A local converse theorem for U(1,1)

TL;DR

This paper introduces a new gamma-factor for generic representations of the unitary group U(1,1) and proves a local converse theorem, also providing a novel proof for the GL_2 case using related gamma-factors.

Contribution

It defines a gamma-factor for U(1,1) representations and establishes a local converse theorem, offering a new proof for the GL_2 case based on Jacquet's gamma-factor.

Findings

Proved a local converse theorem for U(1,1) using the new gamma-factor.

Provided a new proof of the local converse theorem for GL_2.

Defined a gamma-factor for generic representations of U(1,1).

Abstract

In this paper, we define a $\gamma$-factor for generic representations of $\RU(1,1)\times \Res_{E/F}(\GL_1)$ and prove a local converse theorem for $\RU(1,1)$ using the $\gamma$-factor we defined. We also give a new proof of the local converse theorem for $\GL_2$ using a $\gamma$-factor of $\GL_2\times \GL_2$ type which was originally defined by Jacquet in \cite{J}.