The shifted convolution problem for Fourier coefficients of Siegel modular forms of degree $2$

TL;DR

This paper establishes a power-saving bound for shifted convolution sums of Fourier coefficients of Siegel cusp forms of degree 2, advancing understanding of automorphic forms on higher rank groups.

Contribution

It introduces a novel automorphic approach to the delta method, providing the first nontrivial estimate for such sums on higher rank groups like GSp(4).

Findings

First nontrivial estimate for shifted convolution sums on higher rank groups.

Automorphic reinterpretation of the delta method simplifies the analysis.

Bound relies on Fourier coefficients of Siegel Poincare series and Weil bound.

Abstract

We provide a power-saving bound for certain smoothed shifted convolution sums for Fourier coefficients of Siegel cusp forms. This result is the first nontrivial estimate for a shifted convolution sum with two cusp forms on a group of higher rank than $\GL_2$. Our approach is based on a novel automorphic reinterpretation of the delta method of Duke, Friedlander, and Iwaniec. The method reduces the problem to the estimation of Fourier coefficients of Siegel Poincare series, which is ultimately based on the Weil bound.