Abstract

Recently, Kedlaya proved a formula which explicitly describes the Frobenius structure on certain $p$-adic hypergeometric equations. In this paper, we present a generalization of his formula, which is applicable to cases the original does not cover. A striking feature of our generalized formula is that, in these newly covered cases, the Frobenius matrix is expressed by the $p$-adic polygamma values and, consequently, by $p$-adic $L$-values for Dirichlet characters. As an application to $p$-adic geometry, we show that, for a projective smooth family whose Picard–Fuchs equation is a hypergeometric one, the Frobenius matrix on the corresponding log crystalline cohomology is described in terms of some values of the logarithmic function and $p$-adic $L$-functions.

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