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The automorphism group of an Apéry-Fermi K3 surface

Ichiro Shimada

TL;DR

The paper computes the automorphism group Aut(X) of the Apéry–Fermi K3 surface via Borcherds' method, delivering an explicit finite generating set and a Coxeter-like chamber tessellation of the nef-big cone. It embeds NS(X) into a Conway lattice L26, identifies a fundamental chamber D0 with 80 walls, and shows Aut(X) is generated by a dihedral-16 subgroup Aut(X, D0) together with eight additional automorphisms realized by Jacobian fibrations and Mordell–Weil translations. The authors provide a geometric realization of all generators, classify the action on ADE-configurations of smooth rational curves, and present a complete framework of relations among the generators. The work illustrates the power of Borcherds' method for concrete K3 surfaces and yields detailed orbit data for rational curves, with potential applications to related lattice-polarized K3 surfaces.

Abstract

An Apéry-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 Apéry. We compute the automorphism group Aut(X) of the Apéry-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.

The automorphism group of an Apéry-Fermi K3 surface

TL;DR

The paper computes the automorphism group Aut(X) of the Apéry–Fermi K3 surface via Borcherds' method, delivering an explicit finite generating set and a Coxeter-like chamber tessellation of the nef-big cone. It embeds NS(X) into a Conway lattice L26, identifies a fundamental chamber D0 with 80 walls, and shows Aut(X) is generated by a dihedral-16 subgroup Aut(X, D0) together with eight additional automorphisms realized by Jacobian fibrations and Mordell–Weil translations. The authors provide a geometric realization of all generators, classify the action on ADE-configurations of smooth rational curves, and present a complete framework of relations among the generators. The work illustrates the power of Borcherds' method for concrete K3 surfaces and yields detailed orbit data for rational curves, with potential applications to related lattice-polarized K3 surfaces.

Abstract

An Apéry-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 Apéry. We compute the automorphism group Aut(X) of the Apéry-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.

Paper Structure

This paper contains 30 sections, 16 theorems, 149 equations, 15 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Previous studies of the Apéry--Fermi K3 surface
  4. Plan of this paper
  5. Two projective models of an Apéry--Fermi K3 surface
  6. The Fermi surface model
  7. The Mukai--Ohashi quartic
  8. Néron--Severi lattice and automorphism group
  9. Chambers and their faces
  10. The lattice SX
  11. An ample class a32
  12. Embedding Aut(X) into OSXPX
  13. The finite subgroup AutXL32
  14. Borcherds' method
  15. Proof of Corollary \ref{['cor:ratstransitive']}
  16. ...and 15 more sections

Key Result

Theorem 1.1

The automorphism group $\mathord{\mathrm{Aut}}(X_s)$ is generated by a finite subgroup $\mathord{\mathrm{Aut}}(X_s, D_0)$ of order $16$, and eight extra automorphisms.

Figures (15)

  • Figure 2.1: Dual graph of the curves $L_{\gamma_1 \gamma_2 \gamma_3}$ ($L$-cube)
  • Figure 2.2: Sub-configuration containing $2\,\mathrm{II}^*$
  • Figure 2.3: Dual graph of the curves $P_i$, $Q_{ij}$, $T\sp{\prime\prime}_{\nu}$
  • Figure 3.1: Dual graph of $e_1\sp{(\nu)}, \dots, e_8\sp{(\nu)}$
  • Figure 3.2: Basis of $R$
  • ...and 10 more figures

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Definition 3.1
  • Remark 3.2
  • ...and 37 more