Automatic convergence and arithmeticity of modular forms on exceptional groups

TL;DR

This paper establishes an algebraic characterization of cuspidal quaternionic modular forms on exceptional groups, showing their Fourier coefficients are algebraic and proving convergence of their Fourier expansions.

Contribution

It provides the first algebraic description of these modular forms on groups of type F_4 and E_n, including convergence and basis construction.

Findings

Fourier coefficients of these forms are algebraic numbers

Fourier expansions converge absolutely

Forms form a basis with algebraic Fourier coefficients

Abstract

We prove that the space of cuspidal quaternionic modular forms on the groups of type $F_4$ and $E_n$ have a purely algebraic characterization. This characterization involves Fourier coefficients and Fourier-Jacobi expansions of the cuspidal modular forms. The main component of the proof of the algebraic characterization is to show that certain infinite sums, which are potentially the Fourier expansion of a cuspidal modular form, converge absolutely. As a consequence of the algebraic characterization, we deduce that the cuspidal quaternionic modular forms have a basis consisting of forms all of whose Fourier coefficients are algebraic numbers.