Spectral reciprocity for $\mathrm{GL}(n)$ and simultaneous non-vanishing of central $L$-values

TL;DR

This paper establishes simultaneous non-vanishing results for central L-values of automorphic representations on GL(n+1) and GL(n-1), using spectral reciprocity formulas to analyze families of automorphic forms over totally real fields.

Contribution

It introduces a spectral reciprocity formula for the average of products of Rankin--Selberg L-functions, enabling new non-vanishing results for automorphic L-values in higher rank settings.

Findings

Proves simultaneous non-vanishing of L-values for certain automorphic representations.

Establishes a reciprocity formula relating averages of L-functions over automorphic families.

Demonstrates non-vanishing results in the level aspect for GL(n) automorphic forms.

Abstract

Let $F$ be a totally real number field and $n\ge 3$. Let $\Pi$ and $\pi$ be cuspidal automorphic representations for $\mathrm{PGL}_{n+1}(F)$ and $\mathrm{PGL}_{n-1}(F)$, respectively, that are unramified and tempered at all finite places. We prove simultaneous non-vanishing of the Rankin--Selberg $L$-values $L(1/2,\Pi\otimes\widetilde{\sigma})$ and $L(1/2,\sigma\otimes\widetilde{\pi})$ for certain sequences of $\sigma$ varying over cuspidal automorphic representations for $\mathrm{PGL}_n(F)$ with conductor tending to infinity in the level aspect and bearing certain local conditions. Along the way, we also prove a reciprocity formula for the average of the product of Rankin--Selberg $L$-functions $L(1/2,\Pi\otimes\widetilde{\sigma})L(1/2,\sigma\otimes\widetilde{\pi})$ over a conductor aspect family of $\sigma$.