The shifted fourth moment of modular form L-functions in the weight aspect

TL;DR

This paper establishes a reciprocity formula connecting the fourth moment of modular form L-functions in the weight aspect to higher moments of the Riemann zeta function and automorphic L-functions, advancing understanding of their structure.

Contribution

It introduces a new reciprocity formula for the fourth moment of modular form L-functions, enabling analysis of their main term structure for potential generalizations.

Findings

Derived a reciprocity formula linking fourth and eighth moments.

Analyzed the main term structure for higher moments.

Provided insights for extending methods to higher moments.

Abstract

We prove a reciprocity type formula for the fourth moment of L-functions associated to holomorphic primitive cusp forms of level one and large weight which relates it to the eighth moment of the Riemann zeta function and the dual weighted fourth moments of automorphic L-functions (both holomorphic and Maass). The main objective of the paper is to study the structure of the main term for possible generalization of the method to higher moments.