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

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.The Hodge structure on the middle dimensional Betti cohomology group of a Calabi-Yau (CY) three-fold carries important information on the moduli and the one-dimensional (1D) algebraic cycles on the three-fold. However, if the three-fold is easy to define, say by one equation in a (weighted) projective space, the dimension h 3 of this vector space tends to be large. For example, a smooth quintic three-fold in P 4 has h 3 = 204 and a double octic, i.e., a double cover of P 3 branched along a smooth surface of degree 8, has h 3 = 300. Using automorphisms of the three-folds, one can decompose the cohomology into subrepresentations, which give rise to Hodge substructures. In this paper, we consider a double octic with a particularly large automorphism group G of order 16 ·

Quartic surfaces with icosahedral symmetry

“…The fifth of them defines the surface from (3.5). We propose to call the polynomials M i the Maschke quartic polynomials (not to be confused with the Maschke octic polynomial from [2]). It is shown in [17], p. 505, that they satisfy…”

“…Let e 1 , e 2 be a basis in V and (u, v) be the coordinates in V with respect to this basis, i.e. the dual basis of (e 1 , e 2…”

“…This is the equation of the 10-nodal surface S 7/30 from the Hashimoto pencil. 2 Remark 2. The 10 nodes of the symmetric discriminant quartic (also known as a Cayley symmetroid quartic) correspond to the singular lines of 10 reducible quadrics in the web.…”

Modularity of Maschke’s octic and Calabi–Yau threefold

“…Throughout the paper we employ the notation from [BvG], see 1.4 for a brief account. Particularly we are concerned with the following modular forms with notation from [MFIV] indicating the level: f 120 weight 4; f 15C, f 24b, f 120E weight 2 (see [MFIV] or [BvG,7.2] for Fourier coefficients). The associated compatible systems of ℓ-adic Galois representations are denoted by V f,ℓ .…”

Modularity of Calabi–Yau Varieties: 2011 and Beyond

“…Dieulefait-Pacetti-Schütt[12]) The Consani-Scholten Calabi-Yau threefoldX over Q is Hilbert modular. That is, the L-series associated to σ coincides with the L-series of a Hilbert modular newform f on F of weight(2,4) and conductor c f = (30).…”