Picard-Fuchs equations of the generalized Dwork family

TL;DR

This paper derives Picard-Fuchs equations for generalized Dwork families using Katz's method and computes the Frobenius matrix on their rigid cohomology, offering an alternative to previous approaches.

Contribution

It provides a new derivation of Picard-Fuchs equations for the generalized Dwork family and computes the Frobenius matrix via these equations, improving upon earlier methods.

Findings

Derived Picard-Fuchs equations for the family.

Computed Frobenius matrix using these equations.

Provided an alternative to Kloosterman's original method.

Abstract

We determine the Picard-Fuchs equations of the generalized Dwork families by Katz. As an application, we compute the Frobenius matrix on the rigid cohomology of the family. This was originally done by Kloosterman, while we give an alternative computation with use of the Picard-Fuchs equations.