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An explicit Enriques surface with an automorphism of minimum entropy

Simon Brandhorst, Matthias Zach

TL;DR

This work explicitly constructs an Enriques surface $Y_{OY}$ with an automorphism of minimal entropy $\log\tau_8$, grounding the result in lattice and Hodge-theoretic data and making it concrete through explicit equations. The authors identify the covering K3 surface $X_{OY}$ and its Enriques involution via a lattice-guided strategy, then build an explicit Weierstrass model and elliptic fibrations, together with concrete generators for the Néron–Severi lattice. They realize the minimal-entropy automorphism by a sequence of 2-neighbor steps, fibration hops, and carefully controlled conjugations, finally descending to $Y_{OY}$ and verifying the entropy via the Salem polynomial of degree $6$ for the action on $\mathrm{NS}(X_{OY})$. The implementation hinges on the OSCAR framework, enabling rigorous positive-characteristic reductions and explicit computations of intersection numbers, divisors, and automorphism actions, thereby providing reproducible, concrete equations for this notable dynamical example with potential applications to arithmetic dynamics on Enriques surfaces.

Abstract

We derive explicit equations for the Oguiso-Yu automorphism of minimum topological entropy on a complex Enriques surface. The approach is computer aided and makes use of elliptic fibrations.

An explicit Enriques surface with an automorphism of minimum entropy

TL;DR

This work explicitly constructs an Enriques surface with an automorphism of minimal entropy , grounding the result in lattice and Hodge-theoretic data and making it concrete through explicit equations. The authors identify the covering K3 surface and its Enriques involution via a lattice-guided strategy, then build an explicit Weierstrass model and elliptic fibrations, together with concrete generators for the Néron–Severi lattice. They realize the minimal-entropy automorphism by a sequence of 2-neighbor steps, fibration hops, and carefully controlled conjugations, finally descending to and verifying the entropy via the Salem polynomial of degree for the action on . The implementation hinges on the OSCAR framework, enabling rigorous positive-characteristic reductions and explicit computations of intersection numbers, divisors, and automorphism actions, thereby providing reproducible, concrete equations for this notable dynamical example with potential applications to arithmetic dynamics on Enriques surfaces.

Abstract

We derive explicit equations for the Oguiso-Yu automorphism of minimum topological entropy on a complex Enriques surface. The approach is computer aided and makes use of elliptic fibrations.

Paper Structure

This paper contains 36 sections, 15 theorems, 91 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Preliminaries
  4. K3 surfaces
  5. Enriques surfaces
  6. Elliptic Fibrations
  7. Lattices
  8. The Hodge- and lattice theory of the surface
  9. The construction of Oguiso and Yu
  10. The transcendental lattice
  11. Construction of the Enriques surface
  12. The first model
  13. Search for sections
  14. Extracting a Weierstrass model
  15. Specialisation to positive characteristic
  16. ...and 21 more sections

Key Result

Theorem 1.1

The Oguiso-Yu Enriques surface is birational to the surface with equation Its universal cover $X_{OY}$ is the elliptic K3 surface whose weierstrass equation is Translation by the $2$-torsion section $(x,y) = (0,0)$ composed with $(x,y,t)\mapsto (x,y,-t)$ defines an Enriques involution $\iota_{OY}$. Then $Y_{OY} = X_{OY}/\iota_{OY}$ is the Oguiso-Yu Enriques surface. It admits an automorphism $\w

Figures (2)

  • Figure 1: The real points of the affine birational model for the Enriques surface sought for by Oguiso and Yu
  • Figure 2: Liftings of the rational points in (\ref{['eqn:RationalPointsPosChar']}) on $\overline E_1$ from Section \ref{['sec:FindingSections']} to characteristic zero

Theorems & Definitions (35)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • proof
  • Theorem 2.1: Strong Torelli theorem
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • ...and 25 more