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

TL;DR

This paper introduces exceptional theta functions on the split exceptional group G_2 to analyze the arithmetic properties of quaternionic modular forms, showing that certain bases have Fourier coefficients in cyclotomic fields.

Contribution

It develops a new concept of exceptional theta functions on G_2 and proves that bases of cuspidal modular forms have Fourier coefficients in cyclotomic extensions for even weights at least 6.

Findings

Existence of bases with Fourier coefficients in cyclotomic fields

Development of exceptional theta functions on G_2

Arithmeticity results for modular forms

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 $G_2$.