Class group twists and Galois averages of $\operatorname{GL}_n$-automorphic $L$-functions

TL;DR

This paper develops estimates for the central values and nonvanishing of automorphic $L$-functions on $ ext{GL}_n$ over totally real fields, using class group twists, integral representations, and bounds on shifted convolutions, with implications for Deligne's conjecture.

Contribution

It introduces new estimates for twisted automorphic $L$-functions at critical points, including unconditional nonvanishing results for low dimensions, and connects these to rationality theorems and Deligne's conjecture.

Findings

Unconditional nonvanishing for $n \\leq 3$.

New bounds on shifted convolution problems.

Connections to Deligne's conjecture for automorphic motives.

Abstract

Fix $n \geq 2$ an integer, and let $F$ be a totally real number field. We derive estimates for the finite parts of the $L$-functions of irreducible cuspidal $\operatorname{GL}_n({\bf{A}}_F)$-automorphic representations twisted by class group characters or ring class characters of a totally imaginary quadratic extensions $K$ of $F$, evaluated at central values $s=1/2$ or more generally values $s \in {\bf{C}}$ within the strip $\frac{1}{2} - \frac{1}{n^2 + 1} < \Re(s) < 1$. Assuming the generalized Ramanujan conjecture at infinity, we obtain estimates for all arguments in the critical strip $0 < \Re(s) < 1$. We also derive finer nonvanishing estimates for central values $s=1/2$ twisted by ring class characters of $K$. When the dimension $n \leq 4$ is small, these give us nonvanishing estimates depending on the best known approximations towards the generalized Lindel\"of hypothesis for $\operatorname{GL}_2({\bf{A}}_F)$-automorphic forms in the level aspect, and in particular unconditional nonvanishing for $n \leq 3$ (with the case of $n=3$ being new). We derive such estimates via certain exact integral representations for the moments, and in particular for new developments of bounds on the shifted convolution problem in this context. In the setting where the cuspidal representation is cohomological of even rank $n \geq 2$, we also explain how to view these estimates in terms of recent rationality theorems towards Deligne's conjecture for automorphic motives over CM fields.