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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.

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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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Authors and Affiliations

  1. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1, Komaba, Meguro, Tokyo, 153-8914, Japan

    Atsuhira Nagano

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  1. Atsuhira Nagano
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Correspondence to Atsuhira Nagano.

Additional information

This work is supported by Grant in Aid for JSPS Research Fellow (17J04395) The JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers “Mathematical Science of Symmetry, Topology and Moduli, Evolution of International Research Network based on OCAMI,” The Sumitomo Foundation Grant for Basic Science Research Project (No. 150108), and Waseda University Grant for Special Research Project (2015B-191).

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Nagano, A. Icosahedral invariants and a construction of class fields via periods of K3 surfaces. Ramanujan J 46, 201–227 (2018). https://doi.org/10.1007/s11139-017-9924-3

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