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

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.

v5: Some typos corrected

References in corpus (3)

• Twisted L-functions over number fields and Hilbert's eleventh problem
• Local-global principles for representations of quadratic forms
• Local conditions for global representations of quadratic forms

Cited by in corpus (1)

• Distribution of toric periods of modular forms on definite quaternion algebras