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On $p$-integrality of instanton numbers

Frits Beukers, Masha Vlasenko

Abstract

We show integrality of instanton numbers in several key examples of mirror symmetry. Our methods are essentially elementary, they are based on our previous work in the series of papers called Dwork crystals I, II and III.

On $p$-integrality of instanton numbers

Abstract

We show integrality of instanton numbers in several key examples of mirror symmetry. Our methods are essentially elementary, they are based on our previous work in the series of papers called Dwork crystals I, II and III.

Paper Structure

This paper contains 7 sections, 22 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Proof of Remark \ref{['frobenius-in-standard-basis']} and Theorems \ref{['mirror-integrality']}, \ref{['yukawa2']}, \ref{['yukawa3']}
  3. Checking higher Hasse--Witt conditions
  4. Construction of the Frobenius structure
  5. Simplicial family
  6. Hyperoctahedral family
  7. Vanishing of $\alpha_1$

Key Result

Theorem 1.4

Suppose that $L(y)=0$ has a $p$-adic Frobenius structure. Then $\exp(F_1/F_0)\in{\mathbb Z}_p\llbracket t\rrbracket$.

Theorems & Definitions (57)

  • Definition 1.1: Frobenius structure
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: $p$-integrality of the mirror map
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7: $p$-integrality of instanton numbers
  • Proposition 1.8
  • Corollary 1.9
  • Remark 1.10
  • ...and 47 more