K3 surfaces with Picard rank 20

Schütt, Matthias

doi:10.2140/ant.2010.4.335

Cited by 24 publications

(7 citation statements)

References 23 publications

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“…Remark 11.4. In particular, note that there are Enriques surfaces of type VI and VII with full Picard rank over Q, while this is not possible for their canonical cover due to a result of N. D. Elkies (see [Sch10]).…”

Section: Arithmetic Of Enriques Surfaces With Finite Automorphism Groupmentioning

confidence: 99%

Enriques surfaces with finite automorphism group in positive characteristic

Martin¹

2019

Alg. Geom.

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We classify Enriques surfaces with smooth K3 cover and finite automorphism group in arbitrary positive characteristic. The classification is the same as over the complex numbers except that some types are missing in small characteristics. Moreover, we give a complete description of the moduli of these surfaces. Finally, we realize all types of Enriques surfaces with finite automorphism group over the prime fields F p and Q whenever they exist.

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Section: Arithmetic Of Enriques Surfaces With Finite Automorphism Groupmentioning

confidence: 99%

Enriques surfaces with finite automorphism group in positive characteristic

Martin¹

2019

Alg. Geom.

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“…As the discriminant of Pic S 1 is −8, we know by Elkies and Schütt (cf. [26,Section 10]) that the geometric Picard lattice of S 1 is realised over Q, i.e., Pic S 1 = Pic S 1 . This can also be directly observed by noting that all the divisors used are defined over Q.…”

Section: Special Fibersmentioning

confidence: 99%

Bhabha scattering and a special pencil of K3 surfaces

Festi

Straten

2019

Communications in Number Theory and Physics

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We study a pencil of K3 surfaces that appeared in the 2-loop diagrams in Bhabha scattering. By analysing in detail the Picard lattice of the general and special members of the pencil, we identify the pencil with the celebrated Apéry-Fermi pencil, that was related to Apéry's proof of the irrationality of ζ(3) through the work of F. Beukers, C. Peters and J. Stienstra. The same pencil appears miraculously in different and seemingly unrelated physical contexts. arXiv:1809.04970v2 [math.AG]

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“…Then, Elkies and Shütt (see [ES13] and [Sch10]) were lead to the quest of finding all singular K3 surfaces with Picard rank 20 over ℚ, both to answer some questions raised by Shioda and to find a geometric realisation of CM newforms of weight 3 (after a Theorem of Livné [Liv95] we know that if a singular K3 surface is defined over ℚ, then its associated Galois representation is modular). Their results use specialisation of families of K3 surfaces { } with ( ) ≥ 19, the theory of CM elliptic curves and the particular geometry of singular K3 surfaces (in particular their elliptic fibrations).…”

Section: Relation To Other Workmentioning

confidence: 99%

Fields of definition of K3 surfaces with complex multiplication

Valloni¹

2019

Preprint

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Let ∕ℂ be a K3 surface with complex multiplication by the ring of integers of a CM number field . Under some natural conditions on the discriminant of the quadratic form ( ), we produce a model can of over an explicit abelian extension ∕ with the property that ( can ∕ ) = ( ∕ℂ). We prove that can ∕ is canonical in the following sense: if ∕ is another model of such that ( ∕ ) = ( ∕ℂ), then ⊂ and can ≅ . If is fixed, our theorem applies to all but finitely many surfaces with complex multiplication by . In case is not one of those, we still provide necessary and sufficient conditions for a model enjoying the same properties of can to exist. As an application to our work, we give necessary and sufficient conditions for a singular K3 surfaces with CM by the ring of integers of an imaginary quadratic field to have a model with all Picard group defined over , and provide an alternative proof of a finiteness result obtained by Shafarevich and later generalised by Orr and Skorobogatov. CONTENTS 18 6. Applications and open questions 21 References 24

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K3 surfaces with Picard rank 20

Cited by 24 publications

References 23 publications

Enriques surfaces with finite automorphism group in positive characteristic

Enriques surfaces with finite automorphism group in positive characteristic

Bhabha scattering and a special pencil of K3 surfaces

Fields of definition of K3 surfaces with complex multiplication

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