Arithmetic of Fourier coefficients of Gan-Gurevich lifts on $\mathsf{G}_2$

Abstract: Quaternionic modular forms on $\mathsf{G}_2$ carry a surprisingly rich arithmetic structure. For example, they have a theory of Fourier expansions where the Fourier coefficients are indexed by totally real cubic rings. For quaternionic modular forms on $\mathsf{G}_2$ associated via functoriality with certain modular forms on $\mathrm{PGL}_2$, Gross conjectured in 2000 that their Fourier coefficients encode $L$-values of cubic twists of the modular form (echoing Waldspurger's work on Fourier coefficients of half-integral weight modular forms). This talk will report on recent work proving Gross's conjecture when the modular forms are dihedral, giving the first examples for which it is known. Based on joint work with Petar Bakic, Alex Horawa, and Siyan Daniel Li-Huerta.

algebraic geometrynumber theory

Audience: researchers in the topic

MIT number theory seminar

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