TBA

TBA

The arithmetic of Fourier coefficients of Gan-Gurevich lifts on $G_2$ Abstract: Modular forms on exceptional groups carry a surprisingly rich arithmetic structure. For instance, modular forms on $G_2$ have a theory of Fourier expansions, in which the coefficients are indexed by cubic rings (e.g. rings of integers in cubic field extensions of Q). This talk is about the Gan-Gurevich lifts, which are modular forms on $G_2$ arising by Langlands functoriality from classical modular forms on $PGL_2$. Gross conjectured in 2000 that the norm squared of the Fourier coefficients of a Gan-Gurevich lift encode the cubic-twisted L values of the corresponding classical cusp form (echoing Waldspurger's work on Fourier coefficients of half-integral weight modular forms). We prove this conjecture for a large class of Gan-Gurevich lifts coming from CM forms, thus giving the first complete examples of Gross's conjecture. Based on joint work in progress with Petar Bakic, Alex Horawa, and Siyan Daniel Li-Huerta.