Seminar: Non-archimedean geometry

Lucas Gerth (Universität Frankfurt)

Abstract: We study arithmetic analogues of theta series. Given a simplectic vector space V and a Schwartz function f on V, there is a collection of cycles Z(n,f), consisting of CM points, on the Siegel modular variety. Assuming that f satisfies a strong regular semisimple condition at some prime p, we show that the generating series of the degrees of the cycles Z(n,f) is a modular form, We identify it explicitly with a classical theta series for a quaternion unitary similitude group. The proof relies on the p-adic uniformization of the supersingular locus on the Siegel modular variety.