Exceptional theta functions and arithmeticity of modular forms on G₂

2022/11/10 by Pollack, Aaron · 1 citation
#FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2211.05280

Abstract

Quaternionic modular forms on the split exceptional group G₂ = G₂^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 ℓ at least 6, there is a basis of the space of cuspidal modular forms of weight ℓ such that all the Fourier coefficients of elements of this basis are in the cyclotomic extension of Q. Our main tool for proving this is to develop a notion of "exceptional theta functions" on G₂.

Cited by

• Gross's conjecture: the dihedral case

Related

• Computation of Fourier coefficients of automorphic forms of type G₂
• Automatic convergence and arithmeticity of modular forms on exceptional groups
• The Fourier expansion of modular forms on quaternionic exceptional groups
• Modular forms of half-integral weight on exceptional groups
• On the Fourier expansion of Gan-Gurevich lifts on the exceptional group of type G₂