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

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.

20 pages. Revised on 31 March 2017 (Title and Abstract were changed). Typos are Corrected on 5 April 2017

References in corpus (5)

• K3 surfaces and equations for Hilbert modular surfaces
• A theta expression of the Hilbert modular functions for $\sqrt{5}$ via the periods of $K3$ surfaces
• To the Hilbert class field from the hypergeometric modular function
• Double integrals on a weighted projective plane and the Hilbert modular functions for $\mathbb{Q}(\sqrt{5})$
• Icosahedral invariants and Shimura curves

Cited by in corpus (2)

• Inverse period mappings of $K3$ surfaces and a construction of modular forms for a lattice with the Kneser conditions
• Elliptic fibrations on toric $K3$ hypersurfaces and mirror symmetry derived from Fano polytopes