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Modularity of a certain "rank-2 attractor" Calabi-Yau threefold

Neil Dummigan

Abstract

We prove that the 4-dimensional Galois representations associated with a certain Calabi-Yau threefold are reducible, with 2-dimensional composition factors coming from specific modular forms of weights 2 and 4, both level 14. This was essentially conjectured by Meyer and Verrill. It was revisited in its present form by Candelas, de la Ossa, Elmi and van Straten, whose computations of Euler factors in a whole pencil of Calabi-Yau threefolds highlighted this fibre as one of three overwhelmingly likely to be ``rank-2 attractors''.

Modularity of a certain "rank-2 attractor" Calabi-Yau threefold

Abstract

We prove that the 4-dimensional Galois representations associated with a certain Calabi-Yau threefold are reducible, with 2-dimensional composition factors coming from specific modular forms of weights 2 and 4, both level 14. This was essentially conjectured by Meyer and Verrill. It was revisited in its present form by Candelas, de la Ossa, Elmi and van Straten, whose computations of Euler factors in a whole pencil of Calabi-Yau threefolds highlighted this fibre as one of three overwhelmingly likely to be ``rank-2 attractors''.
Paper Structure (9 sections, 23 theorems, 99 equations)

This paper contains 9 sections, 23 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. The image of monodromy mod $5$, and its normaliser
  3. An explicit basis for $5$-integral homology
  4. Identifying a $2$-dimensional mod $5$ Galois representation
  5. Counting points on the Hulek-Verrill manifold
  6. Checking the count for the two-equations construction
  7. Plus/minus $p$-adic $L$-functions and Selmer groups at a supersingular prime
  8. Constructing an element in a Selmer group
  9. The final result

Key Result

Proposition 2.1

The image $J:=\overline{M}(\pi_1({\mathbb P}^1({\mathbb C})-\Sigma))$ in $\mathrm {Sp}_2({\mathbb F}_5)$ is contained in $P$. It contains $U$, and its intersection with $L$ is the subgroup $K$ comprising matrices of the form with $\in\mathrm {SL}_2({\mathbb F}_5)$.

Theorems & Definitions (57)

  • Conjecture 1.1
  • Conjecture 1.2
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Conjecture 3.1
  • ...and 47 more