Modular forms of half-integral weight on exceptional groups

Half-integral weight modular forms are classical objects with many important arithmetic applications.  In terms of automorphic representations, these correspond to objects on the metaplectic double cover of SL(2).  In this talk, I will outline a theory of modular forms of half-integral weight on double covers of exceptional groups, generalizing the integral weight theory developed by Gross-Wallach, Gan-Gross-Savin, and Pollack.  Furthermore, I discuss a particular example of a weight 1/2 modular form on G2 whose Fourier coefficients encode the 2-torsion in the narrow class groups of totally real cubic fields. This is built by studying a remarkable automorphic representation of the double cover of the exceptional group F4.