Virtually abelian symmetry groups of hyperbolic lattices

2025/07/28 by Brandhorst, Simon, Kirschmer, Markus, Mezzedimi, Giacomo
#11H56 #14J28 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.20743

Abstract

We give a classification of integral lattices with virtually abelian symmetry group. As a consequence, we complete the classification of K3 surfaces with virtually abelian automorphism group. In the appendix we formulate an algorithm for weak approximation in orthogonal groups and use it to determine if two indefinite lattices are isometric.

Citations

• Borcherds lattices and K3 surfaces of zero entropy
• K3 surface entropy and automorphism groups
• Explicit Result on Equivalence of Rational Quadratic Forms Avoiding Primes
• K3 Surfaces of zero entropy admitting an elliptic fibration with only\n irreducible fibers
• Connected components of the moduli of elliptic K3 surfaces
• An algorithm to compute automorphism groups of K3 surfaces and an application to singular K3 surfaces
• Automorphs of indefinite binary quadratic forms and K3-surfaces with Picard number 2
• K3 surfaces with interesting groups of automorphisms

Related

• K3 surfaces with interesting groups of automorphisms
• Virtually special embeddings of integral Lorentzian lattices
• Nearest-Neighbor Tight-Binding Realization of Hyperbolic Lattices with ℤ₂ Gauge Structures
• Coxeter groups, Lorentzian lattices, and K3 surfaces
• Simply transitive geodesics and omnipotence of lattices in PSL(2,ℂ)