On the Waldspurger Formula and the Metaplectic Ramanujan Conjecture over Number Fields

TL;DR

This paper extends the Waldspurger formula to arbitrary number fields using Bessel identities and applies it to establish new bounds towards the Ramanujan conjecture for automorphic forms on the metaplectic group over general number fields.

Contribution

It generalizes the Waldspurger formula from totally real fields to all number fields and applies this to derive bounds related to the Ramanujan conjecture for metaplectic automorphic forms.

Findings

Extended Waldspurger formula to arbitrary number fields.

Provided the first non-trivial bounds towards the Ramanujan conjecture in this setting.

Utilized Bessel identities over complex fields in the extension.

Abstract

In this paper, by inputting the Bessel identities over the complex field in previous work of the authors, the Waldspurger formula of Baruch and Mao is extended from totally real fields to arbitrary number fields. This is applied to give a non-trivial bound towards the Ramanujan conjecture for automorphic forms of the metaplectic group $\widetilde{\mathrm{SL}}_2$ for the first time in the generality of arbitrary number fields.