The Enriques surface of minimal entropy

TL;DR

The paper resolves the realizability of Lehmer’s number $λ_{10}$ as a dynamical degree of automorphisms on Enriques surfaces. It shows nonexistence in odd characteristic and, crucially, existence and uniqueness in characteristic 2: there is a unique Enriques surface $X^{oldsymbol{ leftrightarrow}}$ (defined over $F_2$) whose automorphism attains $h= frac{}{} ext{log }λ_{10}$, with $ ext{Pic}^{τ}(X^{oldsymbol{ leftrightarrow}})\congoldsymbol{}$ and a canonical $oldsymbol{}$-torsor $Y$ that is a normal rational surface with a single elliptic singularity. The authors provide explicit equations and construct ten conjugacy classes of such automorphisms, relate them to the $E_{10}$ lattice, and prove uniqueness via lattice, cohomological, and deformation arguments in characteristic 2. These results extend Oguiso’s complex-analytic nonexistence and demonstrate a sharp characteristic-dependent dichotomy for Lehmer dynamics on Enriques surfaces.

Abstract

Lehmer's number $λ_{10}$ is the smallest dynamical degree greater than $1$ that can occur for an automorphism of an algebraic surface. We show that $λ_{10}$ cannot be realized by automorphisms of Enriques surfaces in odd characteristic, extending a result of Oguiso over the complex numbers. In contrast, we prove that in characteristic $2$ there exists a unique Enriques surface that admits an automorphism with dynamical degree $λ_{10}$. We also provide explicit equations for the surface as well as for all conjugacy classes of automorphisms that realize $λ_{10}$.

Figures (1)

• Figure 1: The resolution of singularities of the surface $X_0$ in \ref{['thm: existence main']}.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 2.1
• proof
• Claim 2.2
• proof : Proof of the claim
• Remark 2.3
• Remark 2.4