Periods and nonvanishing of central L-values for GL(2n)

TL;DR

This paper proves a conjecture linking nonvanishing central L-values for GL(2n) automorphic representations to certain period integrals, using a relative trace formula, and applies these results to local representation theory.

Contribution

It establishes the Guo-Jacquet conjecture for GL(2n) and derives new local results on distinguished supercuspidal representations.

Findings

Proved the Guo-Jacquet conjecture under local hypotheses.

Connected nonvanishing L-values to GL(n,E) periods.

Partially confirmed a conjecture of Prasad and Takloo-Bighash.

Abstract

Let $\pi$ be a cuspidal automorphic representation of PGL($2n$) over a number field $F$, and $\eta$ the quadratic idele class character attached to a quadratic extension $E/F$. Guo and Jacquet conjectured a relation between the nonvanishing of $L(1/2,\pi)L(1/2, \pi \otimes \eta)$ for $\pi$ of symplectic type and the nonvanishing of certain GL($n,E$) periods. When $n=1$, this specializes to a well-known result of Waldspurger. We prove this conjecture, and related global results, under some local hypotheses using a simple relative trace formula. We then apply these global results to obtain local results on distinguished supercuspidal representations, which partially establish a conjecture of Prasad and Takloo-Bighash.