The arithmetic of Fourier coefficients of automorphic forms on G2

Shimura (1973) discovered a way to associate a holomorphic half-integral weight modular form h with a classical cusp form f. Subsequently, Waldspurger (1980s) proved a remarkable formula relating squares of the Fourier coefficients of h and quadratic twists of L-values of f. We prove that one can associate "quaternionic" modular forms on the algebraic group G₂ with dihedral cusp forms f whose Fourier coefficients are explicitly related to cubic twists of L-values of f. This gives the first examples where a conjecture of Gross (2000) has been fully verified. (Joint work with Petar Bakić, Siyan Daniel Li-Huerta, and Naomi Sweeting.)