Modular forms of half-integral weight on exceptional groups

TL;DR

This paper introduces a new framework for modular forms of half-integral weight on exceptional groups, analyzes their Fourier coefficients, and links these coefficients to class group torsion in number fields.

Contribution

It defines modular forms of half-integral weight on quaternionic exceptional groups and explores their Fourier coefficients, connecting them to algebraic number theory.

Findings

Fourier coefficients are well-defined up to ±1

The minimal modular form on F_4 is analyzed

Fourier coefficients on G_2 relate to 2-torsion in class groups

Abstract

We define a notion of modular forms of half-integral weight on the quaternionic exceptional groups. We prove that they have a well-behaved notion of Fourier coefficients, which are complex numbers defined up to multiplication by $\pm 1$. We analyze the minimal modular form $\Theta_{F_4}$ on the double cover of $F_4$, following Loke--Savin and Ginzburg. Using $\Theta_{F_4}$, we define a modular form of weight $\frac{1}{2}$ on (the double cover of) $G_2$. We prove that the Fourier coefficients of this modular form on $G_2$ see the $2$-torsion in the narrow class groups of totally real cubic fields.