The second moment of twisted modular L-functions

TL;DR

This paper establishes an asymptotic formula with a power-saving error term for the second moment of twisted L-functions of fixed cusp forms, applicable to most moduli, with implications for non-vanishing and higher moments.

Contribution

It provides a novel asymptotic formula for the second moment of twisted L-functions with power-saving error, using spectral analysis and bounds on Kloosterman sums.

Findings

Asymptotic formula with power-saving error for second moment

Applicable to 99.9% of admissible moduli

Results on non-vanishing and higher moments

Abstract

We prove an asymptotic formula with a power saving error term for the (pure or mixed) second moment of central values of L-functions of any two (possibly equal) fixed cusp forms f, g twisted by all primitive characters modulo q, valid for all sufficiently factorable q including 99.9% of all admissible moduli. The two key ingredients are a careful spectral analysis of a potentially highly unbalanced shifted convolution problem in Hecke eigenvalues and power-saving bounds for sums of products of Kloosterman sums where the length of the sum is below the square-root threshold of the modulus. Applications are given to simultaneous non-vanishing and lower bounds on higher moments of twisted L-functions.