Applications of nonarchimedean developments to archimedean nonvanishing results for twisted $L$-functions

TL;DR

This paper demonstrates the nonvanishing of certain automorphic $L$-function values at the center for many twists, combining nonarchimedean Iwasawa theory with new computational techniques.

Contribution

It introduces a novel approach that merges classical nonarchimedean methods with modern machinery to establish broad nonvanishing results for automorphic $L$-functions.

Findings

Proves nonvanishing for all but finitely many twists by unitary characters.

Extends prior $p$-adic Eisenstein series computations to cases with unramified primes.

Provides a framework connecting nonarchimedean theory with automorphic $L$-functions.

Abstract

We prove the nonvanishing of the twisted central critical values of a class of automorphic $L$-functions for twists by all but finitely many unitary characters in particular infinite families. While this paper focuses on $L$-functions associated to certain automorphic representations of unitary groups, it illustrates how decades-old nonarchimedean methods from Iwasawa theory can be combined with the output of new machinery to achieve broader nonvanishing results. In an appendix, which concerns an intermediate step, we also outline how to extend relevant prior computations for $p$-adic Eisenstein series and $L$-functions on unitary groups to the case where primes dividing $p$ merely needs to be unramified (whereas prior constructions required $p$ to split completely) in the associated reflex field.