Variance of GL(2) Fourier coefficients in arithmetic progressions

TL;DR

This paper enhances understanding of the variance of Fourier coefficients of modular forms in arithmetic progressions by employing advanced bounds on L-functions and convolution sums.

Contribution

It improves previous results on the variance of Fourier coefficients by applying new bounds on Rankin-Selberg L-functions and shifted convolution estimates.

Findings

Enhanced bounds on the variance of Fourier coefficients in arithmetic progressions.

Application of first moment bounds of Rankin-Selberg L-functions.

Non-trivial estimates for shifted convolution sums achieved.

Abstract

We improve a result of Lau and Zhao on the variance of Fourier coefficients of primitive cuspidal modular forms for SL2(Z) in arithmetic progressions. This is achieved by using bounds on the first moment of Rankin-Selberg L-functions in the height aspect and non-trivial estimates for shifted convolution sums.