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

TL;DR

This paper explores how periods of $K3$ surfaces parametrized by Klein's icosahedral invariants can explicitly generate class fields over quartic CM fields, linking complex multiplication theory with geometric structures.

Contribution

It provides an explicit construction of class fields using $K3$ surface periods and Klein's invariants, and derives a canonical model of a Shimura variety in a special case.

Findings

Special values of icosahedral invariants generate class fields over quartic CM fields

Explicit expression of the Shimura variety's canonical model via $K3$ surface periods

Connection established between $K3$ surface geometry and complex multiplication theory

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.