Geometry and arithmetic of Maschke's Calabi-Yau three-fold (Q454873)

The paper under review concerns Maschke's octic: NEWLINE\[NEWLINE F = \sum_{i=0}^3 x_i^8\,+\,14\sum_{i<j}x_i^4x_j^4\,+\,168x_0^2x_1^2x_2^2x_3^2 NEWLINE\]NEWLINE This is the lowest degree invariant of a certain group \(\bar G\) of size 11,520 acting on \(\mathbb{P}^3\).NEWLINENEWLINEThe polynomial F can be used to define two algebraic varieties to start with. First, a smooth surface \(S\) of degree \(8\) in \(\mathbb{P}^3\) (which recently appeared in [\textit{S. Boissière} and \textit{A. Sarti}, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 6, No. 1, 39--52 (2007; Zbl 1150.14013)]). Secondly a Calabi-Yau threefold \(X\) which arises as the double covering of \(\mathbb{P}^3\) branched along \(S\) (a so-called double octic).NEWLINENEWLINEThrough the induced group action, the middle cohomology of these varieties can be decomposed into one-, two- and three-dimensional pieces. This involves a generalisation of a formula due to Chênevert. These can be understood as Hodge structures or as Galois representations (a priori over \(\mathbb{Q}(\sqrt{-1})\)). The authors compute the Picard number of \(S\) and collect numerical evidence for the Galois representations attached to both \(S\) and \(X\) to be modular (as subsequently proved by the reviewer [Commun. Number Theory Phys. 5, No. 4, 827--848 (2011; Zbl 1273.14084)]).NEWLINENEWLINEThe Griffiths intermediate Jacobian of \(X\) has an abelian subvariety \(A\) of dimension 149. The authors work out a family of rational curves on \(X\), parametrised by a curve \(C\), such that the Abel-Jacobi map NEWLINE\[NEWLINE \pi: J(C) \to A NEWLINE\]NEWLINE is non-constant. This provides evidence for the generalised Hodge conjecture for \(X\). The Jacobian of \(C\) is conjectured to be isogenous to a product of 33 elliptic curves so that the image of \(\pi\) might have the maximum dimension of 24.