Borcherds lattices and K3 surfaces of zero entropy

2022/11/17 by Simon Brandhorst, Brandhorst, Simon, Giacomo Mezzedimi +1 · 4 citations
Mathematics · #11H56 #14J28 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2211.09600

openalex publication_date 2022/11/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let L be an even, hyperbolic lattice with infinitely many simple (-2)-roots. We call L a Borcherds lattice if it admits an isotropic vector with bounded inner product with all the simple (-2)-roots. We show that this is the case if and only if L has zero entropy, or equivalently if and only if all symmetries of L preserve some isotropic vector. We obtain a complete classification of Borcherds lattices, consisting of 194 lattices. In turn this provides a classification of hyperbolic lattices of rank ≥ 5 with virtually solvable symmetry group. Finally, we apply these general results to the case of K3 surfaces. We obtain a classification of Picard lattices of K3 surfaces of zero entropy and infinite automorphism group, consisting of 193 lattices. In particular we show that all Kummer surfaces, all supersingular K3 surfaces and all K3 surfaces covering an Enriques surface (with one exception) admit an automorphism of positive entropy.

Cited by

• Virtually abelian symmetry groups of hyperbolic lattices
• On K3 surfaces with hyperbolic automorphism groups
• A Lemma on Leech-like Lattices
• Rank jumps and Multisections of elliptic fibrations on K3 surfaces

Related

• K3 surface entropy and automorphism groups
• Enriques surfaces of zero entropy
• K3 Surfaces of zero entropy admitting an elliptic fibration with only\n irreducible fibers
• The Cayley-Oguiso automorphism of positive entropy on a K3 surface
• K3 surfaces with interesting groups of automorphisms