Elliptic fibrations and symplectic automorphisms on K3 surfaces

TL;DR

This paper explores the classification of symplectic automorphisms on K3 surfaces, focusing on the invariant sublattices and their orthogonal complements using elliptic K3 surfaces, building on Nikulin's foundational work.

Contribution

It computes invariant sublattices and their orthogonal complements for all groups classified by Nikulin, using special elliptic K3 surfaces.

Findings

Explicit computation of invariant sublattices for each group

Identification of orthogonal complements in the K3 lattice

Extension of Nikulin's classification to specific elliptic K3 surfaces

Abstract

Nikulin has classified all finite abelian groups acting symplectically on a K3 surface and he has shown that the induced action on the K3 lattice $U^3\oplus E_8(-1)^2$ depends only on the group but not on the K3 surface. For all the groups in the list of Nikulin we compute the invariant sublattice and its orthogonal complement by using some special elliptic K3 surfaces.