The Modularity of K3 Surfaces with Non-symplectic Group Actions
نویسندگان
چکیده
We consider complex K3 surfaces with a non-symplectic group acting trivially on the algebraic cycles. Vorontsov and Kondō classified those K3 surfaces with transcendental lattice of minimal rank. The purpose of this note is to study the Galois representations associated to these K3 surfaces. The rank of the transcendental lattices is even and varies from 2 to 20, excluding 8 and 14. We show that these K3 surfaces are dominated by Fermat surfaces, and hence they are all of CM type. We will establish the modularity of the Galois representations associated to them. Also we discuss mirror symmetry for these K3 surfaces in the sense of Dolgachev, and show that a mirror K3 surface exists with one exception.
منابع مشابه
Symplectic Automorphisms and the Picard Group of a K3 Surface
Let X be a K3 surface, and let G be a finite group acting on X by automorphisms. The action of G on X induces an action on the cohomology of X . We assume G acts symplectically: that is, G acts as the identity on H(X). In this case, the minimum resolution Y of the quotient X/G is itself a K3 surface. Nikulin classified the finite abelian groups which act symplectically on K3 surfaces by analyzi...
متن کاملec 1 99 5 Non - symplectic involutions of a K 3 surface
Let S be a smooth minimal K3 surface defined over C , G a finite group acting on S. The induced linear action of G on H 0 (ω S) ∼ = C leads to an exact sequence 1 −→ K −→ G −→ N −→ 1 , where the non-symplectic part N is a cyclic group Z m , which acts on the intermediate quotient S/K which is also K3. It is well-known that the Euler number ϕ(m) of m must divide 22 − ρ(S) ([N], Corollary 3.3), i...
متن کاملSymmetric symplectic homotopy K3 surfaces
A study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is shown that an effective action by various maximal symplectic K3 groups forces the...
متن کاملar X iv : 0 80 7 . 37 08 v 2 [ m at h . A G ] 4 S ep 2 00 8 K 3 surfaces with non - symplectic automorphisms of 2 - power order
This paper concerns complex algebraic K3 surfaces with an automorphism which acts trivially on the Néron-Severi group. Complementing a result by Vorontsov and Kondō, we determine those K3 surfaces where the order of the automorphism is a 2-power and equals the rank of the transcendental lattice. We also study the arithmetic of these K3 surfaces and comment on mirror symmetry.
متن کاملOn Symplectic and Non–symplectic Automorphisms of K3 Surfaces
In this paper we investigate when the generic member of a family of complex K3 surfaces admitting a non–symplectic automorphism of finite order admits also a symplectic automorphism of the same order. We give a complete answer to this question if the order of the automorphism is a prime number and we provide several examples and partial results otherwise. Moreover we prove that, under certain c...
متن کامل