Gamma factors of pairs and a local converse theorem in families

TL;DR

This paper establishes a local converse theorem for families of representations of GL(n) over p-adic fields, extending Rankin-Selberg integrals to the family setting, advancing the understanding of automorphic forms and their L-functions.

Contribution

It proves a local converse theorem for l-adic families of GL(n) representations and extends Rankin-Selberg integrals to families, enhancing tools for automorphic representation analysis.

Findings

Proved a local converse theorem for GL(n) x GL(n-1) in families.

Extended Rankin-Selberg integrals to the setting of families.

Provided new methods for studying automorphic L-functions in families.

Abstract

We prove a GL(n)xGL(n-1) local converse theorem for l-adic families of smooth representations of GL(n,F) where F is a finite extension of Q_p and l is different from p. To do so, we also extend the theory of Rankin-Selberg integrals, first introduced by Jacquet, Piatetski-Shapiro, and Shalika, to the setting of families, continuing previous work of the author.