summer 2018 



Jolanta MarzecBallesteros, Uniwersytet im. Adama Mickiewicza w Poznaniu 15.11.2023 11:00, B137 Title: Bounds on Fourier coefficients and global supnorms for Siegel cusp forms of degree 2
Siegel modular forms of degree n are a natural generalisation of classical modular forms (of degree 1). They are holomorphic functions invariant under the action of (subgroups of) Sp(2n,Z), possess Fourier expansion and  when cuspidal  are squareintegrable. One of the most basic and yet unsolved problems concerns the growth of their Fourier coefficients. It is known as RamanujanPetersson conjecture when n=1 and as ResnikoffSaldana conjecture when n>1. Deligne's proof of the first conjecture had a significant impact on many problems in mathematics, including an optimal solution to a supnorm problem given by Xia. When n>1 hardly anything is known. In the talk we focus on the case n=2. We will present a special case of GanGrossPrasad conjecture, proven recently by Furusawa and Morimoto, and explain how it led us to the best current bounds on Fourier coefficients, both unconditionally and under GRH. As a consequence we will derive a global supnorm bound for Siegel cusp forms of degree 2 invariant by Sp(4,Z) in terms of their weight and L^2norm. The aforementioned results are the effect of joint work with Abhishek Saha and Félicien Comtat.


Last modified: 16.11.2022 