\psi$ to be the nontrivial unitary character of $\mathbbm{k}$ given by

[TABLE]

Let $\varphi_{2n,\mu}$ be the character of the Shalika subgroup $S_{2n}$ given by (2.5) using $\eta_{\mu}$ and $\psi$. Then by assumption, we have that ${\mathrm{Hom}}_{S_{2n}}(\pi_{\mu},\varphi_{2n,\mu})\neq\{0\}$. We fix a generator $\lambda_{\pi_{\mu}}$. Similar to (1.3), assume that $\chi$ is a character of $\mathbbm{k}^{\times}$ of the form $\chi=\chi_{\natural}|_{\mathbbm{k}^{\times}}\cdot\chi^{\natural}$, where $\chi_{\natural}=\bigotimes_{\iota\in{\mathcal{E}}_{\mathbbm{k}}}\iota^{\operatorname{d}\!\chi_{\iota}}$ and $\chi^{\natural}$ is quadratic. Using the fixed $\lambda_{\pi_{\mu}}$, as in (2.20), we have the normalized Friedberg-Jacquet integral

[TABLE]

As in [LLS24], we consider the principal series representation $I_{\mu}:={\mathrm{Ind}}^{G_{2n}}_{\overline{B}_{2n}}(\chi_{\mu}\rho_{2n}),$ where $\chi_{\mu}:=(\otimes_{\iota\in{\mathcal{E}}_{\mathbbm{k}}}\iota^{\mu^{\iota}_{1}},\dots,\otimes_{\iota\in{\mathcal{E}}_{\mathbbm{k}}}\iota^{\mu^{\iota}_{2n}})\in(\widehat{\mathbbm{k}^{\times}})^{2n}$ by restriction, and $\rho_{2n}$ is the square root of the modular character of the upper triangular Borel subgroup $B_{2n}$. Then $\chi_{\mu}\rho_{2n}$ is $\eta_{\mu}$-symmetric, and by [LLS24, Lemma 2.2] $I_{\mu}$ has a unique irreducible quotient which is isomorphic to $\pi_{\mu}$. Let $\lambda_{I_{\mu}}$ be the generator of ${\mathrm{Hom}}_{S_{2n}}(I_{\mu},\varphi_{2n,\mu})$ as in Proposition 2.13, so that there is a unique $p_{\mu}\in{\mathrm{Hom}}_{G_{2n}}(I_{\mu},\pi_{\mu})$ such that $\lambda_{\pi_{\mu}}\circ p_{\mu}=\lambda_{I_{\mu}}$.

All the above discussions apply to the zero weight $\mu=0$ case. In such a case $F_{0}$ is trivial. Let $\imath_{\mu}\in{\mathrm{Hom}}_{G_{2n}}(I_{0},I_{\mu}\otimes F_{\mu}^{\vee})$ be the explicit translation given in [LLS24, Section 2.2]. Then there is a unique $\jmath_{\mu}\in{\mathrm{Hom}}_{G_{2n}}(\pi_{0},\pi_{\mu}\otimes F_{\mu}^{\vee})$ making the following diagram commutative:

[TABLE]

Define the character $\xi_{\mu,\chi}:=\chi\boxtimes(\chi^{-1}\eta_{\mu}^{-1})$ of $H_{n}\cong G_{n}\times G_{n}$, and similar to (1.4) define the character $\xi_{\mu,\chi_{\natural}}:=\otimes_{\iota\in{\mathcal{E}}_{\mathbbm{k}}}({\det}^{\operatorname{d}\!\chi_{\iota}}\boxtimes{\det}^{-\operatorname{d}\!\chi_{\iota}-w_{\iota}})$ of $H_{n,{\mathbb{C}}}\cong G_{n,{\mathbb{C}}}\times G_{n,{\mathbb{C}}}$. Note that $\xi_{\mu,\chi}\otimes\xi_{\mu,\chi_{\natural}}^{\vee}=\chi^{\natural}\boxtimes\chi^{\natural}$ as a character of $H_{n}$. In particular $\xi_{\mu,\chi}\otimes\xi_{\mu,\chi_{\natural}}^{\vee}$ only depends on $\chi^{\natural}$. Assume that the $\chi_{\natural}$ is $F_{\mu}$-balanced in the sense of Definition 1.1. Let

[TABLE]

be the generator given in Lemma 8.1. The functional $\operatorname{Z}^{\circ}_{\rm FJ}(\frac{1}{2},\cdot,\chi)\otimes\lambda_{F_{\mu},\chi_{\natural}}$ induces the Archimedean modular symbol

[TABLE]

which is non-vanishing by [JST19, Theorem 3.11]. Here

[TABLE]

Applying Theorem 2.15, we obtain the following theorem, which will be proved in Section 8. It is clear that Theorem 2.16 refines [JST19, Theorem 3.12].

Theorem 2.16** (Archimedean Period Relation).**

Let the notation and assumption be as above. Then one has the following commutative diagram

[TABLE]

where $\Omega_{\mu,\chi_{\natural}}:={\rm i}^{\sum_{\iota\in{\mathcal{E}}_{\mathbbm{k}}}\sum^{n}_{i=1}(\mu^{\iota}_{i}+\operatorname{d}\!\chi_{\iota})}.$

3. Basic Properties of Jacquet-Shalika Integrals

3.1. Preliminaries on Whittaker functions

For preparations, we briefly recall some general results from [BP21]. Let $G$ be a quasi-split connected reductive group over a local field $\mathbbm{k}$. Denote by $A_{G}$ the maximal split torus in the center of $G$, and by $X^{*}(G)$ be the group of algebraic characters of $G$. Put

[TABLE]

Fix a Borel subgroup $B$ of $G$ with Levi decomposition $B=TN$, and write $A_{0}:=A_{T}$, ${\mathcal{A}}_{0}^{*}:={\mathcal{A}}_{T}^{*}$. Denote by $\delta_{B}$ the modular character of $B$. Fix a maximal compact subgroup $K$ of $G$ such that $G=BK$.

Let $\Delta\subset X^{*}(A_{0})$ be the set of simple roots of $A_{0}$ in $N$. As usual, for $\alpha\in\Delta$ denote by $\alpha^{\vee}$ the corresponding simple coroot. Define the closed negative Weyl chamber

[TABLE]

Let $W^{G}=N_{G}(T)/T$ be the Weyl group of $T$. For $\lambda\in{\mathcal{A}}_{0}^{*}$, denote by $|\lambda|$ the unique element in $W^{G}\lambda\cap\overline{({\mathcal{A}}_{0}^{*})^{+}}$. Define a partial order $\prec$ on ${\mathcal{A}}_{0}^{*}$ by

[TABLE]

Fix an algebraic group embedding $\imath:G/A_{G}\hookrightarrow G_{m}$ for some $m\geq 1$, and define the log-norm

[TABLE]

Let $\psi_{N}$ be a generic unitary character of $N$. For every $\lambda\in{\mathcal{A}}_{0}^{*}$, let ${\mathcal{C}}_{\lambda}(N\backslash G,\psi_{N})$ be the LF space of Whittaker functions on $G$ defined as in [BP21, 2.5], whose precise definition will not be recalled here.

We need the following estimate.

Lemma 3.1** (Lemma 2.5.1 of [BP21]).**

Let $\lambda\in{\mathcal{A}}_{0}^{*}$. For any $R,d>0$, there exists a continuous semi-norm $p_{R,d}$ on ${\mathcal{C}}_{\lambda}(N\backslash G,\psi_{N})$ such that

[TABLE]

for every $W\in{\mathcal{C}}_{\lambda}(N\backslash G,\psi_{N})$, $t\in T$ and $k\in K$.

For a standard parabolic subgroup $P=MU$ of $G$, the restriction map $X^{*}(M)\to X^{*}(T)$ induces an embedding ${\mathcal{A}}_{M}^{*}\hookrightarrow{\mathcal{A}}_{0}^{*}$. The restriction $X^{*}(A_{M})\to X^{*}(A_{G})$ induces surjections ${\mathcal{A}}_{M}^{*}\to{\mathcal{A}}_{G}^{*}$ and ${\mathcal{A}}_{M,{\mathbb{C}}}^{*}\to{\mathcal{A}}_{G,{\mathbb{C}}}^{*}$, whose kernels will be denoted by $({\mathcal{A}}^{G}_{M})^{*}$ and $({\mathcal{A}}_{M,{\mathbb{C}}}^{G})^{*}$ respectively. When $M=T$, we also write $({\mathcal{A}}^{G}_{0})^{*}:=({\mathcal{A}}^{G}_{T})^{*}$ and $({\mathcal{A}}^{G}_{0,{\mathbb{C}}})^{*}:=({\mathcal{A}}^{G}_{T,{\mathbb{C}}})^{*}$.

Fix $\tau\in\Pi_{2}(M)$ (or more generally an irreducible tempered representation of $M$), and for $\lambda\in{\mathcal{A}}_{M,{\mathbb{C}}}^{*}$ denote by $\tau_{\lambda}$ the unramified twist of $\tau$ by $\lambda$. Put $\pi_{\lambda}:={\mathrm{Ind}}^{G}_{P}(\tau_{\lambda})$ (normalized smooth induction). As in [BP21, 2.6], assume that $J_{\lambda}\in{\mathrm{Hom}}_{N}(\pi_{\lambda},\psi_{N})$ is a family of Whittaker functionals on $\pi_{\lambda}$, $\lambda\in{\mathcal{A}}_{M,{\mathbb{C}}}^{*}$ such that the map $\lambda\mapsto J_{\lambda}\in(\pi_{\lambda})^{\prime}$ is holomorphic in the sense of [BP21, 2.3]. Then we have a continuous $G$-equivariant linear map $\widetilde{J}_{\lambda}:\pi_{\lambda}\to C^{\infty}(N\backslash G,\psi_{N}),$ where the target is the space of all smooth functions $W:G\to{\mathbb{C}}$ such that $W(ug)=\psi_{N}(u)W(g)$ for any $u\in N$ and $g\in G$.

We recall Proposition 2.6.1 and Corollary 2.7.1 in [BP21] as follows.

Proposition 3.2**.**

Let the notation be as above.

1. (1)

For $\lambda\in{\mathcal{A}}_{M,{\mathbb{C}}}^{*}$ and $\mu\in{\mathcal{A}}_{0}^{*}$ such that $|\Re(\lambda)|\prec\mu$, the image of $\widetilde{J}_{\lambda}$ is contained in ${\mathcal{C}}_{\mu}(N\backslash G,\psi_{N})$ and the resulting linear map

[TABLE]

is continuous. 2. (2)

Let $\mu\in({\mathcal{A}}^{G}_{0})^{*}$ and ${\mathcal{U}}[\prec\mu]:=\set{\lambda\in({\mathcal{A}}^{G}_{M,{\mathbb{C}}})^{*}}{\,|\Re(\lambda)|\prec\mu}$. Then the family of continuous linear maps

[TABLE]

is analytic in the sense that for every analytic section $\lambda\mapsto e_{\lambda}\in\pi_{\lambda}$ (see **[BP21, 2.3]**) the resulting map

[TABLE]

is analytic. 3. (3)

For every $\lambda_{0}\in({\mathcal{A}}^{G}_{M,{\mathbb{C}}})^{*}$ and $W_{\lambda_{0}}\in{\mathcal{W}}(\pi_{\lambda_{0}},\psi_{N})$, there exists a map

[TABLE]

such that

• •

for every $\mu\in{\mathcal{A}}^{*}_{0}$ and $\lambda\in{\mathcal{U}}[\prec\mu]$, we have $W_{\lambda}\in{\mathcal{C}}_{\mu}(N\backslash G,\psi_{N})$ and the resulting map

[TABLE]

is analytic;

• •

$W_{\lambda_{0}}=W$.

3.2. Jacquet-Shalika integrals revisited

From now on assume that $G=G_{m}$. We recall the explicit formulation of Jacquet-Shalika integrals following [JS90, CM15].

Since the element $\tau_{m}$ given by (2.10) is fixed by the MVW involution $h\mapsto{}^{t}h^{-1}$ on $G_{m}$, the involution ${\rm Ad}(\tau_{m})$ and the MVW involution commutes. We introduce the following involution

[TABLE]

It is easy to check that the Shalika subgroup $S_{m}$ is stable under (3.2).

Recall the representation $R_{\varphi_{m}}$ of $S_{m}$ defined in Section 2.1.2. When $m=2n$ is even, as in [JS90] the Jacquet-Shalika integral (2.9) can be explicitly written as

[TABLE]

where $\mathfrak{q}_{n}$ denotes the space of upper triangular matrices in $M_{n}$.

For later use we give the following result.

Proposition 3.3**.**

It holds that $R_{\varphi_{2n}}(\hat{h}){\mathcal{F}}_{\psi}(\phi)=|h|_{\mathbbm{k}}^{\frac{1}{2}}\,{\mathcal{F}}_{\psi}(R_{\varphi_{2n}^{-1}}(h)\phi),$ where $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$, $h\in S_{2n}$ and $\hat{h}$ is given by (3.2).

Proof.

As before write $h=\begin{bmatrix}g&Xg\\ &g\end{bmatrix}$. Then $\hat{h}=\begin{bmatrix}{}^{t}g^{-1}&-{}^{t}X\,{}^{t}g^{-1}\\ &{}^{t}g^{-1}\end{bmatrix}.$ It is easy to check that $\varphi_{2n}(\hat{h})=\varphi_{2n}^{-1}(h)$. The proposition follows from (2.7) and that

[TABLE]

for $v\in\mathbbm{k}^{n}$, where $h.\phi(x):=\phi(x.h)=\phi(xg)$, $x\in\mathbbm{k}^{n}$. ∎

Next we elaborate the odd case. The following is a variant of Propositions 3.1 and 3.2 in [CM15].

Proposition 3.4**.**

(1)* The representation $R_{\varphi_{2n+1}}$ can be realized on the space ${\mathcal{S}}(\mathbbm{k}^{n})$ such that*

[TABLE]

where $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$, $g\in G_{n}$, $X\in M_{n}$, $y\in\mathbbm{k}^{n\times 1}$ and $x,v\in\mathbbm{k}^{1\times n}$.

(2)* It holds that $R_{\varphi_{2n+1}}(\hat{h}){\mathcal{F}}_{\bar{\psi}}(\phi)=|h|_{\mathbbm{k}}^{\frac{1}{2}}\,{\mathcal{F}}_{\bar{\psi}}(R_{\varphi_{2n+1}^{-1}}(h)\phi),$ where $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$, $h\in S_{2n+1}$ and $\hat{h}$ is given by (3.2).*

When $m=2n+1$ is odd, as in [CM15] the Jacquet-Shalika integral (2.9) can be explicitly written as

[TABLE]

To ease the notation, for a subgroup ${\mathcal{G}}$ of $G_{n}$ put

[TABLE]

where for $g\in G_{n}$ we write

[TABLE]

3.3. Convergence and continuity

Apply the discussion in Section 3.1 for the upper triangular Borel subgroup $B_{m}$ of $G_{m}$. Then ${\mathcal{A}}_{0}^{*}={\mathbb{R}}^{m}$ and the closed negative Weyl chamber is

[TABLE]

For $\lambda\in{\mathcal{A}}_{0}^{*}$, we have $|\lambda|=(\lambda_{w(1)},\ldots,\lambda_{w(m)})$ for any permutation $w\in\mathfrak{S}_{m}$ such that $\lambda_{w(1)}\leq\cdots\leq\lambda_{w(m)}$. Similar to (2.2), put $\min\lambda:=\min_{i=1,2,\dots,m}\lambda_{i}.$ We collect some more notation to be used later.

• •

Let $\delta_{m}$ be the modular character of $B_{m}=A_{m}N_{m}$, where $A_{m}$ is the diagonal torus, and let

[TABLE]

• •

Let $\bar{\mathfrak{v}}_{n}$ be the space of strictly lower triangular matrices in $M_{n}$, so that $M_{n}=\mathfrak{q}_{n}\oplus\bar{\mathfrak{v}}_{n}$.

• •

Let $K_{m}$ be the standard maximal compact subgroup ${\mathrm{O}}(m)$, ${\mathrm{U}}(m)$ or ${\mathrm{GL}}_{m}({\mathcal{O}}_{\mathbbm{k}})$ of $G_{m}$, for $\mathbbm{k}\cong{\mathbb{R}},{\mathbb{C}}$ or $\mathbbm{k}$ non-Archimedean with ring of integers ${\mathcal{O}}_{\mathbbm{k}}$, respectively.

• •

Recall the mirabolic $P_{m}$ of $G_{m}$. Let $U_{m}$ be the unipotent radical of $P_{m}$, and let $\overline{U}_{m}={}^{t}U_{m}$. Let $Z_{m}$ be the center of $G_{m}$.

For $W\in C^{\infty}(N_{m}\backslash G_{m},\psi_{m})$ and $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$ with $n=\lfloor m/2\rfloor$, formally define the integral $\operatorname{Z}_{\rm JS}(s,W,\phi,\varphi_{m}^{-1})$ by (2.9). Recall the notation ${\mathcal{H}}_{I}$, $I\subset{\mathbb{R}}$ in (2.18). A vertical strip is a subset of ${\mathbb{C}}$ of the form ${\mathcal{V}}={\mathcal{H}}_{I}$ for a finite closed interval $I\subset{\mathbb{R}}$.

In view of Proposition 3.2, we start from the following result.

Proposition 3.5**.**

Let $\mu\in{\mathcal{A}}_{0}^{*}$, $W\in{\mathcal{C}}_{\mu}(N_{m}\backslash G_{m},\psi_{m})$ and $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$ with $n=\lfloor m/2\rfloor$. Then the following hold.

1. (1)

The integral $\operatorname{Z}_{\rm JS}(s,W,\phi,\varphi_{m}^{-1})$ converges absolutely for all $s\in{\mathcal{H}}_{(\Re(\eta)-2\min\mu,\infty)}$. 2. (2)

The function $s\mapsto\operatorname{Z}_{\rm JS}(s,W,\phi,\varphi_{m}^{-1})$ is holomorphic and bounded in vertical strips on ${\mathcal{H}}_{(\Re(\eta)-2\min\mu,\infty)}$. More precisely, for any vertical strip ${\mathcal{V}}\subset{\mathcal{H}}_{(\Re(\eta)-2\min\mu,\infty)}$, there exist continuous semi-norms $p_{{\mathcal{V}}}$ on ${\mathcal{C}}_{\mu}(N_{m}\backslash G_{m},\psi_{m})$ and $q_{\mathcal{V}}$ on ${\mathcal{S}}(\mathbbm{k}^{n})$ such that $\operatorname{Z}_{\rm JS}(s,W,\phi,\varphi_{m}^{-1})$, with integrand replaced by its absolute value, is bounded by $p_{{\mathcal{V}}}(W)q_{\mathcal{V}}(\phi)$ for any $W\in{\mathcal{C}}_{\mu}(N_{m}\backslash G_{m},\psi_{m})$, $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$ and $s\in{\mathcal{V}}$. In particular the family of functions

[TABLE]

on ${\mathcal{C}}_{\mu}(N_{m}\backslash G_{m},\psi_{m})\times{\mathcal{S}}(\mathbbm{k}^{n})$ indexed by $s\in{\mathcal{V}}$ are equicontinuous.

Proof.

We only prove the case that $m=2n$ is even. The odd case can be proved similarly with suitable modifications using the proof of Proposition 3 in [JS90, Section 9], which will be omitted.

By unramified twists, we may assume that $\eta$ is unitary so that $\Re(\eta)=0$, and that $s\in{\mathbb{R}}$. By the Iwasawa decomposition $G_{n}=N_{n}A_{n}K_{n}$, we need to estimate the integral

[TABLE]

For $X\in M_{n}$, introduce the element

[TABLE]

Then the above integral can be written as

[TABLE]

where for $a=\operatorname{diag}\{a_{1},a_{2},\ldots,a_{n}\}\in A_{n}$ we set

[TABLE]

We write $u_{X}=n_{X}t_{X}k_{X}\in N_{2n}A_{2n}K_{2n}$, where $t_{X}=\operatorname{diag}\{t_{1},\ldots,t_{2n}\}\in A_{2n}$, following the Iwasawa decomposition. The above integral is

[TABLE]

For each $R>0$ we have the following continuous semi-norm on ${\mathcal{S}}(\mathbbm{k}^{n})$,

[TABLE]

It is straightforward to verify that $\delta_{2n}(\tilde{a})^{1/2}=\delta_{n}(a)^{2}$. Thus by Lemma 3.1, we are reduced to estimate

[TABLE]

where we write $|\mu|=(|\mu|_{1},\ldots,|\mu|_{2n})$. After a suitable translation of the $a_{i}$’s, we are reduced to estimate a product of two integrals

[TABLE]

where $\mu_{s}$ is a positive character of $A_{2n}$ depending on $s$ and $\mu$, and

[TABLE]

By Propositions 4 and 5 in [JS90, Section 5], there exists $\alpha>0$ such that

[TABLE]

where $m(X):=\sqrt{1+\|X\|}$ or $\sup(1,\|X\|)$ for $\mathbbm{k}$ Archimedean or non-Archimedean respectively, and $\|\cdot\|$ is the standard norm on $M_{n}$. Note that $m(X)$ can be also replaced by $e^{\bar{\sigma}(u_{X})}$ where $\bar{\sigma}$ is the log-norm (3.1). Since $\mu_{s}(t_{X})$ is of polynomial growth in $X$, given any finite interval $I\subset{\mathbb{R}}$, when $R$ is sufficiently large the integral (3.7) converges uniformly for $s\in I$.

The integral (3.8) can be estimated in the same way as in the proof of [BP21, Lemma 3.3.1]. By the elementary inequality

[TABLE]

and given each $r\in{\mathbb{R}}$ the locally uniform convergence of the integral

[TABLE]

for $R/n-r>s>-r$, we find that (3.8) converges locally uniformly for $R/n-2\max\mu>s>-2\min\mu$.

Combining the discussions for (3.7) and (3.8), the proposition follows easily by noting that separately continuous maps on LF spaces are continuous. ∎

The following result gives the absolute convergence in Theorem 2.2 (1), which holds in general without assuming that $P$ is a Borel subgroup for $\mathbbm{k}$ non-Archimedean.

Proposition 3.6**.**

Let $\pi_{\lambda}={\mathrm{Ind}}^{G_{m}}_{P}(\tau_{\lambda})$ be given by (2.1). Then the following hold.

1. (1)

Proposition 3.5 holds with ${\mathcal{C}}_{\mu}(N_{m}\backslash G_{m},\psi_{m})$ replaced by ${\mathcal{W}}(\pi_{\lambda},\psi)$ and $\min\mu$ replaced by $\min\Re(\lambda)\in{\mathcal{A}}^{*}_{M}\subset{\mathcal{A}}^{*}_{0}$. 2. (2)

If $\pi_{\lambda}\otimes|\eta|^{-\frac{1}{2}}$ is nearly tempered, then there is an $\epsilon>0$ so that $\operatorname{Z}_{\rm JS}(s,W,\phi,\varphi_{m}^{-1})$ converges absolutely and defines a holomorphic function on ${\mathcal{H}}_{(\frac{1}{2}-\epsilon,\infty)}$ bounded in vertical strips, for any $W\in{\mathcal{W}}(\pi_{\lambda},\psi)$ and $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$ with $n=\lfloor m/2\rfloor$.

Proof.

The proof is similar to that of [BP21, Lemma 3.3.2], and we repeat the arguments for completeness.

Let ${\mathcal{V}}\subset{\mathcal{H}}_{(\Re(\eta)-2\min\Re(\lambda),\infty)}$ be a vertical strip. We have $|\Re(\lambda)|\prec|\Re(\lambda)|+\varepsilon\rho$ for every $\varepsilon>0$. Clearly, we have that ${\mathcal{V}}\subset{\mathcal{H}}_{(\Re(\eta)-2\min(\Re(\lambda)+\varepsilon\rho),\infty)}$ for sufficiently small $\varepsilon>0$. Proposition 3.2 implies that ${\mathcal{W}}(\pi_{\lambda},\psi)\subset{\mathcal{C}}_{|\Re(\lambda)|+\varepsilon\rho}(N_{m}\backslash G_{m},\psi_{m}),$ from which (1) follows.

For (2), again by unramified twists we may assume that $\pi$ is nearly tempered and that $\eta$ is unitary, so that $|\Re(\lambda_{i})|<1/4$ for all $i$. The required assertion follows easily from (1) and that $-2\min\Re(\lambda)<1/2$. ∎

3.4. A non-vanishing result

We give the following non-vanishing result.

Proposition 3.7**.**

Let $\pi\in\mathrm{Irr}_{\rm gen}(G_{m})$. For every $s_{0}\in{\mathbb{C}}$, there exist finitely many $W_{i}\in{\mathcal{W}}(\pi,\psi)$ and $\phi_{i}\in{\mathcal{S}}(\mathbbm{k}^{n})$ with $n=\lfloor m/2\rfloor$ indexed by $i\in I$, such that the function

[TABLE]

which is defined for $\Re(s)$ sufficiently large, has a holomorphic extension to ${\mathbb{C}}$ and is non-vanishing at the given $s_{0}\in{\mathbb{C}}$.

Proof.

Again we only give the proof for the case that $m=2n$ is even, which is similar to that of [BP21, Lemma 3.3.3], and omit the odd case.

Note that $P_{n}Z_{n}\overline{U}_{n}\subset G_{n}$ is open dense. By Proposition 3.6, for $W\in{\mathcal{W}}(\pi,\psi)$, $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$ and $\Re(s)$ sufficiently large we have the absolutely convergent integral

[TABLE]

where $u_{X}$ is as in (3.6) and $\omega_{\pi}$ is the central character of $\pi$. For $\varphi_{Z}\in C^{\infty}_{c}(Z_{n})$ and $\varphi_{\overline{U}}\in C^{\infty}_{c}(\overline{U}_{n})$, there is a unique $\phi=\phi_{\varphi_{Z},\varphi_{\overline{U}}}\in C^{\infty}_{c}(\mathbbm{k}^{n})$ such that $\phi(e_{n}z\bar{u})=\varphi_{Z}(z)\varphi_{\overline{U}}(\bar{u})$ for all $(z,\bar{u})\in Z_{n}\times\overline{U}_{n}$. By abuse of notation, view $\varphi_{\overline{U}}$ as a function on $\overline{U}_{n}^{\dagger}$. Then for the above $\phi$ and $\Re(s)$ sufficiently large we have

[TABLE]

where $R(\varphi_{\overline{U}})$ denotes the right regular action. The Tate integral

[TABLE]

converges absolutely for all $s\in{\mathbb{C}}$, and we can choose $\varphi_{Z}$ such that the $\zeta(s_{0},\varphi_{Z})\neq 0$.

It is known that for any $f\in C^{\infty}_{c}(N_{2n}\backslash P_{2n},\psi_{2n})$, there exists $W_{0}\in{\mathcal{W}}(\pi,\psi)$ whose restriction to $P_{2n}$ coincides with $f$. By the Dixmier-Malliavin lemma, there exist finitely many $W_{i}\in{\mathcal{W}}(\pi,\psi)$ and $\varphi_{\overline{U},i}\in C^{\infty}_{c}(\overline{U}_{n})$, indexed by $i\in I$, such that $W_{0}=\sum_{i\in I}R(\varphi_{\overline{U},i})W_{i}$. Put $\phi_{i}:=\phi_{\varphi_{Z},\varphi_{\overline{U},i}}$, $i\in I$. Then for $\Re(s)$ sufficiently large we have that

[TABLE]

noting that $u_{X}\sigma_{2n}p^{\dagger}\in P_{2n}$. The above integrals converge absolutely for all $s\in{\mathbb{C}}$, uniformly on compacta, hence define a holomorphic function on ${\mathbb{C}}$. We can choose $f$ such that

[TABLE]

The holomorphic continuation of $\sum_{i\in I}\operatorname{Z}_{\rm JS}(s,W_{i},\phi_{i},\varphi_{2n}^{-1})$ does not vanish at $s_{0}$, since we have chosen $\varphi_{Z}$ such that $\zeta(s_{0},\varphi_{Z})\neq 0$. ∎

4. Reductions of $({\rm FE}_{m})$

In this short section we make a few reductions of Theorem 2.2, which ultimately lead to Theorem 4.2 for principal series representations.

4.1. Reductions of inducing data

4.1.1. Reduction of spectral parameters

Without loss of generality, assume that $\eta$ is unitary. We first show that for a fixed $\tau\in\Pi_{2}(M)$, Theorem 2.2 for an arbitrary $\pi_{\lambda_{0}}$ can be reduced to the case for nearly tempered representations $\pi_{\lambda}$ with $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{r})\in{\mathcal{A}}_{M,{\mathbb{C}}}^{*}$ satisfying the condition: $\Re(\lambda_{1})<\Re(\lambda_{2})<\cdots<\Re(\lambda_{r}).$ The arguments are the same as in [BP21, 3.10] and we give a sketch for completeness. Note that this reduction holds in general, with no extra assumption on $P$ for $\mathbbm{k}$ non-Archimedean.

We may assume that $\lambda_{0}\in({\mathcal{A}}^{G_{m}}_{M,{\mathbb{C}}})^{*}$. Let $W\in{\mathcal{W}}(\pi_{\lambda_{0}},\psi_{m})$ and $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$. Let $\mu\in{\mathcal{A}}^{*}_{0}$ such that $\lambda_{0}\in{\mathcal{U}}[\prec\mu]$, and choose an analytic section

[TABLE]

as in Proposition 3.2 with $W_{\lambda}\in W(\pi_{\lambda},\psi)$ and $W_{\lambda_{0}}=W$.

There exist constants $u\in{\mathbb{C}}^{\times}$ and $C\in\mathbb{R}^{\times}_{+}$, and a linear form $L$ on $({\mathcal{A}}^{G}_{M,{\mathbb{C}}})^{*}$ such that

[TABLE]

Take a square root $v$ of $u$ and put

[TABLE]

so that $\eta(-1)^{mn}\varepsilon(s,\pi_{\lambda},\wedge^{2}\otimes\eta^{-1},\psi)=\epsilon_{1/2}(s,\pi_{\lambda},\wedge^{2}\otimes\eta^{-1},\psi)^{2}$. Define

[TABLE]

which are a priori partially defined on ${\mathbb{C}}\times({\mathcal{A}}^{G}_{M,{\mathbb{C}}})^{*}$ by Proposition 3.6. Set

[TABLE]

which is a nonempty relatively compact connected open subset of $({\mathcal{A}}^{G_{m}}_{M,{\mathbb{C}}})^{*}$. Then $\pi_{\lambda}$, $\lambda\in U$, are nearly tempered. By Proposition 3.6, $\operatorname{Z}_{+}(s,\lambda)$ and $\operatorname{Z}_{-}(s,\lambda)$ are defined on ${\mathcal{H}}_{[\frac{1}{2},\infty)}\times U$.

Assume that Theorem 2.2 holds for $\pi_{\lambda}$, $\lambda\in U$. Then $\operatorname{Z}_{+}(s,\lambda)$ and $\operatorname{Z}_{-}(s,\lambda)$ admit holomorphic continuations to ${\mathbb{C}}\times U$, which are of finite order in vertical strips in the first variable and locally uniform in the second variable (see [BP21, 2.8]) and satisfy the functional equation

[TABLE]

For a relatively compact connected open subset $U^{\prime}\subset({\mathcal{A}}^{G_{m}}_{M,{\mathbb{C}}})^{*}$ containing $U$, there exists $\mu\in{\mathcal{A}}^{*}_{0}$ such that $U^{\prime}\subset{\mathcal{U}}[\prec\mu]$. By Proposition 3.5, $\operatorname{Z}_{+}(s,\lambda)$ and $\operatorname{Z}_{+}(s,\lambda)$ admit holomorphic continuations to ${\mathcal{H}}_{(D,\infty)}\times U^{\prime}$ for sufficiently large $D\in{\mathbb{R}}$ which are of finite order in vertical strips in the first variable and locally uniform in the second variable. Hence by [BP21, Proposition 2.8.1], $\operatorname{Z}_{+}(s,\lambda)$ and $\operatorname{Z}_{+}(s,\lambda)$ extend to holomorphic functions on ${\mathbb{C}}\times({\mathcal{A}}^{G_{m}}_{M,{\mathbb{C}}})^{*}$ of finite order in vertical strips in the first variable and locally uniform in the second variable such that (4.1) holds on ${\mathbb{C}}\times({\mathcal{A}}^{G_{m}}_{M,{\mathbb{C}}})^{*}$.

By the definitions of $W_{\lambda}$ and $\operatorname{Z}_{\pm}(s,\lambda)$, specializing to $\lambda=\lambda_{0}$ shows that Theorem 2.2 (1), (2) and (3) hold for $\pi_{\lambda_{0}}$. The following general statement implies that Theorem 2.2 (4) holds when $\max\Re(\lambda_{0})<\min\Re(\lambda_{0})+1/2$.

Lemma 4.1**.**

Assume that $\pi_{\lambda}={\mathrm{Ind}}^{G_{m}}_{P}(\tau_{\lambda})$ is as in (2.1) such that

[TABLE]

For $(a,b)=(\Re(\eta)-2\min\Re(\lambda),\Re(\eta)+1-2\max\Re(\lambda))$, if (2.11) holds when $s$ lies in a nonempty open subset of ${\mathcal{H}}_{(a,b)},$ then Theorem 2.2 holds for $\pi_{\lambda}$.

Proof.

By Proposition 3.6 and standard properties of Artin $L$-functions,

[TABLE]

are holomorphic on ${\mathcal{H}}_{(\Re(\eta)-2\min\Re(\lambda),\infty)}$ and ${\mathcal{H}}_{(-\infty,\Re(\eta)+1-2\max\Re(\lambda))}$ respectively, of finite order in vertical strips. Thus Theorem 2.2 (1), (2) and (3) hold by the uniqueness of holomorphic continuation. By Proposition 3.7, for $s_{0}\in{\mathcal{H}}_{(\Re(\eta)-2\min\Re(\lambda),\infty)}$ (resp. $s_{0}\in{\mathcal{H}}_{(-\infty,\Re(\eta)+1-2\max\Re(\lambda))}$), there exist $W\in{\mathcal{W}}(\pi_{\lambda},\psi)$ and $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$ such that

[TABLE]

It follows that Theorem 2.2 (4) holds as well. ∎

4.1.2. Reduction to principal series representations

Next we show that when $\mathbbm{k}$ is Archimedean, Theorem 2.2 can be reduced to the case that $P$ is a Borel subgroup, so that $\pi_{\lambda}$ is isomorphic to a principal series representation of the form $I(\xi)$ with $\xi\in(\widehat{\mathbbm{k}^{\times}})^{m}$.

By the above reduction, we may assume that $\pi_{\lambda}\otimes|\eta|^{-\frac{1}{2}}$ is nearly tempered. Suppose that $P$ is lower triangular of type $(n_{1},n_{2},\ldots,n_{r})$ with $n_{i}=1$ or $2$ for $i=1,2,\dots,r$. We may realize each $\tau_{i}|\cdot|_{\mathbbm{k}}^{\lambda_{i}}$ as a quotient of a principal series representation $I(\xi^{i})$ where $\xi^{i}\in(\widehat{\mathbbm{k}^{\times}})^{n_{i}}$. Then $\pi_{\lambda}$ is isomorphic to a quotient of $I(\xi)$ where $\xi=(\xi^{1},\xi^{2},\dots,\xi^{r})\in(\widehat{\mathbbm{k}^{\times}})^{m}$, and from the irreducibility of $\pi_{\lambda}$ we see that $\pi_{\lambda}^{\vee}$ is isomorphic to a quotient of $I(\tilde{\xi})=I(\tilde{\xi}^{r},\dots,\tilde{\xi}^{2},\tilde{\xi}^{1})$. Using standard results on the admissible representations of $W_{\mathbbm{k}}^{\prime}$ and the local factors in the Archimedean case, it is straightforward to check that

[TABLE]

Let $W\in{\mathcal{W}}(\pi_{\lambda},\psi)={\mathcal{W}}(I(\xi),\psi)$ so that $\widetilde{W}\in{\mathcal{W}}(\pi_{\lambda}^{\vee},\bar{\psi})={\mathcal{W}}(I(\tilde{\xi}),\bar{\psi})$, and let $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$. By Proposition 3.6, there exists $0<\epsilon<\frac{1}{4}$ such that both $\operatorname{Z}_{\rm JS}(s,W,\phi,\varphi_{m}^{-1})$ and $\operatorname{Z}_{\rm JS}(1-s,\tau_{m}.\widetilde{W},\hat{\phi},\varphi_{m})$ converge absolutely when $s\in{\mathcal{H}}_{(\frac{1}{2}-\epsilon,\frac{1}{2}+\epsilon)}$. Moreover, both $\operatorname{L}(s,\pi_{\lambda},\wedge^{2}\otimes\eta^{-1})$ and $\operatorname{L}(1-s,\pi_{\lambda}^{\vee},\wedge^{2}\otimes\eta)$ are holomorphic on ${\mathcal{H}}_{(\frac{1}{2}-\epsilon,\frac{1}{2}+\epsilon)}$. Thus in view of Lemma 4.1 and (4.2), if Theorem 2.2 holds for $I(\xi)$, then it holds for $\pi_{\lambda}$ as well.

4.2. $({\rm MF}_{m})+({\rm FE}^{\prime}_{m})\Rightarrow({\rm FE}_{m})$

By the above reductions, to prove Theorem 2.2 it suffices to consider a principal series representation $I(\xi)$, where $\xi\in(\widehat{\mathbbm{k}^{\times}})^{m}$ such that

[TABLE]

Clearly (4.3) is equivalent to that $\Omega_{\xi,\eta}\neq\varnothing$, and we note that every $\gamma(s,\xi_{i}\xi_{j}\eta^{-1},\psi)$, where $i,j=1,2,\dots,m$, is holomorphic and non-vanishing on $\Omega_{\xi,\eta}$.

In view of Lemma 4.1, to complete the proof of Theorem 2.2 it remains to establish the following result, which will be also referred as (${\rm FE}_{m}$) from now on.

Theorem 4.2** (${\rm FE}_{m}$).**

For $(s,\xi)\in\Omega^{m}_{\eta}$, $f\in I(\xi)$ and $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$ with $n=\lfloor m/2\rfloor$, it holds that

[TABLE]

where

[TABLE]

It is straightforward to verify that Theorem 2.6 (${\rm MF}_{m}$) and Theorem 2.4 (${\rm FE}^{\prime}_{m}$) imply Theorem 4.2 (${\rm FE}_{m}$). These three theorems will be proved in the next two sections.

5. Proof of $({\rm FE}_{m}^{\prime})$

In this section we prove Theorem 2.4 $({\rm FE}_{m}^{\prime})$. To prove the absolute convergence and meromorphic continuation, we use the results for Rankin-Selberg integrals in [LLSS23]. To prove the functional equation, the basic idea is to apply Tate’s thesis for a maximal torus in $S_{m}$ which can be conjugated into $\overline{B}_{m}$ by the element $z_{m}$. The diagonal torus works when $m$ is even, but for the odd case one has to take a conjugation of the diagonal torus in $S_{m}$.

5.1. Convergence and continuation

We first prove that for a standard section $\xi\mapsto f_{\xi}$ on a connected component ${\mathcal{M}}$ of $(\widehat{\mathbbm{k}^{\times}})^{m}$, the integral $\Lambda_{\rm JS}(s,f_{\xi},\phi,\varphi_{m}^{-1})$ given by (2.15) converges absolutely when $(s,\xi)\in\Omega^{m}_{\eta}\cap({\mathbb{C}}\times{\mathcal{M}})$ and has a meromorphic continuation to ${\mathbb{C}}\times{\mathcal{M}}^{\circ}$.

First assume that $m=2n$ is even. Then

[TABLE]

By the standard theory of intertwining operators, when $\xi\in{\mathcal{M}}^{\circ}$ the integral

[TABLE]

converges absolutely hence defines an element of $I(\xi^{1})\,\widehat{\otimes}\,I(\xi^{2})$, where $\xi^{1},\xi^{2}\in(\widehat{\mathbbm{k}^{\times}})^{n}$ are as in Remark 2.10 (4).

It is easy to check that $(\overline{B}_{n},\overline{B}_{n}w_{n},v_{n})$ is a base point of the unique open $G_{n}$-orbit in ${\mathcal{B}}_{n}\times{\mathcal{B}}_{n}\times\mathbbm{k}^{n}$. It follows easily from [LLSS23, Proposition 1.4] that (5.1) converges absolutely when $(s,\xi)\in\Omega^{2n}_{\eta}\cap({\mathbb{C}}\times{\mathcal{M}})$. Moreover by [LLSS23, Theorem 1.6 (a)] and the theory of Rankin-Selberg integrals for $G_{n}\times G_{n}$, (5.1) has a meromorphic continuation to $(s,\xi)\in{\mathbb{C}}\times{\mathcal{M}}^{\circ}$.

The proof for the case $m=2n+1$ is similar, by using [LLSS23, Theorem 1.6 (b)] and the fact that $\left(\overline{B}_{n},\overline{B}_{n+1}\begin{bmatrix}w_{n}&{}^{t}v_{n}\\ &1\end{bmatrix}\right)$ is a base point of the unique open $G_{n}$-orbit in ${\mathcal{B}}_{n}\times{\mathcal{B}}_{n+1}$. We omit the details.

It remains to prove (2.17). We consider the even and odd cases separately.

5.2. The even case

Assume that $m=2n$, in which case $\eqref{eq:FE'}$ is

[TABLE]

where $s\in\Omega_{\xi,\eta}$. By definition and noting that ${}^{t}z_{2n}^{-1}=z_{2n}$, we obtain that

[TABLE]

A direct calculation shows that $w_{2n}z_{2n}\tau_{2n}=z_{2n}$. Thus by a change of variable $h\mapsto\hat{h}$ and using Proposition 3.3, we obtain that

[TABLE]

Recall that $A_{n}$ is the diagonal maximal torus in $G_{n}$. Write (5.2) as an iterated integral $\int_{A_{n}^{\dagger}\backslash S_{2n}}\int_{A_{n}^{\dagger}}$. For $a=\operatorname{diag}\{a_{1},a_{2},\dots,a_{n}\}\in A_{n}$ and $a^{\dagger}=\begin{bmatrix}a\\ &a\end{bmatrix}\in S_{2n}$, using Proposition 3.3 again one can verify that

[TABLE]

By a change of variable $a\mapsto a^{-1}$ and Tate’s thesis, we get that

[TABLE]

where both integrals converge absolutely. In view of the last equation and

[TABLE]

we find that (5.2) equals

[TABLE]

This proves (2.17) in the even case.

5.3. The odd case

Assume that $m=2n+1$, in which case $\eqref{eq:FE'}$ is

[TABLE]

where $s\in\Omega_{\xi,\eta}$. We have that

[TABLE]

where

[TABLE]

In contrast to the even case, the computation in the odd case is much more complicated. We first give the following result regarding the element $z_{2n+1}^{\prime}$.

Lemma 5.1**.**

The element $z_{2n+1}^{\prime}$ as defined in (5.4) belongs to $\overline{N}_{2n+1}z_{2n+1}S_{2n+1}$, where $\overline{N}_{2n+1}$ is the unipotent radical of $\overline{B}_{2n+1}$. More precisely, there exists $u_{0}\in\overline{N}_{2n+1}$ such that $z_{2n+1}^{\prime}=u_{0}z_{2n+1}h_{0}^{-1}$, where

[TABLE]

Proof.

By direct calculation we find that

[TABLE]

where $e_{1}=(1,0,\dots,0)\in\mathbbm{k}^{n}$. It is clear that the above element lies in $\overline{N}_{2n+1}$. ∎

By Lemma 5.1 and Proposition 3.4 (2), and noting that $\det g_{0}=(-1)^{n}$, a change of variable $h\mapsto\hat{h}_{0}\hat{h}$ in (5.3) gives that

[TABLE]

Let us compute the action of $R_{\varphi_{2n+1}}(\hat{h}_{0})$. It is easy to verify that

[TABLE]

so that

[TABLE]

Using Proposition 3.4 (1), we find that for $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$,

[TABLE]

where $v_{n}^{\prime}:=(1,2,\dots,n)\in\mathbbm{k}^{n}$. It follows that

[TABLE]

Because of the diagonal torus $A_{n}$ of $G_{n}$ and (3.5), we have the diagonal torus $A_{n}^{\ddagger}$ of $S_{2n+1}$. Put $A_{n}^{\prime}:=u^{-1}A_{n}^{\ddagger}u$ and $a^{\prime}:=u^{-1}a^{\ddagger}u$ for $a\in A_{n}$, where

[TABLE]

The following result is rather technical but can be verified directly, the proof of which will be omitted.

Lemma 5.2**.**

For $a={\rm diag}\{a_{1},a_{2},\dots,a_{n}\}\in A_{n}$, the element $z_{2n+1}a^{\prime}z_{2n+1}^{-1}$ belongs to $\overline{B}_{2n+1}$ with diagonal entries $a_{1},a_{2},\dots,a_{n},1,a_{n},\dots,a_{2},a_{1}$, which means that

[TABLE]

By Proposition 3.4 (2) again, for $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$ we have that

[TABLE]

Using Proposition 3.4 (1) and

[TABLE]

we find that for $\phi_{1}\in{\mathcal{S}}(\mathbbm{k}^{n})$,

[TABLE]

Similar to the even case, write the integral in (5.5) as an iterated integral $\int_{A_{n}^{\prime}\backslash S_{2n+1}}\int_{A_{n}^{\prime}}$. Applying Lemma 5.2, (5.6) and (5.7), we find that for $a={\rm diag}\{a_{1},a_{2},\dots,a_{n}\}\in A_{n}$,

[TABLE]

By a change of variable $a\mapsto a^{-1}$ and Tate’s thesis, we obtain that

[TABLE]

where in the last step we make a change of variable $a\mapsto-a$ and use the fact that $\gamma(s,\omega,\bar{\psi})=\omega(-1)\gamma(s,\omega,\psi)$ for $\omega\in\widehat{\mathbbm{k}^{\times}}$. Noting that

[TABLE]

and $v_{n}u_{n}=e_{n}$, we have that

[TABLE]

It follows that

[TABLE]

This finishes the proof of (2.17) in the odd case.

6. $({\rm MF}_{m})+({\rm FE}_{m})\Rightarrow({\rm MF}_{m+1})$

In this section we will show that $({\rm MF}_{m})+({\rm FE}_{m})\Rightarrow({\rm MF}_{m+1})$. In view of the discussions in Section 4, this will finish the inductive proof of Theorem 2.2 and Theorem 2.6. The basic idea is to apply the theory of Godement sections for both sides of the functional equation (${\rm MF}_{m+1}$) and perform induction. It turns out that the explicit calculations are rather complicated. In particular $S_{2n-1}$ can not be embedded into $S_{2n}$. In this case one can only conjugate a subgroup of $S_{2n-1}$ into $S_{2n}$ and integrate over an open dense subset of $S_{2n}$. This requires manipulating different base points for the unique open $S_{m}$-orbit in ${\mathcal{X}}_{m}$. Similar strategy has been applied in [LLSS23] for the study of modifying factors for the Rankin-Selberg case, which leads to nice recurrence relations. In contrast, the recurrence relations (6.10), (6.11), (6.18) and (6.19) in the Jacquet-Shalika case are much more involved. As suggested by the method, we prove the absolute convergence and justify the change of order of certain multiple integrals in our calculation by Fubini’s theorem.

6.1. Godement sections

Assume that $({\rm MF}_{m})$ and $({\rm FE}_{m})$ hold, and that

[TABLE]

We need to show that $({\rm MF}_{m+1})$ holds for $I(\xi^{\prime})$, that is,

[TABLE]

where $(s,\xi^{\prime})\in\Omega_{\eta}^{m+1}$, $f^{\prime}\in I(\xi^{\prime})$ and $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$ with $n=\lfloor(m+1)/2\rfloor$, and the integrals of both sides converge absolutely. Note that $(s,\xi^{\prime})\in\Omega_{\eta}^{m+1}$ implies that $(s,\xi)\in\Omega_{\eta}^{m}$.

We first observe that, by Theorem 2.2 (1), Theorem 2.4 (2) and the uniqueness of meromorphic continuation, it suffices to prove (6.1) when $(s,\xi)\in\Omega^{m}_{\eta}$ and $\Re(\xi_{m+1})$ is sufficiently large.

As mentioned above, the method is to use Godement sections, for which we recall some basic results from [J09]. For $f\in I(\xi)$ and $\Phi\in{\mathcal{S}}(\mathbbm{k}^{m\times(m+1)})$, set

[TABLE]

where $h\in G_{m+1}$ and [math] indicates the zero vector in $\mathbbm{k}^{m\times 1}$. This defines an element of $I(\xi^{\prime})$ when the integral converges absolutely. Let

[TABLE]

As in [J09, Section 7.2], there are natural left and right actions of $G_{m+1}$ and $G_{m}$ on ${\mathcal{S}}(\mathbbm{k}^{m\times(m+1)})$ respectively, which are denoted by

[TABLE]

which clearly preserve ${\mathcal{S}}({\mathcal{Y}}_{m})$.

The following are consequences of Propositions 7.1 and 7.2 in [J09].

Proposition 6.1**.**

1. (1)

If $\Re(\xi_{m+1})>\Re(\xi_{i})-1$, $i=1,2,\dots,m$ or $\Phi\in{\mathcal{S}}({\mathcal{Y}}_{m})$, then (6.2) converges absolutely. In this case if $f^{\prime}={\rm g}^{+}_{\Phi,f,\xi^{\prime}}\in I(\xi^{\prime})$, then

[TABLE]

where the integral converges absolutely. 2. (2)

$I(\xi^{\prime})$* is spanned by the functions ${\rm g}^{+}_{\Phi,f,\xi^{\prime}}$ with $f\in I(\xi)$ and $\Phi\in{\mathcal{S}}({\mathcal{Y}}_{m})$.*

Thus to prove (6.1), by Proposition 6.1 (2) we may assume that

[TABLE]

We need to consider the even and odd cases for $m$ separately. To ease the notation, for a subgroup ${\mathcal{G}}$ of $G_{m}$ put ${\mathcal{G}}^{+}:=\set{h^{+}\mid h\in{\mathcal{G}}}\subset G_{m+1},$ where for $h\in G_{m}$ we write $h^{+}:=\begin{bmatrix}h\\ &1\end{bmatrix}\in G_{m+1}$.

6.2. The case $G_{2n}\to G_{2n+1}$

Assume that $m=2n$. We need to prove (6.1) when $(s,\xi)\in\Omega^{2n}_{\eta}$ and $\Re(\xi_{2n+1})$ is sufficiently large, where $f^{\prime}={\rm g}^{+}_{\Phi,f,\xi^{\prime}}$ is as in (6.4).

6.2.1. $\operatorname{Z}_{\rm JS}$-side

We start from $\operatorname{Z}_{\rm JS}(s,W_{f^{\prime}},\phi,\varphi_{2n+1}^{-1})$. Define a subgroup of $S_{2n+1}$ by

[TABLE]

where

[TABLE]

Define that $\overline{S}_{2n+1}^{\prime}:=\sigma_{2n+1}^{-1}N_{2n+1}\sigma_{2n+1}\cap S^{\prime}_{2n+1}\backslash S^{\prime}_{2n+1}$. Then we have a natural identification: $\overline{S}_{2n+1}^{\prime}=\overline{S}_{2n+1}$.

Note from (2.8) that $\sigma_{2n+1}=\sigma_{2n}^{+}$, viewed as permutation matrices. The integral (3.4) can be also written as

[TABLE]

where

[TABLE]

In the same vein, we will write $\Phi_{\phi}$ and $f^{\prime}_{\phi}$ for similar actions of $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$ on $\Phi\in{\mathcal{S}}(\mathbbm{k}^{2n\times(2n+1)})$ and $f^{\prime}\in I(\xi^{\prime})$. By (6.3), for $h\in S_{2n}$ we have that

[TABLE]

We find that $h_{1}[1_{2n}\mid{}^{t}z](\sigma_{2n}h)^{+}=[h_{1}\sigma_{2n}h\mid h_{1}{}^{t}z]$. After change of variables $h_{1}\mapsto h_{1}(\sigma_{2n}h)^{-1}$ and $z\mapsto z\,{}^{t}(\sigma_{2n}h)$, we obtain that

[TABLE]

where $\Phi_{\phi,h_{1}}\in{\mathcal{S}}(\mathbbm{k}^{2n})$ is defined by

[TABLE]

Write $z=(z_{1},z_{2})$ where $z_{1},z_{2}\in\mathbbm{k}^{n}$, and write ${\mathcal{F}}_{\psi^{\prime}}^{1}$, ${\mathcal{F}}_{\psi^{\prime}}^{2}$ for the partial Fourier transforms on ${\mathcal{S}}(\mathbbm{k}^{2n})$ with respect to the variables $z_{1},z_{2}$ and a nontrivial unitary character $\psi^{\prime}$ of $\mathbbm{k}$. Clearly on ${\mathcal{S}}(\mathbbm{k}^{2n})$ one has

[TABLE]

Recall the right action of $h\in S_{2n}$ on $\mathbbm{k}^{n}$ given by (2.6). In terms of the above notation and noting that $e_{2n}h=(0,e_{n}g)=(0,e_{n}.h)$, we obtain that

[TABLE]

Plugging this into (6.7) for $W=W_{f^{\prime}}$ yields an iterated integral

[TABLE]

By Lemma 6.2 below and Fubini’s theorem, we can switch the order of integration and obtain the recurrence relation

[TABLE]

Lemma 6.2**.**

The double integral (6.10) converges absolutely when $(s,\xi)\in\Omega^{2n}_{\eta}$ and $\Re(\xi_{2n+1})$ is sufficiently large.

Proof.

Without loss of generality, assume that $\Phi_{\phi}(X\mid{}^{t}z)=\Phi^{\prime}(X)\phi^{\prime}(z)$ holds with $X\in\mathbbm{k}^{2n\times 2n}$ and $z\in\mathbbm{k}^{2n}$, for some $\Phi^{\prime}\in{\mathcal{S}}(\mathbbm{k}^{2n\times 2n})$ and $\phi^{\prime}\in{\mathcal{S}}(\mathbbm{k}^{2n})$. Then from (6.8) we find that

[TABLE]

Thus by Proposition 3.5 (2) and Proposition 3.6, it suffices to show that given $M>0$, the integral

[TABLE]

converges absolutely for $\Re(\xi_{2n+1})$ sufficiently large, where $\|h_{1}\|_{\rm HC}:=\|h_{1}\|+\|h_{1}^{-1}\|$ for $\|\cdot\|$ the standard norm on $M_{2n}$ (cf. [J09, Section 3.1] for the Archimedean case). This is [J09, Lemma 3.3 (ii)]. ∎

In view of (${\rm FE}_{2n}$) and (6.9), and noting that $s\in\Omega_{\xi,\eta}$, we have that

[TABLE]

Applying (${\rm MF}_{2n}$) for $\tilde{\xi}=(\xi_{2n}^{-1},\dots,\xi_{2}^{-1},\xi_{1}^{-1})$, and noting from Remark 2.5 (1) that $1-s\in\Omega_{\tilde{\xi},\eta^{-1}}$, we obtain that

[TABLE]

Using $\gamma(s,\omega,\psi)\gamma(1-s,\omega^{-1},\bar{\psi})=1$ for $\omega\in\widehat{\mathbbm{k}^{\times}}$, it is straightforward to check that

[TABLE]

From (6.10) and the above calculations, we find that (6.1) for $m=2n$ is reduced to the recurrence relation

[TABLE]

when $(s,\xi)\in\Omega^{2n}_{\xi}$ and $\Re(\xi_{2n+1})$ is sufficiently large.

6.2.2. $\Lambda_{\rm JS}$-side

Let us prove (6.11). Recall that

[TABLE]

where $S_{2n+1}=\Set{u_{y}h^{+}\bar{u}_{x}}{h\in S_{2n},x,y\in\mathbbm{k}^{n}}$ with the element $\bar{u}_{x}$ given by (6.6), and

[TABLE]

Using Proposition 3.4 (1), we find that $R_{\varphi_{2n+1}^{-1}}(u_{y}h^{+}\bar{u}_{x})\phi(0)=\varphi_{2n}^{-1}(h)\phi(x)$ for $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$. It follows that

[TABLE]

By (6.2), we have that

[TABLE]

A direct calculation gives that

[TABLE]

By a change of variable $h_{1}\mapsto h_{1}(z_{2n}h)^{-1}$, and noting that $\det z_{2n+1}=\det z_{2n}$ and $(y,v_{n}){}^{t}(z_{2n}h)^{-1}=(y,v_{n})z_{2n}{}^{t}h^{-1}=(y,v_{n}){}^{t}h^{-1},$ we obtain that

[TABLE]

It is easy to see that we can exchange the order of integration over $h_{1}\in G_{2n}$ in the above integral and that over $y\in\mathbbm{k}^{n}$ in (6.13). Then for any $h\in S_{2n}$ as in (2.6), an affine transform in $y$ yields that

[TABLE]

It follows that

[TABLE]

Assuming the absolute convergence, we can switch the order of integration and obtain that

[TABLE]

On the other hand,

[TABLE]

The same arguments as in the proof of Lemma 6.2 together with $({\rm MF}_{2n})$ show that (6.14) is absolutely convergent. This proves (6.11), hence finishes the proof of (6.1) for $m=2n$.

6.3. The case $G_{2n-1}\to G_{2n}$

Assume that $m=2n-1$. We need to prove (6.1) when $(s,\xi)\in\Omega^{2n-1}_{\eta}$ and $\Re(\xi_{2n})$ is sufficiently large, where $f^{\prime}={\rm g}^{+}_{\Phi,f,\xi^{\prime}}$ is as in (6.4). Although the strategy is similar to the case that $m$ is even, the calculation is much more complicated.

6.3.1. $\operatorname{Z}_{\rm JS}$-side

We first make some group-theoretic preparations. From (2.8) it is easy to verify that

[TABLE]

Consider the subgroup $S_{2n-1}^{\prime}$ of $S_{2n-1}$ as given by (6.5). Put

[TABLE]

Then $T_{n}^{+}\subset S_{2n}$. Moreover if we define $\overline{T}_{n}:=\varsigma_{n}^{-1}\overline{S}_{2n-1}^{\prime}\varsigma_{n}$ and $\overline{T}_{n}^{+}$ in the obvious way, then from (6.15) we see that $\overline{T}_{n}^{+}$ embeds into $\overline{S}_{2n}$. Define a subgroup $R_{n}$ of $G_{n}$ by

[TABLE]

so that $\overline{P}_{n-1,1}:=G_{n-1}^{+}R_{n}$ is the lower triangular maximal parabolic subgroup of $G_{n}$ of type $(n-1,1)$. Following the notation (3.5), it is easy to see that $\overline{P}_{n-1,1}^{\dagger}$ normalizes the unipotent radical of $T_{n}^{+}$, which implies that $T_{n}^{+}R_{n}^{\dagger}$ is a subgroup of $S_{2n}$. Moreover, the multiplication map $T_{n}^{+}\times R_{n}^{\dagger}\to T_{n}^{+}R_{n}^{\dagger}$ is bijective and the multiplication map $\overline{T}_{n}^{+}\times R_{n}^{\dagger}\to\overline{S}_{2n}$ is an embedding with open dense image. It follows that the integral (3.3) can be written as

[TABLE]

where $\varphi_{2n-1}^{\prime}$ is the character of $S_{2n-1}^{\prime}$ given by

[TABLE]

By (6.3), for $f^{\prime}={\rm g}^{+}_{\Phi,f,\xi^{\prime}}$ as in (6.4), $h\in S_{2n-1}^{\prime}$ and $r\in R_{n}$, one has that

[TABLE]

Note that $h_{1}[1_{2n-1}\mid{}^{t}z](\sigma_{2n-1}h\varsigma_{n})^{+}=[h_{1}\sigma_{2n-1}h\varsigma_{n}\mid h_{1}{}^{t}z]$ and change the variables $h_{1}\mapsto h_{1}(\sigma_{2n-1}h\varsigma_{n})^{-1}$ and $z\mapsto z\,{}^{t}(\sigma_{2n-1}h\varsigma_{n})$. For $h$ given by (6.17), a direct calculation shows that $e_{2n-1}\sigma_{2n-1}h\varsigma_{n}=(e_{n},x)\in\mathbbm{k}^{2n-1}$. It follows that

[TABLE]

where $\Phi_{r,h_{1}}\in{\mathcal{S}}(\mathbbm{k}^{2n-1})$ is defined by $\Phi_{r,h_{1}}(z):=\Phi(h_{1}[1_{2n-1}\mid{}^{t}z]r^{\dagger})$ for $z\in\mathbbm{k}^{2n-1}$.

Similar to the even case, write $z=(z_{1},z_{2})$, where $z_{1}\in\mathbbm{k}^{n}$, $z_{2}\in\mathbbm{k}^{n-1}$. Denote by ${\mathcal{F}}_{\psi^{\prime}}^{1}$, ${\mathcal{F}}_{\psi^{\prime}}^{2}$ the partial Fourier transforms on ${\mathcal{S}}(\mathbbm{k}^{2n-1})$ with respect to the variables $z_{1}$, $z_{2}$, where $\psi^{\prime}$ is a nontrivial unitary character of $\mathbbm{k}$. In this way, on ${\mathcal{S}}(\mathbbm{k}^{2n-1})$ one has that ${\mathcal{F}}_{\psi^{\prime}}={\mathcal{F}}_{\psi^{\prime}}^{1}\circ{\mathcal{F}}_{\psi^{\prime}}^{2}={\mathcal{F}}_{\psi^{\prime}}^{2}\circ{\mathcal{F}}_{\psi^{\prime}}^{1}$.

Plugging the above equation for $W_{f^{\prime}}((\sigma_{2n-1}h\varsigma_{n})^{+}r^{\dagger})$ into (6.16) gives that

[TABLE]

Similar to Lemma 6.2, we can switch the order of integration and obtain the recurrence relation

[TABLE]

where we have used (6.7) and (6.17).

Similar to the case that $m$ is even, applying (${\rm FE}_{2n-1}$) for $\xi$ and (${\rm MF}_{2n-1}$) for $\tilde{\xi}$, and noting that $s\in\Omega_{\xi,\eta}$, we find that (6.1) for $m=2n-1$ is reduced to the recurrence relation

[TABLE]

with $\Phi_{r,h_{1}}^{-}(z_{1},z_{2}):=\Phi_{r,h_{1}}(z_{1},-z_{2})$, for $(s,\xi)\in\Omega^{2n-1}_{\eta}$ and $\Re(\xi_{2n})$ sufficiently large.

6.3.2. $\Lambda_{\rm JS}$-side

Let us prove (6.19). Recall the base point $x_{2n}=(\overline{B}_{2n}z_{2n},v_{n})$ of the open $S_{2n}$-orbit in ${\mathcal{X}}_{2n}$ given by (2.13). For convenience we choose a new base point as follows. Recall the element

[TABLE]

as given by (5.4). Let $g_{n}:=\begin{bmatrix}-v_{n-1}&1\\ 1_{n-1}&0\end{bmatrix}\in G_{n}.$ Then one can check that

[TABLE]

and it is clear that $[1_{2n-1}\mid 0]z_{2n}^{\prime}=[z_{2n-1}^{\prime}\varsigma_{n}\mid 0]$. Noting that $\det g_{n}=(-1)^{n-1}$, we have that

[TABLE]

The integral over $S_{2n}$ can be manipulated as follows. Recall the subgroup $T_{n}^{+}R_{n}^{\dagger}$ of $S_{2n}$ and the unipotent radical $U_{n}$ of the mirabolic subgroup $P_{n}$ of $G_{n}$, that is

[TABLE]

Finally let

[TABLE]

Then it is easy to check that the multiplication map

[TABLE]

is an embedding with open dense image. We can take the integral over this image.

Recall that $T_{n}=\varsigma_{n}^{-1}S_{2n-1}^{\prime}\varsigma_{n}$ and consider an element

[TABLE]

associated to the embedding (6.22). Since $U_{n}^{\dagger}T_{n}^{+}V_{n}\subset P_{2n}$, one has that

[TABLE]

where $\varphi_{2n-1}^{\prime}$ is the character of $S_{2n-1}^{\prime}$ given by (6.17). By (6.2) we have

[TABLE]

By direct calculation we find that for $h^{\prime}$ given by (6.23),

[TABLE]

where $u_{y}$ is as in (6.12) and ${}^{t}z_{h^{\prime}}=z_{2n-1}^{\prime}u_{y}h\varsigma_{n}\begin{bmatrix}{}^{t}z\\ 0\end{bmatrix}+\begin{bmatrix}0\\ w_{n-1}{}^{t}y\end{bmatrix}\in\mathbbm{k}^{(2n-1)\times 1}.$ We change the variable $h_{1}\mapsto h_{1}(z_{2n-1}^{\prime}u_{y}h\varsigma_{n})^{-1}$ in the integral representation of $f^{\prime}(z_{2n}^{\prime}h^{\prime})$. At this point, an extensive calculation is required. Write

[TABLE]

as in (6.17). Then by a direct computation we obtain that

[TABLE]

where

[TABLE]

Further make a change of variable $z\mapsto z+(y\,{}^{t}X\,{}^{t}g^{-1},y\,{}^{t}g^{-1}\,{}^{t}x)$ in (6.21). Recall the right action of $S_{2n-1}$ on $\mathbbm{k}^{n-1}$ from (2.12) and the involution in (3.2). It can be verified that $-y\,{}^{t}g^{-1}=0.\widehat{u_{y}h}$.

Using (6.24) and noting that $\det z_{2n}^{\prime}=\det(z_{2n-1}^{\prime}\varsigma_{n})$, after the above change of variables we arrive at

[TABLE]

Assuming the absolute convergence, we can switch the order of integration and obtain that

[TABLE]

On the other hand since $S_{2n-1}=\set{u_{y}h}{h\in S_{2n-1}^{\prime},\,y\in\mathbbm{k}^{n-1}}$, using (5.3) and noting that ${}^{t}\varsigma_{n}^{-1}=\varsigma_{n}$, we find that for any $\phi_{1}\in{\mathcal{S}}(\mathbbm{k}^{n})$,

[TABLE]

For the element $h\in S_{2n-1}^{\prime}$ as above, from Proposition 3.4 (1) it is straightforward to check that

[TABLE]

Now put $\phi_{1}={\mathcal{F}}^{1}_{\bar{\psi}}(\Phi^{-}_{r,h_{1}})(e_{n},\cdot)$. Similar arguments as in the proof of Lemma 6.2 together with $({\rm MF}_{2n-1})$ show that (6.25) is absolutely convergent. This proves (6.19), hence finishes the proof of (6.1) for $m=2n-1$.

7. Friedberg-Jacquet integrals and modifying factors

In this section we prove the results in Section 2.2.

7.1. Proof of Theorem 2.11

By MVW involution, $I(\tilde{\xi})$ has an irreducible generic quotient $\pi(\tilde{\xi})\cong\pi(\xi)^{\vee}$, such that $\pi(\tilde{\xi})\otimes|\eta|^{\frac{1}{2}}$ is nearly tempered. By Theorem 2.2 (4) and that $\operatorname{L}(1-s,\pi(\tilde{\xi}),\wedge^{2}\otimes\eta)$ is holomorphic at $s=0$, it suffices to prove the following lemma.

Lemma 7.1**.**

Under the assumptions of Theorem 2.11, for all $\widetilde{W}\in{\mathcal{W}}(\pi(\tilde{\xi}),\bar{\psi})$ and $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$ with $\phi(0)=0$, it holds that

[TABLE]

Proof.

Since ${\mathcal{W}}(\pi(\tilde{\xi}),\bar{\psi})={\mathcal{W}}(I(\tilde{\xi}),\bar{\psi})$, we may assume that $\widetilde{W}=W_{\tilde{f}}$ for some $\tilde{f}\in I(\tilde{\xi})$. By Theorem 2.4, Theorem 2.6 and meromorphic continuation, it suffices to show that

[TABLE]

for all $\xi^{\prime}\in{\mathcal{M}}^{\circ}$ which is $\eta^{-1}$-symmetric such that $I(\xi^{\prime})\otimes|\eta|^{\frac{1}{2}}$ is nearly tempered, and all $f^{\prime}\in I(\xi^{\prime})$. In this case the integral $\Lambda_{\rm JS}(1,f^{\prime},\hat{\phi},\varphi_{2n})$ is absolutely convergent. Similar to the calculation in Section 5.2,

[TABLE]

Since $\phi(0)=0$ and $\prod^{n}_{i=1}|a_{i}|_{\mathbbm{k}}\operatorname{d}\!a^{\dagger}$ is the restriction of the Haar measure on $\mathbbm{k}^{n}$ to the open dense subset $(\mathbbm{k}^{\times})^{n}\cong A_{n}^{\dagger}$, the last inner integral vanishes. ∎

7.2. Proof of Proposition 2.13

By Theorem 2.2 and Theorem 2.6, for $f\in I(\xi)$ and $\phi\in{\mathcal{S}}(\mathbbm{k}^{n})$ we have

[TABLE]

The proposition follows from Lemma 7.1.

7.3. Proof of Proposition 2.14

Write for short

[TABLE]

where $\xi^{1},\xi^{2}$ are as in Remark 2.10. Without loss of generality we may assume that the restriction $f|_{H_{n}}$ is an element $f_{1}\otimes f_{2}\in I_{1}\otimes I_{2}$, so that

[TABLE]

As mentioned in Section 5.1, $(\overline{B}_{n},\overline{B}_{n}w_{n},v_{n})$ is a base point of the unique open $G_{n}$-orbit in ${\mathcal{B}}_{n}\times{\mathcal{B}}_{n}\times\mathbbm{k}^{n}$. Hence there is a unique element $g^{\prime}\in G_{n}$ taking this base point to the one in [LLSS23, Lemma 1.1]. Then by [LLSS23, Theorem 1.6 (a)], a change of variable $g\mapsto g^{\prime}g$ in the above integral shows that there exists $c\in{\mathbb{C}}^{\times}$ (depending on $g^{\prime},\xi$ and $\eta$) such that

[TABLE]

where

[TABLE]

and $W_{f_{1}}\in{\mathcal{W}}(I_{1},\psi)={\mathcal{W}}(\pi_{1},\psi)$ and $\overline{W}_{f_{2}}\in{\mathcal{W}}(I_{2},\bar{\psi})={\mathcal{W}}(\pi_{2},\bar{\psi})$ are the Whittaker functions associated to $f_{1}$ and $f_{2}$ via Jacquet integrals respectively. Note that both integrals above are first defined in some domains of convergence and then extended meromorphically to $s\in{\mathbb{C}}$.

Recall from Remark 2.10 (4) that $\pi_{2}\cong\pi_{1}^{\vee}\otimes\eta$. It follows from [JPSS83, J09] that there exists $\epsilon=\pm 1$ (depending on $\xi$ and $\eta$) such that

[TABLE]

where

[TABLE]

It is well-known that $\operatorname{L}(s,\pi\times\pi^{\vee})$ is holomorphic at $s=1$ for any $\pi\in\mathrm{Irr}_{\rm gen}(G_{n})$ (see e.g. [FLO12, Appendix A.1]). Since $\operatorname{Z}_{\rm RS}^{\circ}(s,f_{1},f_{2},\phi,\eta^{-1})$ defines a nonzero element in the space ${\mathrm{Hom}}_{G_{n}}(\pi_{1}\,\widehat{\otimes}\pi_{2}\,\widehat{\otimes}\,{\mathcal{S}}(\mathbbm{k}^{n}),\eta|\cdot|_{\mathbbm{k}}^{-s})$ for $\forall s\in{\mathbb{C}}$, we see that $\left(s^{d_{\xi}}\Lambda_{\rm RS}(s,f,\phi,\eta^{-1})\right)_{s=0}=\langle\lambda,f|_{H_{n}}\otimes\phi\rangle$ for a nonzero functional $\lambda\in{\mathrm{Hom}}_{G_{n}}(\pi_{1}\,\widehat{\otimes}\,\pi_{2}\,\widehat{\otimes}\,{\mathcal{S}}(\mathbbm{k}^{n}),\eta)$. Clearly

[TABLE]

hence by the uniqueness of Rankin-Selberg periods ([SZ12, S12]), the functional $\lambda$ factors through $\pi_{1}\,\widehat{\otimes}\,\pi_{2}$. The proposition follows.

7.4. Proof of Theorem 2.15

Following the above proof of Proposition 2.14, write $I_{i}=I(\xi^{i})$, $i=1,2$. Then we have induction in stages: $I(\xi)\cong{\mathrm{Ind}}^{G_{2n}}_{\overline{Q}_{n}}(I_{1}\,\widehat{\otimes}\,I_{2})$ by taking $f\mapsto f^{\prime}$ with $f^{\prime}(g)\in I_{1}\,\widehat{\otimes}\,I_{2}$ for $g\in G_{2n}$, being given by $f^{\prime}(g)(h)=\delta_{\overline{Q}_{n}}^{-1/2}(h)f(hg)$ for $h\in H_{n}$ where $\delta_{\overline{Q}_{n}}$ is the modular character of $\overline{Q}_{n}$. Take $\gamma_{n}$ in (2.23) and let $G_{n}^{\prime}:=\Set{\begin{bmatrix}g\\ &1_{n}\end{bmatrix}}{g\in G_{n}}.$ Then the multiplication map $\overline{Q}_{n}\times\{\gamma_{n}\}\times G_{n}^{\prime}\to\overline{Q}_{n}\gamma_{n}H_{n}$ is a bijection. Hence for $f\in I(\xi)^{\sharp}$, by the support condition we may view the map

[TABLE]

as an element of $C^{\infty}_{c}(G_{n})\,\widehat{\otimes}\,I_{1}\,\widehat{\otimes}\,I_{2}$. From the proof of Proposition 2.14, the functional $\lambda_{I(\xi)}^{\prime}$ given by (2.21) is of the form $\langle\lambda_{I(\xi)}^{\prime},f\rangle=\langle\lambda^{\prime},f^{\prime}(1_{n})\rangle$ for some $\lambda^{\prime}\in{\mathrm{Hom}}_{G_{n}}(I_{1}\,\widehat{\otimes}\,I_{2},\eta)$. Then

[TABLE]

From this (1) and (2) of the theorem follow easily.

Assume that the conditions in (3) hold. We have the twisted Shalika functional $\lambda_{I(\xi)}$. Note that $\overline{Q}_{n}\gamma_{n}H_{n}\subset\overline{Q}_{n}S_{2n}=\overline{Q}_{n}N_{Q_{n}}$, where $N_{Q_{n}}\cong M_{n}$ is the unipotent radical of the upper triangular parabolic subgroup $Q_{n}$ opposite to $\overline{Q}_{n}$, and we have a bijection $\overline{Q}_{n}\times N_{Q_{n}}\to\overline{Q}_{n}N_{Q_{n}}$. In fact one has that $\overline{Q}_{n}\gamma_{n}H_{n}=\overline{Q}_{n}N_{Q_{n}}^{\diamond}$, where

[TABLE]

Hence for $f\in I(\xi)^{\sharp}$ we may view the map $M_{n}\to I_{1}\,\widehat{\otimes}\,I_{2}$ with $X\mapsto f^{\prime}\left(\begin{bmatrix}1_{n}&X\\ &1_{n}\end{bmatrix}\right)$ as an element of $C^{\infty}_{c}(G_{n})\,\widehat{\otimes}\,I_{1}\,\widehat{\otimes}\,I_{2}\subset C^{\infty}_{c}(M_{n})\,\widehat{\otimes}\,I_{1}\,\widehat{\otimes}\,I_{2}$.

From the above discussion and the definitions of $\lambda_{I(\xi)}$ and $\lambda_{I(\xi)}^{\prime}$, we obtain that

[TABLE]

For $\Re(s)$ sufficiently large, we have that

[TABLE]

where we change the variable $X\mapsto gX$ in the last step. By the support condition on $f$ again, we may assume that the function

[TABLE]

lies in the space ${\rm MC}(I_{1}\otimes\chi)\otimes C^{\infty}_{c}(M_{n})$, where ${\rm MC}(I_{1}\otimes\chi)$ denotes the space spanned the matrix coefficients of $I_{1}\otimes\chi$. Then the above inner integral over $M_{n}$ equals ${\mathcal{F}}_{\bar{\psi}}(\Phi)(g,g)$, where ${\mathcal{F}}_{\bar{\psi}}$ indicates the Fourier transform in the variable $X$ with respect to $\bar{\psi}$.

Thus $\operatorname{Z}_{\rm FJ}(s,f,\chi)$ can be viewed as a Godement-Jacquet integral ([GJ72]) for the representation $I_{1}\otimes\chi$ of $G_{n}$. By the functional equation for Godement-Jacquet integrals and the uniqueness of meromorphic continuation, for $-\Re(s)$ sufficiently large we have that

[TABLE]

in view of (7.1). It follows that $\gamma(s,I_{1}\otimes\chi,\psi)\operatorname{Z}_{\rm FJ}(s,f,\chi)=\Lambda_{\rm FJ}(s,f,\chi)$ for all $s\in{\mathbb{C}}$ by the uniqueness of meromorphic continuation.

8. Proof of Archimdedean period relations

In this section we will apply Theorem 2.15 to prove Theorem 2.16, and we retain the notation in Section 2.3. Write $\zeta_{\mu}:=\chi_{\mu}\rho_{2n}=(\zeta_{\mu,1},\zeta_{\mu,2},\dots,\zeta_{\mu,2n})\in(\widehat{\mathbbm{k}^{\times}})^{2n}$, so that $I_{\mu}=I(\zeta_{\mu})$ in the notation of Section 2.2.

Let $v_{\mu}^{\vee}\in(F_{\mu}^{\vee})^{\overline{N}_{2n,{\mathbb{C}}}}$ be the lowest weight vector specified as in [LLS24, Section 2.1], and let $\gamma_{n}^{\prime}:=\begin{bmatrix}1_{n}&1_{n}\\ &w_{n}\end{bmatrix}.$ As in Section 2.3, assume that $\chi_{\natural}$ is $F_{\mu}$-balanced in the sense of Definition 1.1. We specify a generator of ${\mathrm{Hom}}_{H_{n,{\mathbb{C}}}}(F_{\mu}^{\vee},\xi_{\mu,\chi_{\natural}})$ as follows.

Lemma 8.1**.**

There exists a unique $\lambda_{F_{\mu},\chi_{\natural}}\in{\mathrm{Hom}}_{H_{n,{\mathbb{C}}}}(F_{\mu}^{\vee},\xi_{\mu,\chi_{\natural}})$ with the property that $\lambda_{F_{\mu},\chi_{\natural}}(\gamma_{n}^{\prime-1}.v_{\mu}^{\vee})=1$.

Proof.

This follows from the fact that $\overline{B}_{2n,{\mathbb{C}}}\,\gamma_{n}^{\prime}\,H_{n,{\mathbb{C}}}\subset G_{2n,{\mathbb{C}}}$ is Zariski open dense. ∎

Define

[TABLE]

which is holomorphic and non-vanishing on $I_{\mu}$ for each $s\in{\mathbb{C}}$. Put

[TABLE]

which a priori depends on $\chi^{\natural}$ (in the real case) and is meromorphic. Similar to the proof of [LLS24, Proposition 4.7], using the standard results for the Archimedean local factors it is straightforward to verify that

Lemma 8.2**.**

$\Xi_{\mu,\chi_{\natural}}(s)\equiv\Omega_{\mu,\chi_{\natural}}^{-1},$* where $\Omega_{\mu,\chi_{\natural}}$ is the constant in Theorem 2.16.*

Therefore in view of (2.25), Theorem 2.16 is reduced to the following result.

Proposition 8.3**.**

The following diagram is commutative:

[TABLE]

Proof.

Following [LLS24, Section 2.2], we realize $I_{\mu}\otimes F_{\mu}^{\vee}$ as a space of $F_{\mu}^{\vee}$-valued functions $\varphi$ on $G_{2n}$, on which $h\in G_{2n}$ acts by $h.\varphi(x):=h.(\varphi(xh))$ for $x\in G_{2n}$. Then the translation $\imath_{\mu}:I_{0}\to I_{\mu}\otimes F_{\mu}^{\vee}$ is given by

[TABLE]

Clearly $\imath_{\mu}$ maps $I_{0}^{\sharp}$ into $I_{\mu}^{\sharp}\otimes F_{\mu}^{\vee}$, where $I_{\mu}^{\sharp}=I(\zeta_{\mu})^{\sharp}$ is defined by (2.24).

By the uniqueness of twisted linear periods ([CS20]) and holomorphic continuation, in view of Theorem 2.15 it suffices to prove the commutativity of following diagram:

[TABLE]

By definition, for $f\in I_{0}^{\sharp}$ we have that

[TABLE]

where $\lambda_{I_{\mu}}^{\prime}$ is given by (2.21) and $\gamma_{n}$ is given by (2.23). We find that

[TABLE]

where $\phi$ is an arbitrary element of ${\mathcal{S}}(\mathbbm{k}^{n})$ with $\phi(0)=0$, and the last integral is interpreted in the sense of meromorphic continuation via standard sections. Noting that $z_{2n}\gamma_{n}=\gamma_{n}^{\prime}$ and

[TABLE]

from Lemma 8.1 and (8.1) it is easy to check that

[TABLE]

Recall that by definition $d_{\zeta_{\mu}}$ is the order of

[TABLE]

at $s_{1}=0$. It is straightforward to verify that $d_{\zeta_{\mu}}=d_{\zeta_{0}}$, hence

[TABLE]

Plugging the last equation into (8.3) shows that

[TABLE]

which verifies the commutativity of (8.2). ∎

9. Cohomology groups and modular symbols

In this section we introduce certain cohomology groups and modular symbols, which are needed for the proof of Theorem 1.4 in the next section. We turn to the global setting and retain the notation from the Introduction.

9.1. Preliminaries on cohomology groups

For convenience write $G:={\mathrm{GL}}_{2n}$ in the sequel. We have the regular algebraic irreducible cuspidal automorphic representation $\Pi=\Pi_{f}\otimes\Pi_{\infty}$ of $G({\mathbb{A}})$, which is of symplectic type and has a coefficient system $F_{\mu}$ with $\mu$ being now a pure weight in $({\mathbb{Z}}^{2n})^{{\mathcal{E}}_{\mathrm{k}}}$.

Recall that $\boldsymbol{\eta}$ is a character of ${\mathrm{k}}^{\times}\backslash{\mathbb{A}}^{\times}$ such that $\operatorname{L}(s,\Pi,\wedge^{2}\otimes\boldsymbol{\eta}^{-1})$ has a pole at $s=1$. Define a nontrivial unitary character $\boldsymbol{\psi}$ of ${\mathrm{k}}\backslash{\mathbb{A}}$ by the composition

[TABLE]

where ${\mathbb{A}}_{\mathbb{Q}}$ is the adele ring of ${\mathbb{Q}}$, $\widehat{\mathbb{Z}}$ is the profinite completion of ${\mathbb{Z}}$ and $\psi_{\mathbb{R}}(x)=e^{2\pi{\rm i}x}$, $x\in{\mathbb{R}}$. Denote by $S={\mathrm{GL}}_{n}^{\dagger}\ltimes N$ the Shalika subgroup of ${\mathrm{GL}}_{2n}$, where ${\mathrm{GL}}_{n}^{\dagger}$ is the diagonal image of ${\mathrm{GL}}_{n}$ in $H={\mathrm{GL}}_{n}\times{\mathrm{GL}}_{n}$, and $N\cong{\rm Mat}_{n\times n}$ is the unipotent radical of $S$. Similar to the local case, we have a character $\boldsymbol{\eta}\otimes\boldsymbol{\psi}$ of $S({\mathrm{k}})\backslash S({\mathbb{A}})$ defined as in [JST19, Section 2.3].

Fix the measure on $N({\mathrm{k}})\backslash N({\mathbb{A}})$ to be induced from the self-dual Haar measure on ${\mathrm{k}}\backslash{\mathbb{A}}$ with respect to $\boldsymbol{\psi}$, and fix once for all an ${\mathrm{GL}}_{n}^{\dagger}({\mathbb{A}})$-invariant positive Borel measure on $({\mathrm{GL}}_{n}^{\dagger}({\mathrm{k}}){\mathbb{R}}^{\times}_{+})\backslash{\mathrm{GL}}_{n}^{\dagger}({\mathbb{A}})$. This gives an $S({\mathbb{A}})$-invariant positive Borel measure on $(S({\mathrm{k}}){\mathbb{R}}^{\times}_{+})\backslash S({\mathbb{A}})$, and thereby fixes a Shalika functional

[TABLE]

Fix a factorization $\lambda_{\mathbb{A}}=\lambda_{f}\otimes\lambda_{\infty}$ thanks to the uniqueness of Shalika models. Using $\lambda_{f}$ we embed $\Pi_{f}$ into ${\mathrm{Ind}}^{G({\mathbb{A}}_{f})}_{S({\mathbb{A}}_{f})}(\boldsymbol{\eta}_{f}\otimes\boldsymbol{\psi}_{f})$. Using cyclotomic characters as in [JST19, Section 3.1], each $\sigma\in{\mathrm{Aut}}({\mathbb{C}})$ gives a $\sigma$-linear isomorphism ${\mathrm{Ind}}^{G({\mathbb{A}}_{f})}_{S({\mathbb{A}}_{f})}(\boldsymbol{\eta}_{f}\otimes\boldsymbol{\psi}_{f})\to{\mathrm{Ind}}^{G({\mathbb{A}}_{f})}_{S({\mathbb{A}}_{f})}({}^{\sigma}\boldsymbol{\eta}_{f}\otimes\boldsymbol{\psi}_{f})$, which restricts to a $\sigma$-linear isomorphism $\sigma:\Pi_{f}\to{}^{\sigma}\Pi_{f}$.

Recall that $H={\mathrm{GL}}_{n}\times{\mathrm{GL}}_{n}\subset G$. We introduce

[TABLE]

where $K_{\infty}$ and $C_{\infty}$ are the standard maximal compact subgroups of $G_{\infty}:=G({\mathrm{k}}_{\infty})$ and $H_{\infty}:=H({\mathrm{k}}_{\infty})$ respectively. Then the natural inclusion $\imath:{\mathcal{X}}_{H}\hookrightarrow{\mathcal{X}}_{G}$ is a proper map. Define a real vector space $\mathfrak{q}_{\infty}:=(\mathfrak{c}_{\infty}\oplus{\mathbb{R}})\backslash\mathfrak{h}_{\infty}$, where as usual gothic letters denote the Lie algebras of the corresonding real Lie groups, and ${\mathbb{R}}$ indicates the Lie algebra of ${\mathbb{R}}^{\times}_{+}$. Put $d_{\infty}:=\dim\mathfrak{q}_{\infty}=\sum_{v\mid\infty}d_{{\mathrm{k}}_{v}}+r-1,$ where $d_{{\mathrm{k}}_{v}}$ is as in (2.27) and $r$ is the number of Archimedean places of ${\mathrm{k}}$. As in [Cl90], we have the canonical isomorphism

[TABLE]

where $\operatorname{H}^{*}_{c}$ denotes the Betti cohomology with compact support. As is known (see e.g. [LLS24, Section 6.3]), (9.1) is $G^{\natural}$-equivariant, where $G^{\natural}:=G({\mathbb{A}}_{f})\times\pi_{0}({\mathrm{k}}_{\infty}^{\times}).$

Denote by $\mathfrak{m}:=\mathfrak{m}_{f}\otimes\mathfrak{m}_{\infty}$ the one-dimensional space of invariant measures on $H({\mathbb{A}})$. Let ${\mathrm{GL}}_{n}^{\prime}:={\mathrm{GL}}_{n}\times\{1\}\subset H$, and denote by $\mathfrak{m}^{\prime}:=\mathfrak{m}_{f}^{\prime}\otimes\mathfrak{m}_{\infty}^{\prime}$ the one-dimensional space of invariant measures on ${\mathrm{GL}}_{n}^{\prime}({\mathbb{A}})$. Recall that we have fixed a positive Borel measure on $({\mathrm{GL}}_{n}^{\dagger}({\mathrm{k}}){\mathbb{R}}^{\times}_{+})\backslash{\mathrm{GL}}_{n}^{\dagger}({\mathbb{A}})$. This enables us to identify $\mathfrak{m},\mathfrak{m}_{f}$ and $\mathfrak{m}_{\infty}$ with $\mathfrak{m}^{\prime},\mathfrak{m}^{\prime}_{f}$ and $\mathfrak{m}^{\prime}_{\infty}$ respectively.

Let $\omega_{\infty}:=(\wedge^{d_{\infty}}\mathfrak{q}_{\infty})\otimes_{\mathbb{R}}{\mathbb{C}}$, and let $\mathfrak{O}_{\infty}$ be the complex orientation space of $\omega_{\infty}$. It is clear that $\pi_{0}({\mathrm{k}}_{\infty}^{\times})$ acts on $\omega_{\infty}$ and $\mathfrak{O}_{\infty}$ trivially. Similar to [LLS24, Section 3.1], we have an identification: $\mathfrak{m}_{\infty}=\omega_{\infty}^{*}\otimes\mathfrak{O}_{\infty},$ where a superscript $*$ indicates the linear dual. Then we have that

[TABLE]

where we use $(\mathfrak{h}_{\infty},{\mathbb{R}}^{\times}_{+}C_{\infty}^{\circ})$-cohomology in the last equality.

Recall that we have an algebraic Hecke character $\chi$ of ${\mathrm{k}}^{\times}\backslash{\mathbb{A}}^{\times}$, with coefficient system $\chi_{\natural}$. Define the character $\xi_{\boldsymbol{\eta},\chi}:=\chi\boxtimes(\chi^{-1}\boldsymbol{\eta}^{-1})$ of $H({\mathbb{A}})$. Then we have the factorization $\xi_{\boldsymbol{\eta},\chi}=\xi_{\boldsymbol{\eta}_{f},\chi_{f}}\otimes\xi_{\boldsymbol{\eta}_{\infty},\chi_{\infty}}.$ Recall the character $\xi_{\mu,\chi_{\natural}}$ of $H({\mathrm{k}}\otimes_{\mathbb{Q}}{\mathbb{C}})$ given by (1.4), which is the coefficient system of $\xi_{\boldsymbol{\eta},\chi}$. To ease the notation, write

[TABLE]

Likewise, write

[TABLE]

Without further explanation, similar notation applies to the $\sigma$-twist with $\sigma\in{\mathrm{Aut}}({\mathbb{C}})$.

9.2. Modular symbols and a commutative diagram

We define global and (normalized) local modular symbols.

9.2.1. Global modular symbol

When $\chi_{\natural}$ is $F_{\mu}$-balanced, fix a generator

[TABLE]

as in Lemma 8.1 (by abuse of notation). Recall the space of measures $\mathfrak{m}_{f}$ on $H({\mathbb{A}}_{f})$ and the orientation space $\mathfrak{O}_{\infty}$. Put $\mathfrak{m}^{\natural}:=\mathfrak{m}_{f}\otimes\mathfrak{O}_{\infty}.$ In the notation of (9.3), we have the global modular symbol

[TABLE]

where $\int_{{\mathcal{X}}_{H}}$ is the pairing with the fundamental class (see e.g. [JST19, Section 4.2] for details).

9.2.2. Archimedean modular symbol

Recall the Shalika functional $\lambda_{\mathbb{A}}=\lambda_{f}\otimes\lambda_{\infty}$. Similar to the local case, using $\lambda_{\infty}$ we have the normalized Friedbert-Jacquet periods

[TABLE]

where we have identified $\mathfrak{m}_{\infty}$ with $\mathfrak{m}^{\prime}_{\infty}$ as in Section 9.1. As above assume that $\chi_{\natural}$ is $F_{\mu}$-balanced. Introduce the normalized Archimedean modular symbol

[TABLE]

where the first arrow is induced by restriction and the functional

[TABLE]

and the last equality is (9.2).

We mention that the above formulation is more canonical, while in the Archimedean modular symbol given by (2.26) we have fixed the measure on ${\mathrm{GL}}_{n}(\mathbbm{k})$ for simplicity.

9.2.3. Non-Archimedean modular symbol

We further factorize $\lambda_{f}=\otimes_{v\nmid\infty}\lambda_{v}$ and $\mathfrak{m}_{f}=\mathfrak{m}_{f}^{\prime}=\otimes_{v\nmid v}\mathfrak{m}_{v}^{\prime}$, and introduce the normalized non-Archimedean modular symbol

[TABLE]

where $\wp^{\circ}_{v}:\Pi_{v}\otimes\xi_{\boldsymbol{\eta}_{v},\chi_{v},\frac{1}{2}}\otimes\mathfrak{m}_{v}^{\prime}\to{\mathbb{C}}$ is given by

[TABLE]

In the above, ${\mathcal{G}}(\chi_{v})$ is the local Gauss sum defined using $\boldsymbol{\psi}_{v}$ as in [JST19, Section 2.2].

9.2.4. A commutative diagram

The following is a consequence of [FJ93, Proposition 2.3], which relates the local Friedberg-Jacquet periods and the global period

[TABLE]

where $Z$ is the center of $G$. They are interpreted in terms of the global and local modular symbols as follows.

Proposition 9.1**.**

The following diagram is commutative:

[TABLE]

where the left vertical arrow is induced by (9.1).

10. Shalika periods and the Blasius-Deligne conjecture

In this section we are ready to define the canonical family of Shalika periods under Assumption 1.3 and prove Theorem 1.4.

10.1. The kernels of modular symbols

Recall that $\widehat{\pi_{0}({\mathrm{k}}_{\infty}^{\times})}$ acts on $\operatorname{H}(\Pi)$ and $\operatorname{H}(\Pi_{\infty})$, and we shall write their $\varepsilon^{\prime}$-isotypic components as $\operatorname{H}(\Pi)[\varepsilon^{\prime}]$ and $\operatorname{H}(\Pi_{\infty})[\varepsilon^{\prime}]$ respectively for every $\varepsilon^{\prime}\in\widehat{\pi_{0}({\mathrm{k}}_{\infty}^{\times})}$. We now make the identification

[TABLE]

For the modular symbol $\wp_{\infty}^{\circ}$ given by (9.5), it is clear that the map $\operatorname{H}(\Pi_{\infty})\to{\mathbb{C}}$ with $\kappa\mapsto\wp^{\circ}_{\infty}(\kappa\otimes 1)$ is supported on $\operatorname{H}(\Pi_{\infty})[\varepsilon]$, and we denote its restriction by

[TABLE]

Recall that $\Pi_{\infty}\cong\pi_{\mu}:=\widehat{\otimes}_{v\mid\infty}\pi_{\mu_{v}},$ where $\mu_{v}:=\{\mu^{\iota}\}_{\iota\in{\mathcal{E}}_{{\mathrm{k}}_{v}}}$, and we have a Shalika functional $\lambda_{\infty}$ on $\Pi_{\infty}$. Let $\pi_{0}\in\mathrm{Irr}(G_{\infty})$ be the specialization of $\pi_{\mu}$ at $\mu=0$, and we fix a nonzero Shalika functional $\lambda_{0,\infty}$ on $\pi_{0}$. There is a map $\jmath_{\mu}:\pi_{0}\to\Pi_{\infty}\otimes F_{\mu}^{\vee}$, which is $G_{\infty}$-equivariant, uniquely determined by $\lambda_{\infty}$ and $\lambda_{0,\infty}$ as in (2.25), and induces an isomorphism

[TABLE]

Specializing at $\mu=0$ and $\chi_{\infty}=\varepsilon$ in (10.2), we obtain a map

[TABLE]

Lemma 10.1**.**

The map $\wp^{\circ}_{\varepsilon}$ in (10.2) and the kernel ${\mathrm{Ker}}\,\wp^{\circ}_{\varepsilon}\subset\operatorname{H}(\Pi_{\infty})[\varepsilon]$, which is a codimension one subspace, depend only on $\varepsilon$, but not on the character $\chi_{\infty}$ with $\chi^{\natural}=\varepsilon$.

Proof.

By the Archimedean period relation in Theorem 2.16 and the proof of [JST19, Proposition 4.9], we have a commutative diagram

[TABLE]

where the bottom arrow is (10.4). The lemma follows easily. ∎

Let $\sigma\in{\mathrm{Aut}}({\mathbb{C}})$. Recall that $\Pi_{f}$ is realized as a space of Shalika functions on $G({\mathbb{A}}_{f})$, and we have a $\sigma$-linear isomorphism $\sigma:\Pi_{f}\to{}^{\sigma}\Pi_{f}$. We also have a $\sigma$-linear isomorphism on the Betti cohomology

[TABLE]

which via (9.1) restricts to a $\sigma$-linear isomorphism $\sigma:\operatorname{H}(\Pi)\to\operatorname{H}({}^{\sigma}\Pi).$ Since (10.5) intertwines the actions of $\pi_{0}({\mathrm{k}}_{\infty}^{\times})$, we have a further restriction (cf. [LLS24, Proposition 6.2]): $\sigma:\operatorname{H}(\Pi)[\varepsilon]\to\operatorname{H}({}^{\sigma}\Pi)[\varepsilon]$. This induces a $\sigma$-linear isomorphism $\sigma:\operatorname{H}(\Pi_{\infty})[\varepsilon]\to\operatorname{H}({}^{\sigma}\Pi_{\infty})[\varepsilon]$ making the following diagram commutative:

[TABLE]

Introduce a family of representations ${}^{\sigma}\Pi^{\natural}:={}^{\sigma}\Pi_{f}\otimes\varepsilon$ of $G^{\natural}=G({\mathbb{A}}_{f})\times\pi_{0}({\mathrm{k}}_{\infty}^{\times})$, where $\varepsilon$ is realized as the $\sigma$-twist of (10.1), noting that ${}^{\sigma}\chi^{\natural}=\chi^{\natural}$ (cf. [LLS24, Remark 6.3]). We equip $\mathfrak{m}_{f}$ with a natural ${\mathbb{Q}}$-rational structure as in [LLS24, Section 5.2].

For all the modular symbols on the cohomologies of $\sigma$-twists, we will also put a left superscript $\sigma$ for clarity. By (9.7), (10.6) and the well-known ${\mathrm{Aut}}({\mathbb{C}})$-equivariance of global modular symbols, we have a commutative diagram

[TABLE]

Here we have used the facts that ${}^{\sigma}F_{\mu}=F_{{}^{\sigma}\mu}$ with ${}^{\sigma}\mu:=\{\mu^{\sigma^{-1}\circ\iota}\}_{\iota\in{\mathcal{E}}_{\mathrm{k}}}$, and that

[TABLE]

The following result is crucial for the definition of Shalika periods.

Lemma 10.2**.**

Under Assumption 1.3 when ${\mathrm{k}}$ has a complex place, the $\sigma$-linear isomorphism $\sigma:\operatorname{H}(\Pi_{\infty})[\varepsilon]\to\operatorname{H}({}^{\sigma}\Pi_{\infty})[\varepsilon]$ restricts to a $\sigma$-linear isomorphism

[TABLE]

Proof.

First note that if ${\mathrm{k}}$ is totally real, then $\dim\operatorname{H}(\Pi_{\infty})[\varepsilon]=1$ so that ${\mathrm{Ker}}\,\wp^{\circ}_{\varepsilon}=\{0\}$, in which case the assertion is trivial.

In view of Lemma 10.1, the assertion follows easily from a diagram chasing in (10.7) for the data $\sigma^{\prime}$ and $\chi^{\prime}$ satisfying Assumption 1.3 when ${\mathrm{k}}$ has a complex place. ∎

10.2. Shalika periods and the end of proof

We now give the definition of Shalika periods. Recall from [JST19, Proposition 4.4] that $\Pi_{f}$ has a unique ${\mathbb{Q}}(\Pi,\boldsymbol{\eta})$-rational structure such that the modular symbol $\wp^{\circ}_{f}$ in (9.6) is defined over $\mathbb{Q}(\Pi,\boldsymbol{\eta},\chi)$ for all algebraic Hecke characters $\chi$. Moreover we have the non-Archimedean period relation

[TABLE]

It is clear that there is a $\kappa_{\varepsilon}\in\operatorname{H}(\Pi_{\infty})[\varepsilon]\setminus{\mathrm{Ker}}\,\wp^{\circ}_{\varepsilon}$ such that the map $\omega_{\Pi^{\natural}}:\Pi_{f}\to\operatorname{H}(\Pi)[\varepsilon]$ by $\phi_{f}\mapsto\iota_{\rm can}(\kappa_{\varepsilon}\otimes\phi_{f})$ belongs to ${\mathrm{Hom}}_{G({\mathbb{A}}_{f})}(\Pi_{f},\operatorname{H}(\Pi)[\varepsilon])^{{\mathrm{Aut}}({\mathbb{C}}/{\mathbb{Q}}(\Pi,\boldsymbol{\eta}))}.$ For $\sigma\in{\mathrm{Aut}}({\mathbb{C}})$ put ${}^{\sigma}\kappa_{\varepsilon}:=\sigma(\kappa_{\varepsilon})\in\operatorname{H}({}^{\sigma}\Pi)[\varepsilon],$ so that the map $\sigma(\omega_{\Pi^{\natural}})$ is ${\mathrm{Aut}}({\mathbb{C}}/{\mathbb{Q}}({}^{\sigma}\Pi,{}^{\sigma}\boldsymbol{\eta}))$-invariant, i.e., it belongs to the space ${\mathrm{Hom}}_{G({\mathbb{A}}_{f})}({}^{\sigma}\Pi_{f},\operatorname{H}({}^{\sigma}\Pi)[\varepsilon])^{{\mathrm{Aut}}({\mathbb{C}}/{\mathbb{Q}}({}^{\sigma}\Pi,{}^{\sigma}\boldsymbol{\eta}))}$, and is given by

[TABLE]

Definition 10.3**.**

Under the Assumption 1.3 when ${\mathrm{k}}$ has a complex place, for every $\sigma\in{\mathrm{Aut}}({\mathbb{C}})$ define the Shalika period

[TABLE]

We justify that $\Omega_{\varepsilon}({}^{\sigma}\Pi,{}^{\sigma}\boldsymbol{\eta})$ is well-defined through the following steps:

• •

By Lemma 10.2, in Definition 10.3 we have that ${}^{\sigma}\kappa_{\varepsilon}\in\operatorname{H}({}^{\sigma}\Pi_{\infty})[\varepsilon]\setminus{\mathrm{Ker}}\,{}^{\sigma}\wp^{\circ}_{\varepsilon},$ hence ${}^{\sigma}\wp^{\circ}_{\varepsilon}({}^{\sigma}\kappa_{\varepsilon})\neq 0$.

• •

By Lemma 10.1, $\Omega_{\varepsilon}({}^{\sigma}\Pi,{}^{\sigma}\boldsymbol{\eta})$ only depends on $\varepsilon$, not on $\chi$.

• •

By definition it is clear that if ${}^{\sigma}\Pi\cong\Pi$ and ${}^{\sigma}\boldsymbol{\eta}\cong\boldsymbol{\eta}$, then $\Omega_{\varepsilon}({}^{\sigma}\Pi,{}^{\sigma}\boldsymbol{\eta})=\Omega_{\varepsilon}(\Pi,\boldsymbol{\eta})$.

• •

For every $\sigma\in{\mathrm{Aut}}({\mathbb{C}})$, there exists a unique class in ${\mathbb{C}}^{\times}/{\mathbb{Q}}({}^{\sigma}\Pi,{}^{\sigma}\boldsymbol{\eta})^{\times}$ given by the Shalika period $\Omega_{\varepsilon}({}^{\sigma}\Pi)$. More precisely we have the following result.

Remark 10.4**.**

We expect that Lemma 10.2 holds without the Assumption 1.3. If this is the case, the Shalika periods $\{\Omega_{\varepsilon}({}^{\sigma}\Pi,{}^{\sigma}\boldsymbol{\eta})\}_{\sigma\in{\mathrm{Aut}}(\mathbb{C})}$ is similarly defined without the Assumption 1.3.

Lemma 10.5**.**

If $\kappa^{\prime}_{\varepsilon}\in\operatorname{H}(\Pi_{\infty})[\varepsilon]\setminus{\mathrm{Ker}}\,\wp^{\circ}_{\varepsilon}$ is another class such that the map

[TABLE]

also belongs to ${\mathrm{Hom}}_{G({\mathbb{A}}_{f})}(\Pi_{f},\operatorname{H}(\Pi)[\varepsilon])^{{\mathrm{Aut}}({\mathbb{C}}/{\mathbb{Q}}(\Pi,\boldsymbol{\eta}))}$, then the resulting Shalika period $\Omega^{\prime}_{\varepsilon}({}^{\sigma}\Pi)$ satisfies that $\Omega^{\prime}_{\varepsilon}({}^{\sigma}\Pi)=c\cdot\Omega_{\varepsilon}({}^{\sigma}\Pi,{}^{\sigma}\boldsymbol{\eta})$ for some $c\in{\mathbb{Q}}({}^{\sigma}\Pi,{}^{\sigma}\boldsymbol{\eta})^{\times}$.

Proof.

By (10.6) and Lemma 10.2, the quotient space $\operatorname{H}({}^{\sigma}\Pi_{\infty})[\varepsilon]/{\mathrm{Ker}}\,{}^{\sigma}\wp^{\circ}_{\varepsilon}$, which is one-dimensional, is defined over ${\mathbb{Q}}({}^{\sigma}\Pi,{}^{\sigma}\boldsymbol{\eta})$. By assumption, the images of ${}^{\sigma}\kappa_{\varepsilon}$ and ${}^{\sigma}\kappa_{\varepsilon}^{\prime}:=\sigma(\kappa_{\varepsilon})$ in the above quotient space differ by a scalar in ${\mathbb{Q}}({}^{\sigma}\Pi,{}^{\sigma}\boldsymbol{\eta})^{\times}$. Hence the assertion is clear by the definition of Shalika periods. ∎

Finally, we finish the proof of the Blasius-Deligne conjecture as follows.

Proof.

(of Theorem 1.4) In view of (10.7) and (10.9), we have a commutative diagram

[TABLE]

Chase the diagram from the top-left corner to the penultimate copy of ${\mathbb{C}}$ in the right column, along the boundary of the diagram in two different directions. From (10.8) and Definition 10.3, we deduce that

[TABLE]

This proves (1.5), from which (1.6) follows directly. ∎

Acknowledgements

D. Jiang is supported in part by the Simons Grants: SFI-MPS-SFM-00005659 and SFI-MPS-TSM-00013449. D. Liu is supported in part by National Key R & D Program of China No. 2022YFA1005300 and Zhejiang Provincial Natural Science Foundation of China under Grant No. LZ22A010006. B. Sun is supported in part by National Key R & D Program of China No. 2022YFA1005300 and New Cornerstone Science Foundation. F. Tian is supported in part by National Key R & D Program of China No. 2022YFA1005304.