Explicit modularity of K3 surfaces with complex multiplication of large degree

TL;DR

This paper investigates the explicit modularity of certain K3 surfaces with complex multiplication, linking their transcendental motives to algebraic Hecke characters and abelian threefolds, providing evidence for a Kuga-Satake correspondence.

Contribution

It establishes explicit connections between the transcendental motives of K3 surfaces with CM and algebraic Hecke characters, advancing understanding of their modularity and Kuga-Satake relations.

Findings

Matching of transcendental motives to algebraic Hecke characters

Evidence for Kuga-Satake correspondence involving abelian threefolds

Explicit algebraic descriptions of CM K3 surfaces

Abstract

We consider the transcendental motive of three K3 surfaces $X$ conjectured to have complex multiplication (CM). Under this assumption, we match these to explicit algebraic Hecke quasi-characters $\psi_X$, and CM abelian threefolds $A$. This provides substantial evidence that a power of $A$ corresponds to $X$ under the Kuga-Satake correspondence.