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 ${\mathrm{\Theta }}_{F_4}$ on the double cover of $F_4$, following Loke–Savin and Ginzburg. Using ${\mathrm{\Theta }}_{F_4}$, 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.