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Hasse-Witt invariants of Calabi-Yau varieties

Jin Cao, Mohamed Elmi, Hossein Movasati

Abstract

We define the Hasse-Witt invariant of Calabi-Yau varieties in two different ways. The first method is through Cartier operator and the second method is through the theory of Calabi-Yau modular forms developed by the third author. We conjecture that these two definitions are equivalent and provide many examples of Calabi-Yau varieties in support of this conjecture.

Hasse-Witt invariants of Calabi-Yau varieties

Abstract

We define the Hasse-Witt invariant of Calabi-Yau varieties in two different ways. The first method is through Cartier operator and the second method is through the theory of Calabi-Yau modular forms developed by the third author. We conjecture that these two definitions are equivalent and provide many examples of Calabi-Yau varieties in support of this conjecture.
Paper Structure (4 sections, 4 theorems, 73 equations, 2 tables)

This paper contains 4 sections, 4 theorems, 73 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Cartier operator and Hasse-Witt invariant
  3. Mirror quintic
  4. Calabi-Yau Operators

Key Result

Theorem 1

17102025bimsa-1, 17112025ymsc and 02112025huairou are true for the first $200$ primes and to $O(q^{200})$ for four families of hypergeometric Calabi-Yau threefolds in Morrison:1991cd. We have ${\sf S}_\mathbb{Q}={\rm Spec}(\mathbb{Q}[t_1, t_k, \frac{1}{k(t_1^k-t_k)}])$. The universal family of hyper The differential $3$-form is given by $\alpha=\frac{dx_1\wedge dx_2\wedge dx_3\wedge dx_4}{df}$, wh

Theorems & Definitions (16)

  • Conjecture 1
  • Conjecture 2
  • Conjecture 3
  • Theorem 1
  • Remark 1
  • Definition 1
  • Remark 2
  • Definition 2
  • Remark 3
  • Theorem 2
  • ...and 6 more