The modularity of K3 surfaces with non-symplectic group actions

TL;DR

This paper studies special K3 surfaces with non-symplectic group actions, showing they are dominated by Fermat surfaces, are of CM type, and their Galois representations are modular, with implications for mirror symmetry.

Contribution

It demonstrates that these K3 surfaces are dominated by Fermat surfaces, are of CM type, and establishes the modularity of their Galois representations, extending understanding of their symmetry and arithmetic properties.

Findings

K3 surfaces are dominated by Fermat surfaces.

All such K3 surfaces are of CM type.

Galois representations associated are modular.

Abstract

We consider complex K3 surfaces with a non-symplectic group acting trivially on the algebraic cycles. Vorontsov and Kondo classified those K3 surfaces with transcendental lattice of minimal rank. The purpose of this note is to study the Galois representations associated to these K3 surfaces. The rank of the transcendental lattices is even and varies from 2 to 20, excluding 8 and 14. We show that these K3 surfaces are dominated by Fermat surfaces, and hence they are all of CM type. We will establish the modularity of the Galois representations associated to them. Also we discuss mirror symmetry for these K3 surfaces in the sense of Dolgachev, and show that a mirror K3 surface exists with one exception.