Automorphisms on K3 Surfaces with Small Picard Number

In this paper, we demonstrate via an example a variety of techniques both general and ad hoc that can be used to find the group of automorphisms of a K3 surface. Introduction Given a K3 surface X/k over a number field k, what is its group of automorphisms A = Aut(X/k)? In this paper, we offer some ideas of how to answer this natural question, and demonstrate these ideas by applying them to a particular example. This paper grew out of a talk given at the Banff International Research Station in December 2008. There are three main general tools. (1) Every automorphism σ induces a linear action on the Picard lattice Pic(X ) that preserves the intersection pairing; (2) the intersection pairing on the Picard lattice is a Lorentz product, so induces a hyperbolic structure on H = L+/R+, where L+ is the light cone; and (3) a fundamental result due to Pjateckĭi-S̆apiro and S̆afarevic̆ , which establishes a correspondence between A and a particular subgroup of the lattice preserving isometries of H. We apply these results, together with some ad hoc results, to a class of K3 surfaces, and come up with a group of finite index in A. Though not complete, we consider this answer to be sufficient; completing the problem likely depends on the arithmetic and not the geometry. That is, it depends on X and not just on Pic(X ). It is clear to the author that these techniques are applicable to many classes of K3 surfaces, particularly those with small Picard number. The hyperbolic space H of a surface with Picard number n is n − 1 dimensional, hence the difficulty of dealing with surfaces with large Picard number includes our difficulty imagining hyperbolic spaces of large dimension. 1. Background Let X/k be a K3 surface defined over a number field k. Let n be the dimension of the Picard lattice Pic(X ), and let {D1, ..., Dn} be a basis over Z, so Pic(X ) = D1Z⊕ · · · ⊕DnZ. Let J = [Di ·Dj] be the intersection matrix for the basis D. By the Hodge index theorem, J has signature (1, n − 1). That is, it has one positive eigenvalue and 2000 Mathematics Subject Classification. 14J28, 14J50, 14G05, 11G50.