Exceptional theta functions and arithmeticity of modular forms on $G_2$

Abstract

Quaternionic modular forms on the split exceptional group $G_2=G_2^s$ were defined by Gan-Gross-Savin. A remarkable property of these automorphic functions is that they have a robust notion of Fourier expansion and Fourier coefficients, similar to the classical holomorphic modular forms on Shimura varieties. In this paper we prove that in even weight $\ell$ at least $6$, there is a basis of the space of cuspidal modular forms of weight $\ell$ such that all the Fourier coefficients of elements of this basis are in the cyclotomic extension of $\mathbf{Q}$. Our main tool for proving this is to develop a notion of "exceptional theta functions" on $G_2$.

Main result substantially improved. We now prove that the cuspidal quaternionic modular forms of even weight at least $6$ have an algebraic structure, defined in terms of Fourier coefficients