A local converse Theorem for U(2,2)

TL;DR

This paper proves a local converse theorem for generic representations of the unitary group U(2,2) over p-adic fields, extending to symplectic groups under certain conditions, using purely local and analytic methods.

Contribution

It establishes the local converse theorem for U(2,2) in unramified cases and odd residue characteristic, also applying to symplectic groups with the same approach.

Findings

Proves the local converse theorem for U(2,2) when E/F is unramified or residue characteristic is odd.

Extends the method to prove the theorem for Sp_4(F) and its metaplectic cover under odd residue characteristic.

Uses a purely local and analytic approach, avoiding global methods.

Abstract

Let $F$ be a $p$-adic field and $E/F$ be a quadratic extension. In this paper, we prove the local converse theorem for generic representations of $\textrm{U}_{E/F}(2,2)$ if $E/F$ is unramified or the residue characteristic of $F$ is odd. Our method is purely local and analytic, and the same method also gives the local converse theorem for $\textrm{Sp}_4(F)$ and $\widetilde {\textrm{Sp}}_4(F)$ if the residue characteristic of $F$ is odd.