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Hasse--Witt matrices and mirror toric pencils

Adriana Salerno, Ursula Whitcher

Abstract

Mirror symmetry suggests unexpected relationships between arithmetic properties of distinct families of algebraic varieties. For example, Wan and others have shown that for some mirror pairs, the number of rational points over a finite field matches modulo the order of the field. In this paper, we obtain a similar result for certain mirror pairs of toric hypersurfaces. We use recent results by Huang, Lian, Yau and Yu describing the relationship between the Picard-Fuchs equations of these varieties and their Hasse--Witt matrices, which encapsulate information about the number of points. The result allows us to compute the number of points modulo the order of the field explicitly. We illustrate this by computing K3 surface examples related to hypergeometric functions.

Hasse--Witt matrices and mirror toric pencils

Abstract

Mirror symmetry suggests unexpected relationships between arithmetic properties of distinct families of algebraic varieties. For example, Wan and others have shown that for some mirror pairs, the number of rational points over a finite field matches modulo the order of the field. In this paper, we obtain a similar result for certain mirror pairs of toric hypersurfaces. We use recent results by Huang, Lian, Yau and Yu describing the relationship between the Picard-Fuchs equations of these varieties and their Hasse--Witt matrices, which encapsulate information about the number of points. The result allows us to compute the number of points modulo the order of the field explicitly. We illustrate this by computing K3 surface examples related to hypergeometric functions.

Paper Structure

This paper contains 13 sections, 13 theorems, 56 equations, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Toric hypersurfaces and Picard--Fuchs equations
  3. Batyrev mirrors and special pencils
  4. Kernel pairs in two and three dimensions
  5. Hasse-Witt matrices for Calabi-Yau toric hypersurfaces
  6. Proof of the Main Theorem
  7. Elliptic curves in $\mathbb{P}^1 \times \mathbb{P}^1$
  8. Fermat quartic in $\mathbb{P}^3$
  9. Sextic in $\mathbb{P}(1,1,1,3)$
  10. Group I
  11. Group II
  12. Symmetric square roots
  13. Special hypergeometric parameters

Key Result

Corollary 1

Let $\Delta$ and $\Gamma$ be a kernel pair of $n$-dimensional reflexive polytopes, and suppose $\Delta^\circ$ and $\Gamma^\circ$ are also a kernel pair. Let $V_\Delta$ and $V_\Gamma$ be smooth toric varieties determined by maximal simplicial refinements of the fans over the faces of $\Delta$ and $\G

Theorems & Definitions (29)

  • Corollary : See Corollary \ref{['C:pointCountsModp']}
  • Corollary : See Corollary \ref{['C:4multisets']}
  • Example 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Theorem 3.3: HLYY (Theorem 1.2)
  • ...and 19 more