The Theta Correspondence, Periods of Automorphic Forms and Special Values of Standard Automorphic L-Functions

TL;DR

This paper explores the deep connections between the zeros and poles of automorphic L-functions, theta lifts, Fourier coefficients, and periods of automorphic forms, establishing a new formula for special values of these L-functions.

Contribution

It demonstrates that Fourier coefficients of automorphic forms are linked to periods of their theta lifts, leading to a new formula for special L-values in terms of these coefficients and periods.

Findings

Link between zeros/poles of L-functions and theta lifts.

Fourier coefficients relate to periods of automorphic forms.

Proved a new special value formula for automorphic L-functions.

Abstract

The zeros and poles of standard automorphic $L$-functions attached to representations of classical groups are linked to the nonvanishing of lifts in the theory of the theta correspondence. The results of this paper show that when a cuspidal representation $\sigma$ of a symplectic group $G$ lifts to a cuspidal representation $\pi = \theta_{\psi}(\sigma)$ of an orthogonal group $H$ attached to a quadratic space $V$ of dimension $m$, the Fourier coefficients of automorphic forms in $\sigma$ are linked to periods of automorphic forms in $\pi$. Consequently, when our results are combined with the Rallis inner product formula in the convergent range or the second term range, we prove a special value formula for the standard automorphic $L$-function $L(s,\sigma,\chi_V)$ attached to $\sigma$ (and twisted by the character $\chi_V$ of $V$) at the point $s_{m,2n}=m/2-(2n+1)/2$ in terms of the Fourier coefficients of automorphic forms in $\sigma$ and periods of forms in $\pi$.