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Unimodular Hunting

Gaëtan Chenevier

Abstract

We develop a method initiated by Bacher and Venkov, and based on a study of the Kneser neighbors of the standard lattice Z^n, which allows to classify the integral unimodular Euclidean lattices of rank n. As an application, of computational flavour, we determine the isometry classes of unimodular lattices of rank 26 and 27.

Unimodular Hunting

Abstract

We develop a method initiated by Bacher and Venkov, and based on a study of the Kneser neighbors of the standard lattice Z^n, which allows to classify the integral unimodular Euclidean lattices of rank n. As an application, of computational flavour, we determine the isometry classes of unimodular lattices of rank 26 and 27.

Paper Structure

This paper contains 55 sections, 47 theorems, 81 equations, 13 tables.

Table of Contents

  1. Introduction
  2. The classification of unimodular lattices
  3. The cyclic $d$-neighbors of ${\rm I}_n$
  4. Theoretical exhaustion
  5. First examples
  6. The general method
  7. The splitting root system by root system
  8. The visible part of a $d$-neighbor of ${\rm I}_n$
  9. The Biased Neighbor Enumeration algorithm
  10. An example : the root system $10\,{\bf A}_1$ in ${\rm X}_{26}$
  11. A coupon collector's problem and choices of invariants
  12. Proofs of Theorems \ref{['X2627']} and \ref{['thm:listuninei']} : the full lists
  13. General conventions and notations
  14. Cyclic Kneser neighbors of unimodular lattices
  15. Definitions and notations
  16. ...and 40 more sections

Key Result

Theorem A

We have $|{\rm X}_{26}|=2566$ and $|{\rm X}_{27}|=17059$.

Theorems & Definitions (88)

  • Theorem A
  • Theorem B
  • Definition 1
  • Theorem C
  • Remark 2
  • Corollary D
  • Remark 1.7
  • Definition 1.9
  • Example 3
  • Theorem E
  • ...and 78 more