Fields of definition of singular K3 surfaces

Schütt, Matthias

doi:10.4310/cntp.2007.v1.n2.a2

Cited by 30 publications

(62 citation statements)

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“…Since the Jacobian fibration φ is not unique, the isomorphism class of T − X is not unique in general (see [26] and [29]). However, for the eleven cases in Table 10.1, T − X is uniquely determined by the condition…”

Section: Singular K3 Surfacesmentioning

confidence: 99%

An Algorithm to Compute Automorphism Groups of K3 Surfaces and an Application to Singular K3 Surfaces

Shimada

2015

International Mathematics Research Notices

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Abstract. Let X be a complex algebraic K3 surface or a supersingular K3 surface in odd characteristic. We present an algorithm by which, under certain assumptions on X, we can calculate a finite set of generators of the image of the natural homomorphism from the automorphism group of X to the orthogonal group of the Néron-Severi lattice of X. We then apply this algorithm to certain complex K3 surfaces, among which are singular K3 surfaces whose transcendental lattices are of small discriminants. IntroductionThe automorphism group Aut(X) of an algebraic K3 surface X is an important and interesting object. Suppose that X is defined over the complex number field C, or is supersingular in odd characteristic. Then, thanks to the Torelli-type theorem due to and Ogus [21], [22], we can study Aut(X) by the Néron-Severi lattice S X of X. We denote by O(S X ) the orthogonal group of S X . Then we have a natural homomorphism ϕ X : Aut(X) → O(S X ).It is known that this homomorphism has only a finite kernel. Using the reduction theory for arithmetic subgroups of O(S X ), Sterk [36] and Lieblich and Maulik [16] proved that Aut(X) is finitely generated. The Néron-Severi lattices for which Aut(X) are finite were classified by Nikulin [18], [19] and Vinberg [40]. On the other hand, when Aut(X) is infinite, it is in general a difficult problem to give a set of generators.We also have the following related problem. Let Nef(X) denote the nef cone of X; that is, the cone of S X ⊗ R consisting of vectors x ∈ S X ⊗ R such that x, C ≥ 0 holds for any curve C on X, where , is the intersection form on S X . In order to classify various geometric objects on X (for example, smooth rational curves, Jacobian fibrations, or polarizations of a fixed degree) modulo Aut(X), it is useful to describe explicitly a fundamental domain of the action of Aut(X) on Nef(X). Several authors have studied these problems by using the idea of Borcherds [4], [5] to embed S X into an even unimodular hyperbolic lattice of rank 26. In these works, however, they required that S X should satisfy a certain strong condition (see Section 1.1 below for the details), and hence the range of applications is limited.The purpose of this paper is to present an algorithm (Algorithm 6.1) that calculates, under assumptions on S X milder than the preceding works, a finite set of generators of the image of ϕ X and a closed domain F of Nef(X) with the following properties:(i) For any v ∈ Nef(X), there exists an element g ∈ Aut(X) such that v g ∈ F .(ii) The domain F is tiled by a finite number of convex cones, which we call chambers. Each chamber is bounded by a finite number of hyperplanes and its stabilizer subgroup in Aut(X) is finite.See Remark 6.5 for the relation of F with a fundamental domain of the action of Aut(X) on Nef(X). The detailed description of the assumptions we impose on S X will be given in Section 8. The algorithm can be applied to a wide class of K3 surfaces. We give two examples. Example 1.1. As an example of a K3 surface with small Picard number and an i...

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Section: Singular K3 Surfacesmentioning

confidence: 99%

An Algorithm to Compute Automorphism Groups of K3 Surfaces and an Application to Singular K3 Surfaces

Shimada

2015

International Mathematics Research Notices

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“…After the first version of this paper appeared on the e-print archive, Schütt [24] has succeeded in removing the assumptions in Theorem 3(T) and Corollary 4 that D = disc(NS(S)) be a fundamental discriminant, and that [T (S)] be in L * D . Interesting examples of singular K3 surfaces defined over number fields are also given in [24, §7].…”

Section: F (P) Holds and The Set {[ L(x P) ] | P ∈ S P (X )} Coincmentioning

confidence: 99%

“…Applications of Theorem 3(T) to topology and its generalization by Schütt [24] are given in [27] and [28].…”

Section: F (P) Holds and The Set {[ L(x P) ] | P ∈ S P (X )} Coincmentioning

confidence: 99%

Transcendental lattices and supersingular reduction lattices of a singular 𝐾3 surface

Shimada

2008

Trans. Amer. Math. Soc.

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Abstract. A K3 surface X defined over a field k of characteristic 0 is called singular if the Néron-Severi lattice NS(X) of X ⊗ k is of rank 20. Let X be a singular K3 surface defined over a number field F . For each embedding σ : F → C, we denote by T (X σ ) the transcendental lattice of the complex K3 surface X σ obtained from X by σ. For each prime p of F at which X has a supersingular reduction X p , we define L(X, p) to be the orthogonal complement of NS(X) in NS(X p ). We investigate the relation between these lattices T (X σ ) and L(X, p). As an application, we give a lower bound for the degree of a number field over which a singular K3 surface with a given transcendental lattice can be defined. IntroductionFor a smooth projective surface X defined over a field k, we denote by Pic(X) the Picard group of X, and by NS(X) the Néron-Severi lattice of X ⊗k, wherek is the algebraic closure of k. When X is a K3 surface, we have a natural isomorphism Pic(X ⊗k) ∼ = NS(X). We say that a K3 surface X in characteristic 0 is singular if NS(X) is of rank 20, while a K3 surface X in characteristic p > 0 is supersingular if NS(X) is of rank 22. It is known ([17], [30], [31]) that every complex singular K3 surface is defined over a number field.For a number field F , we denote by Emb(F ) the set of embeddings of F into C, by Z F the integer ring of F , and by π F : Spec Z F → Spec Z the natural projection. Let X be a singular K3 surface defined over a number field F , and let X → U be a smooth proper family of K3 surfaces over a non-empty open subset U of Spec Z F such that the generic fiber is isomorphic to X. We put d(X) := disc(NS(X)).Remark that we have d(X) < 0 by the Hodge index theorem. For σ ∈ Emb(F ), we denote by X σ the complex analytic K3 surface obtained from X by σ. The transcendental lattice T (X σ ) of X σ is defined to be the orthogonal complement of NS(X) ∼ = NS(X σ ) in the second Betti cohomology group H 2 (X σ , Z), which we regard as a lattice by the cup-product. Then T (X σ ) is an even positive-definite lattice of rank 2 with discriminant −d (X). For a closed point p of U , we denote by X p the reduction of X at p. Then X p is a K3 surface defined over the finite field For each p ∈ S p (X ), we have the specialization homomorphism ρ p : NS(X) → NS(X p ), which preserves the intersection pairing (see [2, Exp. X], [11, §4] or [12, §20.3]), and hence is injective. We denote by L(X , p) the orthogonal complement of NS(X) in NS(X p ), and call L(X , p) the supersingular reduction lattice of X at p. Then L(X , p) is an even negative-definite lattice of rank 2. We will see that, if p | 2d(X), then the discriminant of L(X , p) is −p 2 d(X). For an odd prime integer p not dividing x ∈ Z, we denote by The first main result of this paper, which will be proved in §6.5, is as follows:There exists a finite set N of prime integers containing the prime divisors of 2d(X) such that the following holds:We put Z ∞ := R. Let R be Z or Z l , where l is a prime integer or ∞. An R-lattice is a free R-module Λ of finit...

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“…By [27], X has a model over the Hilbert class field H (−24) = Q( √ 2, √ −3). We will establish a model with the following properties:…”

Section: An Explicit Singular K3 Surfacementioning

confidence: 99%

Enriques surfaces and Jacobian elliptic K3 surfaces

Hulek

Schütt

2010

Math. Z.

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This paper proposes a new geometric construction of Enriques surfaces. Its starting point are K3 surfaces with Jacobian elliptic fibration which arise from rational elliptic surfaces by a quadratic base change. The Enriques surfaces obtained in this way are characterised by elliptic fibrations with a rational curve as bisection which splits into two sections on the covering K3 surface. The construction has applications to the study of Enriques surfaces with specific automorphisms. It also allows us to answer a question of Beauville about Enriques surfaces whose Brauer groups show an exceptional behaviour. In a forthcoming paper, we will study arithmetic consequences of our construction.

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Fields of definition of singular K3 surfaces

Abstract: This paper gives upper and lower bounds for the degree of the field of definition of a singular K3 surface, generalising a recent result by Shimada. We use work of ShiodaMitani and Shioda-Inose and classical theory of complex multiplication.

Cited by 30 publications

References 14 publications

An Algorithm to Compute Automorphism Groups of K3 Surfaces and an Application to Singular K3 Surfaces

An Algorithm to Compute Automorphism Groups of K3 Surfaces and an Application to Singular K3 Surfaces

Transcendental lattices and supersingular reduction lattices of a singular 𝐾3 surface

Enriques surfaces and Jacobian elliptic K3 surfaces

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