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

TL;DR

This paper proves that certain Galois representations from a specific Calabi-Yau threefold are reducible, with components linked to modular forms, confirming a conjecture and highlighting the modularity properties of this geometric object.

Contribution

It establishes the reducibility of Galois representations for a particular Calabi-Yau threefold and identifies their modular form components, advancing understanding of their arithmetic structure.

Findings

Galois representations are reducible with specific modular form factors.

The modular forms involved have weights 2 and 4, both level 14.

Supports the conjecture by Meyer and Verrill about this 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''.