Communications in Number Theory and Physics

Volume 5 (2011)

Number 4

Geometry and arithmetic of Maschke’s Calabi–Yau three-fold

Pages: 779 – 826

DOI: http://dx.doi.org/10.4310/CNTP.2011.v5.n4.a2

Authors

Gilberto Bini (Dipartimento di Matematica, Università di Milano, Italy)

Bert van Geemen (Dipartimento di Matematica, Università di Milano, Italy)

Abstract

Maschke’s Calabi–Yau three-fold 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 (4D) vector space. Using this group, we show that the middle Betti cohomology group of the three-fold decomposes into the direct sum of 150 2D Hodge substructures.We exhibit 1D families of rational curves on the three-fold 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 Néron–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 three-fold (these have now been proven by M. Schütt) and to a genus 33 curve, which parametrizes rational curves in the three-fold.

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