On families of K3 surfaces with real multiplication

TL;DR

The paper advances the understanding of real multiplication (RM) on K3 surfaces by developing both abstract lattice/theory-based and explicit geometric constructions. It proves the existence of large, in some cases maximal, RM families for several degrees m of totally real fields, including quadratic and cyclotomic subfields, and provides concrete families arising from purely non-symplectic automorphisms and from dihedral-cover deformations via Dickson polynomials. The results integrate Hodge-theoretic endomorphism algebras with moduli via the Torelli theorem and period map, delivering explicit RM on transcendental lattices and detailing how deformations preserve Picard data. Additionally, the paper demonstrates how isogenies between elliptic K3 surfaces yield RM and CM phenomena, enriching the set of tools for constructing RM K3 surfaces and highlighting connections to CM theory and arithmetic geometry.

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.