CM newforms with rational coefficients

Schütt, Matthias

doi:10.1007/s11139-008-9147-8

Cited by 38 publications

(67 citation statements)

References 11 publications

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“…In particular, the trace a p of F p in this representation is: The a p are also the Fourier coefficients of an elliptic modular form f of weight three (cf. [28]). In this case f is one of the two newforms with Dirichlet character p → −15 p , it has level N = 15.…”

Section: The Galois Representation On H 2 Et ( W Q )mentioning

confidence: 99%

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 Galois Representation On H 2 Et ( W Q )mentioning

confidence: 99%

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

Bini

Geemen

2011

Communications in Number Theory and Physics

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“…The classification in [Schütt 2009] . In terms of the associated newform f , this corresponds to the quadratic twist in (2).…”

Section: Modularity Of Singular K3 Surfaces Over ‫ޑ‬mentioning

confidence: 99%

“…The fibre is split multiplicative, so the Picard rank of the surface over ‫ޑ‬ is already 20. The corresponding Hecke eigenform has level 19 by [Schütt and Top 2006] (see [Schütt 2009, Table 1]).…”

Section: K3 Surfaces Of Picard Rank 20 Over ‫:ޑ‬ Examplesmentioning

confidence: 99%

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K3 surfaces with Picard rank 20

Schütt

2010

ANT

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We determine all complex K3 surfaces with Picard rank 20 over ‫.ޑ‬ Here the Néron-Severi group has rank 20 and is generated by divisors which are defined over ‫.ޑ‬ Our proof uses modularity, the Artin-Tate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that Mordell-Weil rank 18 over ‫ޑ‬ is impossible for an elliptic K3 surface. We apply our methods to general singular K3 surfaces, that is, those with Néron-Severi group of rank 20, but not necessarily generated by divisors over ‫.ޑ‬

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“…CM newforms with rational coefficients have been classified in [14]. In weight 4, they correspond to imaginary-quadratic fields with class group exponent one or three.…”

Section: Introductionmentioning

confidence: 99%