# Twisted moments of L-functions and spectral reciprocity

## TL;DR

This paper establishes a spectral reciprocity formula linking twisted fourth moments of automorphic L-functions across different levels, leading to new bounds and simplified proofs for moments in analytic number theory.

## Contribution

It introduces a novel reciprocity formula connecting moments of automorphic L-functions of different levels, enabling new bounds and simpler proofs.

## Key findings

- Derived subconvexity bounds in the level aspect.
- Provided a short proof for an upper bound of the fifth moment.
- Established a spectral reciprocity formula for twisted moments.

## Abstract

A reciprocity formula is established that expresses the fourth moment of automorphic L-functions of level q twisted by the ell-th Hecke eigenvalue as the fourth moment of automorphic L-functions of level ell twisted by the q-th Hecke eigenvalue. Direct corollaries include subconvexity bounds for L-functions in the level aspect and a short proof of an upper bound for the fifth moment of automorphic L-functions.