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Picard-Fuchs equations of the generalized Dwork family

Ryo Negishi

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.

Picard-Fuchs equations of the generalized Dwork family

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.
Paper Structure (13 sections, 16 theorems, 102 equations)

This paper contains 13 sections, 16 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definition of the generalized Dwork family
  4. Group action and eigendecomposition
  5. Review of Katz's results
  6. Picard-Fuchs equations of the generalized Dwork family
  7. Jacobian rings of hypersurfaces
  8. Picard-Fuchs equations of the generalized Dwork family
  9. $\mathcal{D}$-module structure of $\mathrm{Prim}^{n-2}_{\mathrm{dR}}$.
  10. Application to $p$-adic cohomology theory
  11. Differential modules and Matrix calculus
  12. Deformation matrix of Dwork families
  13. Example (dwork-padiccycles, kloosterman)

Key Result

Lemma 2.1

The morphism $\pi$ is smooth on $U:= \operatorname{Spec} R_0[\lambda, (1-\lambda^d)^{-1}]$.

Theorems & Definitions (29)

  • Lemma 2.1: katz
  • Lemma 2.2: katz
  • Lemma 2.3: katz
  • Lemma 2.4
  • Theorem 3.1: Griffiths, gri
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • Definition 3.4
  • Proposition 3.5
  • ...and 19 more