Martin Westerholt-Raum, Chalmers Technological University
Title: Harmonic weak Siegel Maass forms
Abstract: Harmonic weak Maass forms for SL(2,R) where defined by Bruinier and Funke more
than ten years ago. They have been successfully applied to combinatorial
problems thanks to their overlap with indefinite theta functions. Their genuine
arithmetic, on the other hand, is linked to derivatives of L-series, as
demonstrated by Bruinier and Ono. We start by revisiting these connections.
Siegel modular forms are modular forms for the group Sp(g,R). If g=1 they
coincide with elliptic modular forms, but in the general case they are more
intricate. In joint work with Bringmann and Richter, the speaker studied real
analytic Siegel modular forms and connected their Fourier Jacobi coefficients in
the case of g=2 to harmonic weak Maass forms for SL(2,R). We showcase this
connection, and exhibit the importance of Fourier Jacobi coefficients in the
study of Siegel modular forms.
Finally, we discuss the existence of harmonic weak Siegel Maass forms. Using
the connection of Eisenstein series and principal series representations, one
manages to obtain sufficiently tight control of Dolbeault cohomology to show
that every non-holomorphic Saito-Kurokawa lift can be further lifted to a
harmonic weak Siegel Maass form.
