Siegel theta series for indefinite quadratic forms
Heilbronn Number Theory Seminar
10th February 2021, 4:00 pm – 5:00 pm
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×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.