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The classic enumeration of genera of even definite lattices

A mix of human-written and AI-generated textHuman understanding: all partsmath.NT — Number Theorycs.SC — Symbolic Computationcs.MS — Mathematical Software

Contributed by Mathieu Dutour Sikiric ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

We revisit the enumeration of a genus of even positive definite lattices by Kneser neighbours, certified by the mass. We work at primes dividing the determinant, where the projective spaces are smallest, schedule the exploration of the neighbour graph by a random walk and by orders based on a Hecke bookkeeping, and compute canonical forms through the Weyl chamber of the root system. Genera of rank up to 19 with up to 40,617 classes are enumerated in minutes.

Author comments: 22 pages. The algorithms are implemented in the program GENUS_Enumerate of polyhedral_common, https://github.com/MathieuDutSik/polyhedral_common

Tools used

Anthropic
Claude Opus 5.5 (via Claude Code)

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Version history

  1. v1 · 22 pagesSubmitted by Mathieu Dutour Sikiric22 pages. The algorithms are implemented in the program GENUS_Enumerate of polyhedral_common, https://github.com/MathieuDutSik/polyhedral_commonCurrentNov 11, 2026