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 ± 1 . We analyze the minimal modular form Θ F₄ on the double cover of F₄ , following Loke–Savin and Ginzburg. Using Θ F₄ , we define a modular form of weight 1/2 on (the double cover of) G₂ . We prove that the Fourier coefficients of this modular form on G₂ see the $2$ -torsion in the narrow class groups of totally real cubic fields.