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October 10, 2025Open Access

Gross's conjecture: the dihedral case

Key Points

Gross's conjecture is proven for dihedral modular forms, marking a significant advancement in understanding their arithmetic properties.
The research confirms that the Fourier coefficients of these forms encode L-values, specifically of cubic twists of the modular form.
This work builds on prior knowledge of quaternionic modular forms and their associated Fourier expansions via functoriality.
Providing these first examples under the conjecture highlights the intersection of modular forms and number theory.

Abstract

Quaternionic modular forms on G₂ 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 G₂ associated via functoriality with certain modular forms on PGL₂, 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). We prove Gross's conjecture when the modular forms are dihedral, giving the first examples for which it is known.

Gross's conjecture: the dihedral case

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Abstract

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