Toward explicit formulas for higher representation numbers of quadratic forms

TL;DR

This paper develops explicit formulas for higher representation numbers of quadratic forms by expressing average Siegel theta series as linear combinations of Eisenstein series, advancing understanding of quadratic form representations.

Contribution

It constructs maps that make average Siegel theta series eigenforms and explicitly expresses these series as linear combinations of Eisenstein series.

Findings

Average Siegel theta series are eigenforms under certain maps.

Explicit linear combinations of Eisenstein series represent average theta series.

Provides a method to evaluate the action of maps on Eisenstein series without Fourier coefficients.

Abstract

It is known that average Siegel theta series lie in the space of Siegel Eisenstein series. Also, every lattice equipped with an even integral quadratic form lies in a maximal lattice. Here we consider average Siegel theta series of degree 2 attached to maximal lattices; we construct maps for which the average theta series is an eigenform. We evaluate the action of these maps on Siegel Eisenstein series of degree 2 (without knowing their Fourier coefficients), and then realise the average theta series as an explicit linear combination of the Eisenstein series.