On families of K3 surfaces with real multiplication

TL;DR

This paper constructs extensive families of K3 surfaces exhibiting real multiplication, employing lattice theory, Torelli theorem, period map, dihedral covers, and isogenies to both theoretically and explicitly demonstrate their existence.

Contribution

It introduces new methods to explicitly and abstractly construct large families of K3 surfaces with real multiplication.

Findings

Large families of K3 surfaces with real multiplication are constructed.

Both abstract lattice-theoretic and explicit geometric methods are used.

The existence of these families is established through Torelli theorem and period map analysis.

Abstract

We exhibit large families of K3 surfaces with real multiplication, both abstractly using lattice theory, the Torelli theorem and the surjectivity of the period map, as well as explicitly using dihedral covers and isogenies.