International Seminar on Automorphic Forms

Naomi Sweeting (Princeton University)

The arithmetic of Fourier coefficients of Gan-Gurevich lifts on G2

The arithmetic of Fourier coefficients of Gan-Gurevich lifts on G2 Abstract: Modular forms on exceptional groups carry a surprisingly rich arithmetic structure. For instance, modular forms on G2 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 G2 arising by Langlands functoriality from classical modular forms on PGL2. 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.

https://tu-darmstadt.zoom.us/j/68048280736

The password is the first Fourier coefficient of the modular j-function (as digits)