11 April 2016 to 15 July 2016
Institut Henri Poincaré
Europe/Paris timezone


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.