Unimodular lattices of rank 29 and related even genera of small determinant

TL;DR

This paper classifies unimodular lattices of rank 29 using an efficient inductive method and determines even lattices of rank up to 28 with small prime determinants, providing new invariants and insights.

Contribution

Develops an elementary, efficient inductive approach to classify unimodular lattices of rank 29 and related even lattices with small determinants, expanding the scope of known lattice classifications.

Findings

Classified unimodular lattices of rank 29.

Determined isometry classes of even lattices up to rank 28 with small determinants.

Introduced new invariants for lattice verification.

Abstract

We classify the unimodular Euclidean integral lattices of rank 29 by developing an elementary, yet very efficient, inductive method. As an application, we determine the isometry classes of even lattices of rank at most 28 and prime (half-)determinant at most 7. We also provide new isometry invariants allowing for independent verification of the completeness of our lists, and we give conceptual explanations of some unique orbit phenomena discovered during our computations. Some of the genera classified here are orders of magnitude larger than any genus previously classified. In a forthcoming companion paper, we use these computations to study the cohomology of GL_n(Z).