Siegel theta series for indefinite quadratic forms

Abstract: In this talk, we will give an insight into the field of Siegel modular forms. As they occur as a generalization of elliptic modular forms, some results can be transferred from the well-known theory developed for these functions. We examine a result by Vignéras, who showed that 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.

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 we can even determine a basis of Schwartz functions that generate series which transform like modular forms.

number theory

Audience: researchers in the topic

Dublin Algebra and Number Theory Seminar

Series comments: Passcode: The 3-digit prime numerator of Riemann zeta at -11