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arXiv · 2307.08940

Frobenius structure on hypergeometric equations, p-adic polygamma values and p-adic L-values

Masanori Asakura·Kei Hagihara
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Abstract

Recently, Kedlaya proves certain formula describing explicitly the Frobenius structure on a hypergeometric equation. In this paper, we give a generalization of it. In our case, the Frobenius matrix is no longer described by p-adic gamma function, and then we describe it by the p-adic polygamma functions. Since the p-adic polygamma values are linear combinations of p-adic L-values of Dirichlet characters, it turns out that the Frobenius matrix is described by p-adic L-values. Our result has an application to the study on Frobenius on p-adic cohomology. We show that, for a projective smooth family such that the Picard-Fuchs equation is a hypergeometric equation, the Frobenius matrix on the log-crystalline cohomology is described by some values of the logarithmic function and p-adic L-functions of Dirichlet characters.

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Masanori Asakura, Kei Hagihara. 2023-07-18. Frobenius structure on hypergeometric equations, p-adic polygamma values and p-adic L-values. https://doi.org/10.2140/tunis.2026.8.453

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