Modularity of Maschke’s octic and Calabi–Yau threefold

Schütt, Matthias

doi:10.4310/cntp.2011.v5.n4.a3

Cited by 2 publications

(4 citation statements)

References 6 publications

(19 reference statements)

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“…The representations σ 5 , σ 9 are expected to be Tate twists of Galois representations associated to elliptic curves defined over Q, these curves should become isogeneous, over C, to the curves E 5 , E 9 from Section 6.2. This is now proven in [29]. Hence, by Wiles' theorem, these Galois representations correspond to newforms of weight two on Γ 0 (N i ) (i = 5, 9) for certain integers N i divisible only by primes where Y has bad reduction.…”

Section: The Heisenberg Quotient Y Of Xmentioning

confidence: 85%

“…where V g, denotes the -adic Galois representation associated to the newform g. The conjecture was recently proved by M. Schütt [29]. To find the a p , we assumed that σ 5 and σ 9 are Tate twists of Galois representations, in particular that tr(F p |W i, ) is a multiple of p for i = 5, 9.…”

Section: The Galois Representation On H 3 Et ( Y Q )mentioning

confidence: 99%

“…Putting all conjectures together, and given the proofs of them in [29], we have the following formula:…”

Section: The Galois Representation On H 3 Et (X Q )mentioning

confidence: 99%

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

Bini

Geemen

2011

Communications in Number Theory and Physics

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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 ·

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Section: The Heisenberg Quotient Y Of Xmentioning

confidence: 85%

Section: The Galois Representation On H 3 Et ( Y Q )mentioning

confidence: 99%

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

Bini

Geemen

2011

Communications in Number Theory and Physics

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“…A recent example due to Bini and van Geemen [2], and Schütt [57] is the Calabi-Yau threefold called Maschke's double octic, which arises as the double covering of P 3 branched along Maschke's surface S. The Maschke octic surface S is defined by the homogeneous equation…”

Section: Galois Representationsmentioning

confidence: 99%

Modularity of Calabi–Yau Varieties: 2011 and Beyond

Yui

2013

Fields Institute Communications

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This paper presents the current status on modularity of Calabi-Yau varieties since the last update in 2003. We will focus on Calabi-Yau varieties of dimension at most three. Here modularity refers to at least two different types: arithmetic modularity and geometric modularity. These will include: (1) the modularity (automorphy) of Galois representations of Calabi-Yau varieties (or motives) defined over Q or number fields, (2) the modularity of solutions of Picard-Fuchs differential equations of families of Calabi-Yau varieties, and mirror maps (mirror moonshine), (3) the modularity of generating functions of invariants counting certain quantities on Calabi-Yau varieties, and (4) the modularity of moduli for families of Calabi-Yau varieties. The topic (4) is commonly known as geometric modularity.Discussions in this paper are centered around arithmetic modularity, namely on (1), and (2), with a brief excursion to (3).

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Modularity of Maschke’s octic and Calabi–Yau threefold

Abstract: We prove the modularity of Maschke's octic and two Calabi-Yau threefolds derived from it as double octic and quotient thereof by a suitable Heisenberg group, as conjectured by Bini and van Geemen. The proofs rely on automorphisms of the varieties and isogenies of K3 surfaces.

Cited by 2 publications

References 6 publications

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

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

Modularity of Calabi–Yau Varieties: 2011 and Beyond

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