Non-vanishing of L-functions, the Ramanujan conjecture, and families of Hecke characters

TL;DR

This paper establishes a non-vanishing result for families of Rankin-Selberg L-functions twisted by Hecke characters, leading to simplified bounds towards the Generalized Ramanujan Conjecture for GL_n cusp forms.

Contribution

It proves a non-vanishing theorem for GL_n x GL_n L-functions with twists by Hecke characters and simplifies existing bounds towards the Ramanujan conjecture.

Findings

Non-vanishing of L-functions in the critical strip for twisted families.

Simplified proof of bounds towards the Ramanujan conjecture.

Use of Arakelov ray class groups for regularizing units.

Abstract

We prove a non-vanishing result for families of $\GL_n\times\GL_n$ Rankin-Selberg $L$-functions in the critical strip, as one factor runs over twists by Hecke characters. As an application, we simplify the proof, due to Luo, Rudnick, and Sarnak, of the best known bounds towards the Generalized Ramanujan Conjecture at the infinite places for cusp forms on $\GL_n$. A key ingredient is the regularization of the units in residue classes by the use of an Arakelov ray class group.