# New bounds towards Fourier coefficients of Siegel modular forms

Lead Research Organisation:
Queen Mary, University of London

Department Name: Sch of Mathematical Sciences

### Abstract

Automorphic forms are highly symmetric functions that constitute one of the most important concepts in modern mathematics. For instance, Sir Andrew Wiles' proof of Fermat's Last Theorem in 1995 relied on a deep connection between modular forms (an example of automorphic forms) and elliptic curves. Together with their associated L-functions, automorphic forms are also central objects in the Langlands programme - a vast web of theorems and conjectures connecting algebra, geometry, number theory, and analysis - which is one of the most active areas of mathematical research today.

A key way in which automorphic forms can be understood is via their Fourier coefficients. Basic questions about Fourier coefficients of automorphic forms can contain an incredible amount of deep mathematics and can be extremely hard. For example, Ramanujan's conjecture (made in 1916) regarding an upper bound for the size of Fourier coefficients of modular forms was finally proved by Deligne in 1974, as a consequence of his deep, Fields medal winning work in arithmetic geometry. A very natural generalization of the (classical) modular forms is given by the Siegel modular forms, which were first investigated by Carl Ludwig Siegel in the 1930s. They are of great importance in number theory and the Langlands programme, and also have applications to physics and information technology. To give an example, Wiles' proof of Fermat's last theorem relies on a deep connection between modular forms and elliptic curves; the generalization of this to one dimension up (the so-called paramodular conjecture, which is a hot topic currently) involves Siegel modular forms.

The main goal of this project is to prove new bounds towards the Fourier coefficients of (cuspidal) Siegel modular forms and thus make progress towards the famous Resnikoff-Saldana conjecture, a problem that has been open for almost 50 years. The successful completion of this project will lead to new improved understanding of Siegel modular forms, and it will demonstrate for the first time deep links between the Resnikoff-Saldana conjecture and other central conjectures in number theory. This will open up many avenues of further exploration.

A key way in which automorphic forms can be understood is via their Fourier coefficients. Basic questions about Fourier coefficients of automorphic forms can contain an incredible amount of deep mathematics and can be extremely hard. For example, Ramanujan's conjecture (made in 1916) regarding an upper bound for the size of Fourier coefficients of modular forms was finally proved by Deligne in 1974, as a consequence of his deep, Fields medal winning work in arithmetic geometry. A very natural generalization of the (classical) modular forms is given by the Siegel modular forms, which were first investigated by Carl Ludwig Siegel in the 1930s. They are of great importance in number theory and the Langlands programme, and also have applications to physics and information technology. To give an example, Wiles' proof of Fermat's last theorem relies on a deep connection between modular forms and elliptic curves; the generalization of this to one dimension up (the so-called paramodular conjecture, which is a hot topic currently) involves Siegel modular forms.

The main goal of this project is to prove new bounds towards the Fourier coefficients of (cuspidal) Siegel modular forms and thus make progress towards the famous Resnikoff-Saldana conjecture, a problem that has been open for almost 50 years. The successful completion of this project will lead to new improved understanding of Siegel modular forms, and it will demonstrate for the first time deep links between the Resnikoff-Saldana conjecture and other central conjectures in number theory. This will open up many avenues of further exploration.