Zeroes of Rankin-Selberg $L$-functions and the trace formula

TL;DR

This paper links the zeros of Rankin--Selberg L-functions to the spectral analysis of automorphic forms via the trace formula, extending known results from GL(2) to higher rank groups and deriving new bounds and relations.

Contribution

It generalizes the connection between zeros of L-functions and spectral contributions from GL(2) to GL(n), providing new bounds and base change relations.

Findings

Established a lower bound for the sum of zeros of Rankin--Selberg L-functions.

Derived a base change relation for zeros in cyclic prime degree extensions.

Extended previous GL(2) results to GL(n) in the context of the trace formula.

Abstract

We express the contribution of certain maximal parabolic Eisenstein series to the spectral side of the Arthur--Selberg trace formula for GL$(n)$ in terms of zeroes of Rankin--Selberg $L$-functions, generalizing previous results for GL(2) and Hecke $L$-functions. As applications, we prove a lower bound for the sum of these zeroes, and a base change relation between the zeroes in the case $n=2$ for cyclic extensions of prime degree.