Elliptic K3 surfaces with abelian and dihedral groups of symplectic automorphisms

TL;DR

This paper studies elliptic K3 surfaces with specific automorphism groups, analyzing their quotient structures, torsion sections, and lattice isometries, revealing how certain automorphism groups influence the surface's properties.

Contribution

It constructs quotient elliptic fibrations with preserved torsion sections and describes lattice isometries, advancing understanding of automorphism groups on K3 surfaces.

Findings

Quotient fibrations preserve torsion sections.

Lattice isometries relate different automorphism group actions.

Certain subgroups imply larger automorphism groups on K3 surfaces.

Abstract

We analyze K3 surfaces admitting an elliptic fibration $E$ and a finite group $G$ of symplectic automorphisms preserving this elliptic fibration. We construct the quotient elliptic fibration $E/G$ comparing its properties to the ones of $E$. We show that if $E$ admits an $n$-torsion section, its quotient by the group of automorphisms induced by this section admits again an $n$-torsion section. Considering automorphisms coming from the base of the fibration, we can describe the Mordell--Weil lattice of a fibration described by Kloosterman. We give the isometries between lattices described by the author and Sarti and lattices described by Shioda and by Griees and Lam. Moreover we show that for certain groups $H$ of $G$, $H$ subgroups of $G$, a K3 surface which admits $H$ as group of symplectic automorphisms actually admits $G$ as group of symplectic automorphisms.