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

Gaëtan Chenevier, Olivier Taïbi

TL;DR

The paper delivers a dramatic advance in the classification of unimodular lattices by proving the exact count of isometry classes in rank $29$ and by introducing robust invariants and an inductive orbit-method framework that scales from lower to higher ranks. It develops a comprehensive toolkit—groupoids, gluing, residues, and marked BV invariants—and demonstrates their effectiveness through large-scale computations, including the rank-$29$ classification and detailed studies of genera ${\mathcal G}_{n,p}$ for small primes $p$. The BV and marked BV invariants play a central role in distinguishing lattices and validating completeness via mass formulas, enabling the independent verification of results that would be computationally prohibitive with previous methods. The work also uncovers structural phenomena for exceptional vectors and fertile root-extensions, connects to gluing constructions, and provides methodological groundwork with substantial implications for automorphic forms and the cohomology of ${\rm GL}_n(\mathbb{Z})$, paving the way for further exploration of high-rank lattice genera. Practical impact includes new classification data, improved algorithms (e.g., for BV computation and orbit enumeration), and a framework applicable to related questions in lattice theory and arithmetic geometry. The companion paper will leverage these findings to study local orbital integrals and cusp cohomology in GL$_n$ over $\mathbb{Z}$.

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).

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

TL;DR

The paper delivers a dramatic advance in the classification of unimodular lattices by proving the exact count of isometry classes in rank and by introducing robust invariants and an inductive orbit-method framework that scales from lower to higher ranks. It develops a comprehensive toolkit—groupoids, gluing, residues, and marked BV invariants—and demonstrates their effectiveness through large-scale computations, including the rank- classification and detailed studies of genera for small primes . The BV and marked BV invariants play a central role in distinguishing lattices and validating completeness via mass formulas, enabling the independent verification of results that would be computationally prohibitive with previous methods. The work also uncovers structural phenomena for exceptional vectors and fertile root-extensions, connects to gluing constructions, and provides methodological groundwork with substantial implications for automorphic forms and the cohomology of , paving the way for further exploration of high-rank lattice genera. Practical impact includes new classification data, improved algorithms (e.g., for BV computation and orbit enumeration), and a framework applicable to related questions in lattice theory and arithmetic geometry. The companion paper will leverage these findings to study local orbital integrals and cusp cohomology in GL over .

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).
Paper Structure (23 sections, 39 theorems, 19 equations, 1 figure, 20 tables)

This paper contains 23 sections, 39 theorems, 19 equations, 1 figure, 20 tables.

Table of Contents

  1. Introduction
  2. The classification of unimodular lattices
  3. Even lattices of small and prime (half-)determinant $p$
  4. Unique orbit property for "exceptional" vectors
  5. Motivations and perspectives
  6. General preliminaries on lattices
  7. Basic notations, conventions, and terminology
  8. Groupoids and masses
  9. Residues and gluing constructions
  10. Characteristic vectors and even sublattices
  11. Marked ${\rm BV}$ invariant of depth $d$
  12. Lattice algorithms
  13. Unimodular lattices having a pair of orthogonal roots
  14. An application: the classification of rank 29 unimodular lattices
  15. Fertile weights, extensions of root systems and gluing
  16. ...and 8 more sections

Key Result

Theorem A

There are exactly $38\,966\,352$ classes in ${\rm X}_{29}$, all distinguished by their BV invariants, and with Gram matrices given in chetaiweb.

Figures (1)

  • Figure 1: Unimodular lattices rule! ( Leitfaden for the proof of Theorem \ref{['thmintro:hdiscptable']})

Theorems & Definitions (72)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Example 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 2.7
  • Remark 2.8
  • ...and 62 more