K3 surfaces,Picard numbers and Siegel disks |
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| Affiliation: | 1. Department of Mathematics, Faculty of Science, Hokkaido University, Kita 10, Nishi 8, Kita-ku, Sapporo 060-0810 Japan;2. Department of Mathematics, Graduate School of Science, Hokkaido University, Kita 10, Nishi 8, Kita-ku, Sapporo 060-0810 Japan;1. University of Missouri-Columbia, Mathematics Department, Columbia, MO, USA;2. College of the Ozarks, Mathematics Department, Point Lookout, MO, USA;1. CIEM-FaMAF, CONICET-Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina;2. Guangdong Technion Israel Institute of Technology, 241 Daxue Road, Jinping District, Shantou, Guandong Province, China |
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| Abstract: | If a K3 surface admits an automorphism with a Siegel disk, then its Picard number is an even integer between 0 and 18. Conversely, using the method of hypergeometric groups, we are able to construct K3 surface automorphisms with Siegel disks that realize all possible Picard numbers. The constructions involve extensive computer searches for appropriate Salem numbers and computations of algebraic numbers arising from holomorphic Lefschetz-type fixed point formulas and related Grothendieck residues. |
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| Keywords: | K3 surface Picard number Siegel disk Salem number Hypergeometric group Lefschetz-type fixed point formula |
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