The modularity of K3 surfaces with non-symplectic group actions

Ron Livné; Matthias Schütt; Noriko Yui

doi:10.1007/s00208-009-0475-9

The modularity of K3 surfaces with non-symplectic group actions

Ron Livné, Matthias Schütt^*, Noriko Yui

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

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.

Original language	English
Pages (from-to)	333-355
Number of pages	23
Journal	Mathematische Annalen
Volume	348
Issue number	2
DOIs	https://doi.org/10.1007/s00208-009-0475-9
State	Published - 2010

Bibliographical note

Funding Information:
R. Livné was partially supported by an ISF grant. M. Schütt was supported by Deutsche Forschungsgemeinschaft (DFG) under grant Schu 2266/2-2. N. Yui was supported in part by Discovery Grant of the Natural Sciences and Engineering Research Council (NSERC) of Canada.

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abstract = "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.",

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The modularity of K3 surfaces with non-symplectic group actions

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