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

TL;DR

This paper computes the automorphism group of a special K3 surface linked to Apéry's proof of zeta(3) irrationality, revealing its structure and action on rational curves.

Contribution

It explicitly determines the automorphism group of the Apéry-Fermi K3 surface and describes its action on rational curves, a novel analysis in this context.

Findings

Aut(X) is generated by specific elements and relations.

Aut(X) acts transitively on smooth rational curves.

Pairs of disjoint rational curves are partitioned into two orbits.

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.