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

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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 \(d \rightarrow \infty \) for the average over T of the number of representations of T by an integral positive definite quaternary quadratic form and obtain bounds for averages of Fourier coefficients of linear combinations of Siegel theta series. We also find an asymptotic estimate 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.

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  1. Universität des Saarlandes (Geb. E2.4), Postfach 151150, 66041, Saarbrücken, Germany

    Rainer Schulze-Pillot

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  1. Rainer Schulze-Pillot
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Correspondence to Rainer Schulze-Pillot.

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To the memory of Hiroshi Saito

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Schulze-Pillot, R. Averages of Fourier coefficients of Siegel modular forms and representation of binary quadratic forms by quadratic forms in four variables. Math. Ann. 368, 923–943 (2017). https://doi.org/10.1007/s00208-016-1448-4

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  • DOI: https://doi.org/10.1007/s00208-016-1448-4

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