Abstract

We compute the Fourier coefficients of a basis of the space of degree two Siegel–Eisenstein series of square-free level $$N$$ transforming with the trivial character. We then apply these formulæ to present some explicit examples of higher representation numbers attached to non-unimodular quadratic forms.

Highlights

  • Eisenstein series have played an important rôle in the theory of Siegel modular forms, dating back to Siegel’s Hauptsatz which expresses the genus-average Siegel theta series as a linear combination of Eisenstein series

  • An important example of this is that the Fourier coefficients of the Siegel–Eisenstein series remain unknown in many cases

  • Maass [11,12] obtained a formula for the Fourier coefficients of degree 2 Siegel–Eisenstein series by explicitly computing the local densities in Siegel’s formula

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Summary

Introduction

Eisenstein series have played an important rôle in the theory of Siegel modular forms, dating back to Siegel’s Hauptsatz which expresses the genus-average Siegel theta series as a linear combination of Eisenstein series. Using a convenient level 1 formula, namely that of [5], one can argue by induction to obtain Fourier coefficients for a full basis of the Eisenstein subspace in the case of squarefree level and trivial character This is carried out in Lemma 3.1: the main bulk of the computation is placing these in a more elucidating form as stated in Theorem 3.3. There is a finite number of integral quadratic forms which have a single-class genus, and from the viewpoint of degree 2 representation numbers only the 8-dimensional ones have dimension large enough to study via Siegel–Eisenstein series (i.e. the Eisenstein series of degree 2 and weight 4 converges) of even weight (since odd weight Eisenstein series are problematic to define with trivial character). The remaining 4 have small prime level and for these we will note how the methods of this paper give new closed formulæ for their degree 2 representation numbers

Siegel–Eisenstein series and Hecke operators
Calculation of the Fourier coefficients
Applications to representation numbers of quadratic forms
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