Wojciech J. Gajda

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 square-integrable. One of the most basic and yet unsolved problems concerns the growth of their Fourier coefficients. It is known as Ramanujan-Petersson conjecture when n=1 and as Resnikoff-Saldana 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 sup-norm 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 Gan-Gross-Prasad 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 sup-norm bound for Siegel cusp forms of degree 2 invariant by Sp(4,Z) in terms of their weight and L^2-norm. The aforementioned results are the effect of joint work with Abhishek Saha and Félicien Comtat.