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

Masanori Asakura, Kei Hagihara

TL;DR

This work generalizes Kedlaya's explicit Frobenius description for hypergeometric differential equations by replacing the $p$-adic gamma residue data with $p$-adic polygamma values, thereby expressing Frobenius matrix entries in terms of $p$-adic L-values. The authors develop a generalized residue formula using a limiting construction that yields polygamma- and $L_p$-value dependent expressions, and extend this to scalar extensions and Frobenius twists. They apply the formula to the Frobenius action on log-crystalline cohomology for families with hypergeometric Picard-Fuchs equations, connecting arithmetic invariants to geometric degeneration data, including Katz pencils and Dwork-type families. The results unify local arithmetic (polygamma/L-values) with global geometry (log-crystalline and rigid cohomology) and yield explicit computations of syntomic regulators for Milnor symbols, offering new evidence for $p$-adic Beilinson-type conjectures. Significance lies in providing a tractable framework to read off $p$-adic regulators and $L$-values from Frobenius data in degenerating families of Calabi-Yau type and related hypergeometric systems.

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.

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

TL;DR

This work generalizes Kedlaya's explicit Frobenius description for hypergeometric differential equations by replacing the -adic gamma residue data with -adic polygamma values, thereby expressing Frobenius matrix entries in terms of -adic L-values. The authors develop a generalized residue formula using a limiting construction that yields polygamma- and -value dependent expressions, and extend this to scalar extensions and Frobenius twists. They apply the formula to the Frobenius action on log-crystalline cohomology for families with hypergeometric Picard-Fuchs equations, connecting arithmetic invariants to geometric degeneration data, including Katz pencils and Dwork-type families. The results unify local arithmetic (polygamma/L-values) with global geometry (log-crystalline and rigid cohomology) and yield explicit computations of syntomic regulators for Milnor symbols, offering new evidence for -adic Beilinson-type conjectures. Significance lies in providing a tractable framework to read off -adic regulators and -values from Frobenius data in degenerating families of Calabi-Yau type and related hypergeometric systems.

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.
Paper Structure (21 sections, 32 theorems, 246 equations)

This paper contains 21 sections, 32 theorems, 246 equations.

Table of Contents

  1. Introduction
  2. $p$-adic gamma, $p$-adic polygamma and $p$-adic $L$-functions
  3. Kedlaya's residue formula revisited
  4. Notation
  5. Frobenius intertwiners
  6. Construction of the Frobenius intertwiners on hypergeometric equations
  7. Construction of Frobenius intertwiner, Ke.
  8. Basis $\widehat{\omega}_k(\ul a;\ul b)$
  9. Definition of $\widehat{\omega}_k(\ul a;\ul b)$.
  10. Kedlaya's residue formula
  11. Generalization of Kedlaya's residue formula
  12. Definition of $\widehat{\omega}(m)$
  13. Generalized Kedlaya's residue formula
  14. Proof of Theorem \ref{['main.formula1']}
  15. Scalar extension and Changing the $p$-th Frobenius
  16. ...and 6 more sections

Key Result

Theorem 1.2

Let $X\to S=\operatorname{Spec} F[z,(z-z^2)^{-1}]$ be a projective smooth $F$-morphism which has an extension ${\mathscr X}\to\operatorname{Spec} R[[z]]$ at $z=0$ which fits into the above setting. Let $M\subset H^{n-1}_{\mathrm{d\space R}}(X/S)$ be a sub ${\mathscr D}_F=F[z,(z-z^2)^{-1},\frac{d}{dz where $\ul1:=(1,\ldots,1)$ and $I$ is a finite set of parameters $\ul a=(a_1,\ldots,a_n)$ such that

Theorems & Definitions (67)

  • Theorem 1.2: cf. Theorem \ref{['degenerate.cor']}
  • Remark 2.1
  • Theorem 2.2: New
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • ...and 57 more