Elliptic K3 Surfaces with Abelian and Dihedral Groups of Symplectic Automorphisms

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, and we describe the coarse moduli space of K3 surfaces with a given finite group contained in the Mordell–Weil group.

Considering automorphisms coming from the base of the fibration, we find the Mordell–Weil lattice of a fibration described by Kloosterman, and we find K3 surfaces with dihedral groups as group of symplectic automorphisms. We prove the isometries between lattices described by the author and Sarti and lattices described by Shioda and by Greiss and Lam.

Key Words:

• Elliptic fibrations
• K3 surfaces
• Lattices
• Symplectic automorphisms

2010 Mathematics Subject Classification:

• 14J28
• 14J27
• 14J50
• 52C07

ACKNOWLEDGMENTS

I would like to thank Bert van Geemen for his invaluable help, Jaap Top for useful discussions, Tetsuji Shioda for the lectures he held in Milan, which aided me greatly in working on this article, Matthias Schütt, Stefano Vigni, and the referee for many helpful comments.

Notes

Communicated by R. Piene.