Siegel theta series for indefinite quadratic forms

Abstract: Due to a result by Vignéras from 1977, there is a quite simple way to determine whether a certain theta series admits modular transformation properties. To be more specific, she showed that solving a differential equation of second order serves as a criterion for modularity. We generalize this result for Siegel theta series of arbitrary genus $n$. In order to do so, we construct Siegel theta series for indefinite quadratic forms by considering functions that solve an $n\times n$-system of partial differential equations. These functions do not only give examples of Siegel theta series, but build a basis of the family of Schwartz functions that generate series that transform like modular forms.

number theory

Audience: researchers in the topic

Heilbronn number theory seminar

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