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
Summary
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
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