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The modularity of K3 surfaces with non-symplectic group actions

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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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Authors and Affiliations

  1. Institute of Mathematics, Hebrew University of Jerusalem Givat-Ram, 91904, Jerusalem, Israel

    Ron Livné

  2. Institute for Algebraic Geometry, Leibniz University Hannover, Welfengarten 1, 30167, Hannover, Germany

    Matthias Schütt

  3. Department of Mathematics and Statistics, Queen’s University, Kingston, ON, K7L 3N6, Canada

    Noriko Yui

Authors
  1. Ron Livné
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  2. Matthias Schütt
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  3. Noriko Yui
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Corresponding author

Correspondence to Matthias Schütt.

Additional 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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Livné, R., Schütt, M. & Yui, N. The modularity of K3 surfaces with non-symplectic group actions. Math. Ann. 348, 333–355 (2010). https://doi.org/10.1007/s00208-009-0475-9

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  • DOI: https://doi.org/10.1007/s00208-009-0475-9

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