Averages of Fourier coefficients of Siegel modular forms and representation of binary quadratic forms by quadratic forms in four variables

TL;DR

This paper establishes asymptotic formulas for the average number of representations of binary quadratic forms by quaternary forms and analyzes Fourier coefficients of Siegel theta series, revealing a positive proportion of forms are represented as discriminant grows.

Contribution

It provides new asymptotic results on representation counts and Fourier coefficients for quadratic forms, extending understanding of their distribution and representation properties.

Findings

Asymptotic formula for average representations as discriminant increases

Lower bounds on the number of forms represented by a fixed quaternary form

Positive proportion of binary forms represented for large discriminants

Abstract

Let $-d$ be a a negative discriminant and let $T$ vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant $-d$. We prove an asymptotic formula for $d \to \infty$ for the average over $T$ of the number of representations of $T$ by an integral positive definite quaternary quadratic form and obtain results on averages of Fourier coefficients of linear combinations of Siegel theta series. We also find an asymptotic bound from below on the number of binary forms of fixed discriminant $-d$ which are represented by a given quaternary form. In particular, we can show that for growing $d$ a positive proportion of the binary quadratic forms of discriminant $-d$ is represented by the given quaternary quadratic form.