Fourier coefficients of Siegel modular forms of block-diagonal index

Research Project, 2024 – 2027

The aim of the proposed work is to advance our understanding of Fourier coefficients of Siegel modular forms. They appear as generating series of representation numbers of quadratic forms in discrete geometry and special cycles in the Kudla Program in algebraic geometry, among others. The goal is to demonstrate that Fourier coefficients of diagonal and block-diagonal index do not vanish, which implies the non-vanishing of corresponding representation and intersection numbers. Additionally, we want to explore how Siegel modular forms can be connected to integer combinatorics and apply our results to this field, too. The proposed work, beyond its own scope, sets the scene for future investigations of indivisibility as opposed to non-vanishing, which is a main topic in integer combinatorics.Fourier coefficients of Siegel modular forms are difficult to study, as they largely escape the powerful machinery of the Langlands Program. The state-of-the-art is to leverage results on the simplest special case, elliptic modular forms. The proposed project goes beyond this and targets deeper statements on relations among parts of the Fourier expansion that can only be accessed through genuine properties of Siegel modular forms. The key tools to achieve our goals come from analytic number theory and finite dimensional representation theory.The project consists of three parts, each with three sub-goals, which serve as the basis for the time plan and feasibility of the research.