The automorphism group of an Ap\'ery-Fermi K3 surface

Abstract

An Ap\'ery-Fermi K3 surface is a complex K3 surface of Picard number 19 that is birational to a general member of a certain one-dimensional family of affine surfaces related to the Fermi surface in solid-state physics. This K3 surface is also linked to a recurrence relation that appears in the famous proof of the irrationality of zeta(3) by Ap\'ery. We compute the automorphism group Aut(X) of the Ap\'ery-Fermi K3 surface X using Borcherds' method. We describe Aut(X) in terms of generators and relations. Moreover, we determine the action of Aut(X) on the set of ADE-configurations of smooth rational curves on X for some ADE-types. In particular, we show that Aut(X) acts transitively on the set of smooth rational curves, and that it partitions the set of pairs of disjoint smooth rational curves into two orbits.