Geometry and Arithmetic of Maschke's Calabi-Yau Threefold

TL;DR

This paper investigates the geometric, arithmetic, and modular properties of Maschke's Calabi-Yau threefold, revealing its cohomological structure, rational curves, and conjectures on associated Galois representations and modularity.

Contribution

It provides a detailed decomposition of the cohomology, constructs rational curves, and formulates conjectures on the modularity of Galois representations related to Maschke's threefold.

Findings

Decomposition of middle Betti cohomology into 150 Hodge substructures

Existence of rational curves with non-trivial Abel-Jacobi map

Determination of Néron-Severi group rank and Galois representations

Abstract

Maschke's Calabi-Yau threefold is the double cover of projective three space branched along Maschke's octic surface. This surface is defined by the lowest degree invariant of a certain finite group acting on a four dimensional vector space. Using this group, we show that the middle Betti cohomology group of the threefold decomposes into the direct sum of 150 two-dimensional Hodge substructures. We exhibit one dimensional families of rational curves on the threefold and verify that the associated Abel-Jacobi map is non-trivial. By counting the number of points over finite fields, we determine the rank of the N\'eron-Severi group of Maschke's surface and the Galois representation on the transcendental lattice of some of its quotients. We also formulate precise conjectures on the modularity of the Galois representations associated to Maschke's threefold and to a genus 33 curve which parametrizes rational curves in the threefold.