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This is a Experimental Study study.

October 10, 2025Open Access

Gross's conjecture: the dihedral case

Puntos clave

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.

Resumen

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.

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Cite This Study

Bakić et al. (Fri,) studied this question.

synapsesocial.com/papers/68e861b07ef2f04ca37e4c29 https://doi.org/https://doi.org/10.48550/arxiv.2510.03476

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