Certain Fourier operators and their associated Poisson summation formulae on $\mathrm{GL}_1$

Dihua Jiang and Zhilin Luo

Vol. 326 (2023), No. 2, 301–372

DOI: 10.2140/pjm.2023.326.301

Abstract

We explore the possibility of using harmonic analysis on ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_1$ to understand Langlands automorphic $L$-functions in general, as a vast generalization of the PhD Thesis of J. Tate in 1950. For a split reductive group $G$ over a number field $k$, let $G^{\vee }(ℂ)$ be its complex dual group and $\rho$ be an $n$-dimensional complex representation of $G^{\vee }(ℂ)$. For any irreducible cuspidal automorphic representation $\sigma$ of $G(\mathbb{𝔸})$, where $\mathbb{𝔸}$ is the ring of adeles of $k$, we introduce the space ${\mathcal{𝒮}}_{\sigma ,\rho }({\mathbb{𝔸}}^{\times })$ of $(\sigma ,\rho )$-Schwartz functions on ${\mathbb{𝔸}}^{\times }$ and $(\sigma ,\rho )$-Fourier operator ${\mathcal{ℱ}}_{\sigma ,\rho ,\psi }$ that takes ${\mathcal{𝒮}}_{\sigma ,\rho }({\mathbb{𝔸}}^{\times })$ to ${\mathcal{𝒮}}_{\tilde{\sigma },\rho }({\mathbb{𝔸}}^{\times })$, where $\tilde{\sigma }$ is the contragredient of $\sigma$. By assuming the local Langlands functoriality for the pair $(G,\rho )$, we show that the $(\sigma ,\rho )$-theta functions ${\Theta }_{\sigma ,\rho }(x,\varphi ):={\sum <!--FUNCTION\; APPLICATION-->}_{\alpha \in k^{\times }}\varphi (\alpha x)$ converge absolutely for all $\varphi \in {\mathcal{𝒮}}_{\sigma ,\rho }({\mathbb{𝔸}}^{\times })$. We state conjectures on the $(\sigma ,\rho )$-Poisson summation formula on ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_1$, and prove them in the case where $G={\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_n$ and $\rho$ is the standard representation of ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_n(ℂ)$. This is done with the help of results of Godement and Jacquet (1972). As an application, we provide a spectral interpretation of the critical zeros of the standard $L$-functions $L(s,\pi \times \chi )$ for any irreducible cuspidal automorphic representation $\pi$ of ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_n(\mathbb{𝔸})$ and idele class character $\chi$ of $k$, extending theorems of C. Soulé (2001) and A. Connes (1999). Other applications are in the introduction.

Keywords

invariant distribution, Fourier operator, Poisson summation formula, automorphic representation, automorphic $L$-function, representation of real and $p$-adic reductive groups, spectral interpretation of critical zeros of automorphic $L$-functions

Mathematical Subject Classification

Primary: 11F66, 43A32, 46S10

Secondary: 11F70, 22E50, 43A80

Milestones

Received: 10 May 2023

Revised: 18 November 2023

Accepted: 17 December 2023

Published: 9 January 2024