K3 Surfaces, Picard Numbers and Siegel Disks

TL;DR

This paper explores the relationship between automorphisms with Siegel disks on K3 surfaces and their Picard numbers, providing constructions for all possible Picard numbers using hypergeometric groups and computational methods.

Contribution

It establishes a complete characterization of Picard numbers for K3 surfaces with automorphisms having Siegel disks and introduces computational techniques for constructing such automorphisms.

Findings

Automorphisms with Siegel disks occur only for even Picard numbers between 0 and 18.

All possible Picard numbers for such automorphisms are realizable through explicit constructions.

Extensive computer searches and algebraic computations underpin the constructions.

Abstract

If a K3 surface admits an automorphism with a Siegel disk, then its Picard number is an even integer between $0$ and $18$. Conversely, using the method of hypergeometric groups, we are able to construct K3 surface automorphisms with Siegel disks that realize all possible Picard numbers. The constructions involve extensive computer searches for appropriate Salem numbers and computations of algebraic numbers arising from holomorphic Lefschetz-type formulas and related Grothendieck residues.