Single-Class Genera of Positive Integral Lattices

Abstract

We give an enumeration of all positive definite primitive Z-lattices in dimension >= 3 whose genus consists of a single isometry class. This is achieved by using bounds obtained from the Smith-Minkowski-Siegel mass formula to computationally construct the square-free determinant lattices with this property, and then repeatedly calculating pre-images under a mapping first introduced by G. L. Watson. We hereby complete the classification of single-class genera in dimensions 4 and 5 and correct some mistakes in Watson's classifications in other dimensions. A list of all single-class primitive Z-lattices has been compiled and incorporated into the Catalogue of Lattices.

References in corpus (1)

• Enumerating maximal definite quadratic forms of bounded class number over Z in n >= 3 variables

Cited by in corpus (3)

• Bulk-Edge Correspondence in 2+1-Dimensional Abelian Topological Phases
• Universal quadratic forms and indecomposables in number fields: A survey
• Definite orders with locally free cancellation