The Ramanujan Journal

, Volume 46, Issue 1, pp 201–227 | Cite as

Icosahedral invariants and a construction of class fields via periods of K3 surfaces

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Abstract

In the theory of complex multiplication, it is important to construct class fields over CM fields. In this paper, we consider explicit K3 surfaces parametrized by Klein’s icosahedral invariants. Via the periods and the Shioda–Inose structures of K3 surfaces, the special values of icosahedral invariants generate class fields over quartic CM fields. Moreover, we give an explicit expression of the canonical model of the Shimura variety for the simplest case via the periods of K3 surfaces.

Keywords

Class fields K3 surfaces Shimura varieties Abelian varieties Complex multiplication Hilbert modular functions Quartic fields 

Mathematics Subject Classification

Primary 11G45 Secondary 14J28 14G35 11F46 11G15 11R16 
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Notes

Acknowledgements

The author would like to thank Professor Hironori Shiga for helpful advice and valuable suggestions, and also to Professor Kimio Ueno for kind encouragement.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

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