A Twisted Motohashi Formula and Weyl-Subconvexity for $L$-functions of Weight Two Cusp Forms

TL;DR

This paper develops a new twisted Motohashi formula for cubic moments of $L$-functions of weight two cusp forms, leading to subconvex bounds and progress on the Ramanujan-Petersson conjecture.

Contribution

It introduces a novel twisted Motohashi formula for cubic moments of $L$-functions, enabling new bounds and conjecture estimates.

Findings

Weyl-subconvex bounds for twisted $L$-functions

Estimates towards Ramanujan-Petersson conjecture for weight 3/2 forms

New formula connecting moments of $L$-functions

Abstract

We derive a Motohashi-type formula for the cubic moment of central values of $L$-functions of level $q$ cusp forms twisted by quadratic characters of conductor $q$, previously studied by Conrey and Iwaniec and Young. Corollaries of this formula include Weyl-subconvex bounds for $L$-functions of weight two cusp forms twisted by quadratic characters, and estimates towards the Ramanujan-Petersson conjecture for Fourier coefficients of weight 3/2 cusp forms.