The arithmetic of Fourier coefficients of automorphic forms on G2Arithmetic Geometry and Representation Theory Research Seminar

by Aleksander Horawa (University of Bonn)

Friday Nov 28, 2025, 2:05 PM → 4:00 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Seminar Room (Max Planck Institute for Mathematics)

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.)