Modular forms of half-integral weight on exceptional groups

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 $\tfrac {1}{2}$ 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.