A Converse Theorem for Metaplectic Eisenstein Series on $SL_2(\mathbb{A})$

TL;DR

This paper establishes a converse theorem linking double Dirichlet series satisfying certain functional equations to metaplectic Eisenstein series on the double cover of SL_2, enhancing understanding of their analytic properties.

Contribution

It provides a new converse theorem characterizing metaplectic Eisenstein series via their associated double Dirichlet series and functional equations.

Findings

Double Dirichlet series satisfy functional equations

These series are Mellin transforms of Eisenstein series

The work advances the theory of automorphic forms on metaplectic groups

Abstract

The purpose of this work is to produce a converse theorem for adelic Eisenstein series on the double metaplectic cover of the group $SL_2(\mathbb{A})$. We show that the double Dirichlet series, which satisfy the natural functional equations required in our theorem, are in fact Mellin transforms at infinity of metaplectic Eisenstein series.