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arXiv · 2510.03476

Gross's conjecture: the dihedral case

Petar Bakić·Aleksander Horawa·Siyan Daniel Li-Huerta·Naomi Sweeting
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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). We prove Gross's conjecture when the modular forms are dihedral, giving the first examples for which it is known.

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Petar Bakić, Aleksander Horawa, Siyan Daniel Li-Huerta, Naomi Sweeting. 2025-10-03. Gross's conjecture: the dihedral case. https://arxiv.org/abs/2510.03476

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