Modular forms and K3 surfaces

TL;DR

This paper establishes a correspondence between certain Hecke eigenforms of weight 3 with rational eigenvalues and associated K3 surfaces over QQ, answering a longstanding question in the field.

Contribution

It constructs explicit K3 surfaces for all known such eigenforms, advancing the understanding of the link between modular forms and algebraic geometry.

Findings

Every known Hecke eigenform of weight 3 with rational eigenvalues has an associated K3 surface over QQ.

The proof utilizes a classification of CM forms by the second author.

The work confirms a question posed by Mazur and van Straten.

Abstract

For every known Hecke eigenform of weight 3 with rational eigenvalues we exhibit a K3 surface over QQ associated to the form. This answers a question asked independently by Mazur and van Straten. The proof builds on a classification of CM forms by the second author.