[Paper Review] Elliptic fibrations on K3 surfaces and Salem numbers of maximal degree

Xun Yu|arXiv (Cornell University)|May 30, 2016

Algebraic Geometry and Number Theory18 references7 citations

TL;DR

This paper establishes a characterization of the maximal Salem degree of automorphisms on K3 surfaces using elliptic fibrations with infinite automorphism groups. It proves that for K3 surfaces in characteristic ≠ 2,3, the maximal Salem degree is determined by the rank of the sublattice generated by such fibrations—equal to the rank if even, or one less if odd—and applies this to show that every supersingular K3 surface in odd characteristic admits an automorphism with entropy equal to the logarithm of a degree-22 Salem number.

ABSTRACT

We study the maximal Salem degree of automorphisms of K3 surfaces via elliptic fibrations. By generalizing \cite{EOY14}, we establish a characterization of such maximum in terms of elliptic fibrations with infinite automorphism groups. As an application, we show that any supersingular K3 surface in odd characteristic has an automorphism the entropy of which is the natural logarithm of a Salem number of degree $22$.

Motivation & Objective

• To characterize the maximal Salem degree of automorphisms on K3 surfaces using elliptic fibrations with infinite automorphism groups.
• To generalize results from [EOY14] to establish a rank-based criterion for maximal Salem degree.
• To prove that every supersingular K3 surface in odd characteristic admits an automorphism with entropy equal to the logarithm of a degree-22 Salem number.
• To clarify the relationship between the exceptional sublattice, elliptic fibrations with infinite automorphism groups, and the maximality of Salem degrees.
• To provide a comparison principle for maximal Salem degrees between K3 surfaces with isometric Néron-Severi lattices and shared fibrational structure.

Proposed method

• Define the sublattice $ L_{ u}(X) riangleq \langle \text{elliptic fibrations with infinite automorphism groups} \rangle \subset \mathrm{NS}(X) $.
• Establish a rank-based formula: if $ d = \mathrm{rk}(L_{ u}(X)) $, then maximal Salem degree is $ d $ if $ d $ even, $ d-1 $ if $ d $ odd.
• Use the action of $ \mathrm{Aut}(X) $ on $ \mathrm{NS}(X) \otimes \mathbb{Q} $ to relate irreducibility of representations to the density of $ \mathbb{Q}\langle \mathcal{E}_{\infty}(X) \rangle $.
• Apply the exceptional sublattice $ E(\mathrm{NS}(X)) $ to characterize when $ \mathbb{Q}\langle \mathcal{E}_{\infty}(X) \rangle = \mathrm{NS}(X) \otimes \mathbb{Q} $, linking it to maximality of Salem degree.
• Leverage known results on supersingular K3 surfaces (e.g., $ \mathrm{NS}(X) $ isometric to sublattices of Artin invariant 1 surfaces) and Theorem 1.2 to extend existence of degree-22 Salem number entropy.
• Use the finiteness of certain Néron-Severi lattices (e.g., $ \mathcal{SEK}3' $) to constrain the set of K3 surfaces with bounded maximal Salem degree.

Experimental results

Research questions

• RQ1 What is the maximal possible Salem degree of an automorphism on a K3 surface, and how is it determined by the geometry of elliptic fibrations?
• RQ2 How does the presence of elliptic fibrations with infinite automorphism groups influence the structure of the Néron-Severi lattice and the dynamics of automorphisms?
• RQ3 Can the existence of a degree-22 Salem number entropy automorphism be guaranteed for all supersingular K3 surfaces in odd characteristic?
• RQ4 What is the relationship between the exceptional sublattice $ E(\mathrm{NS}(X)) $, the set of fibrations with infinite automorphism groups, and the maximality of Salem degrees?
• RQ5 Under what conditions does the maximal Salem degree of one K3 surface dominate that of another with isometric Néron-Severi lattice?

Key findings

• For a K3 surface $ X $ over an algebraically closed field of characteristic ≠ 2,3, the maximal Salem degree of automorphisms is $ d $ if the rank $ d $ of the sublattice $ L_{ u}(X) $ generated by fibrations with infinite automorphism groups is even, and $ d-1 $ if $ d $ is odd.
• The maximal Salem degree of an automorphism on a K3 surface $ X $ is bounded above by $ \rho(X) $, and equality holds if and only if $ \mathbb{Q}\langle \mathcal{E}_{\infty}(X) \rangle = \mathrm{NS}(X) \otimes \mathbb{Q} $.
• Any supersingular K3 surface in odd characteristic admits an automorphism whose entropy is the logarithm of a Salem number of degree 22.
• The exceptional sublattice $ E(\mathrm{NS}(X)) $ vanishes for all supersingular K3 surfaces in odd characteristic, which implies the existence of a full rational span of fibrational classes.
• The set of Néron-Severi lattices of K3 surfaces with $ E(\mathrm{NS}(X)) \neq \{0\} $ and bounded maximal Salem degree is finite, implying finitely many such K3 surfaces with maximal Salem degree ≤ 18.
• For the Kummer surface $ \mathrm{Km}(E \times F) $ with non-isogenous elliptic curves $ E, F $, the maximal Salem degree is exactly 10, as confirmed by both the rank of $ L_{ u}(X) $ and the negative definiteness of $ E(\mathrm{NS}(X)) $.