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Invertible Calabi-Yau Orbifolds over Finite Fields II

Marco Aldi, Andrija Peruničić

Abstract

We state a conjecture about the zeta function of crepant resolutions of Berglund--Hübsch orbifold hypersurfaces over a finite field. In addition to numerical evidence, we show that our conjectural zeta function satisfies the Weil conjectures and we elucidate its connection with Monsky--Washnitzer cohomology.

Invertible Calabi-Yau Orbifolds over Finite Fields II

Abstract

We state a conjecture about the zeta function of crepant resolutions of Berglund--Hübsch orbifold hypersurfaces over a finite field. In addition to numerical evidence, we show that our conjectural zeta function satisfies the Weil conjectures and we elucidate its connection with Monsky--Washnitzer cohomology.
Paper Structure (7 sections, 4 theorems, 72 equations, 1 figure, 17 tables)

This paper contains 7 sections, 4 theorems, 72 equations, 1 figure, 17 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conjectures
  4. The Weil Conjectures for $\zeta_p(A,G,t)$
  5. K3 Surfaces
  6. The Fermat Quintic and Its Quotients
  7. Alternative Proof via Monsky--Washnitzer Cohomology

Key Result

Proposition 2.2

Let $A$ be a Berglund--Hübsch matrix over $\mathbb F_p$. Then

Figures (1)

  • Figure 1: Fundamental region of the lattice $\mathbb Z^3 /\mathbb Z^3 A$ ($W_A = x_1^2 x_2 + x_2^2 x_3 + x_3^3$).

Theorems & Definitions (30)

  • Definition 2.1
  • Proposition 2.2: AP
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Conjecture 3.1
  • Remark 3.2
  • Definition 3.3
  • Remark 3.4
  • ...and 20 more