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

TL;DR

This paper generalizes Kedlaya's explicit Frobenius structure formula for hypergeometric equations, expressing the Frobenius matrix via p-adic polygamma and L-values, with applications to p-adic cohomology.

Contribution

It extends the description of Frobenius matrices from p-adic gamma functions to p-adic polygamma and L-values, broadening understanding of p-adic cohomology.

Findings

Frobenius matrix described by p-adic L-values.

Application to Frobenius on p-adic cohomology.

Explicit formulas for Frobenius in hypergeometric cases.

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.