The arithmetic of quaternionic modular forms on G2

Surprisingly, quaternionic modular forms on G2 have rich arithmetic properties. For example, they enjoy Fourier expansions where the Fourier coefficients are indexed by totally real cubic rings. When the quaternionic modular form arises by functoriality from a (classical) modular form f, Gross conjectured in 2000 that its Fourier coefficients encode the L-values of cubic twists of f (echoing Waldspurger's work on the Fourier coefficients of modular forms of half-integral weight). We prove Gross's conjecture when f is dihedral, giving the first examples for which it is known. This is joint with P. Bakić, A. Horawa, and N. Sweeting.