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Efficient enumeration of quadratic lattices

Eran Assaf, Victor Chen, Rohan Garg, Benny Wang

TL;DR

The work addresses the problem of enumerating isometry classes of integral quadratic lattices of fixed rank $n$ and determinant $D$, presenting an algorithm that combines Brandhorst-style maximal overlattices with local modifications and contrasting it with the Dubey–Holenstein approach, augmented by Hanke’s improvements. It derives explicit running-time bounds by counting genus symbols via $p$-adic analysis and Meinardus-type asymptotics, yielding bounds of the form $O\left(n^4 \lambda^{e_2} f(D) \frac{\log D}{\log \log D}\right)$ and refined bounds across divisors using Dirichlet-series methods. The paper provides rigorous complexity analyses for both Brandhorst and Hanke-based maximal overlattice constructions and local modification steps, and includes publicly available SageMath implementations to facilitate practical enumeration. Together, these results advance efficient tabulation of lattice genera and isometry classes, with potential impact on catalogs of quadratic forms and related arithmetic applications.

Abstract

We present an algorithm to enumerate isometry classes of integral quadratic lattices of a given rank and determinant, and analyze its running time by giving bounds on the number of genus symbols for a fixed rank and determinant. We build on previous work of Brandhorst, Hanke, and Dubey and Holenstein. We analyze the running times of their respective algorithms and compare the practical performance of their implementations with our own. Our implementations are publicly available.

Efficient enumeration of quadratic lattices

TL;DR

The work addresses the problem of enumerating isometry classes of integral quadratic lattices of fixed rank and determinant , presenting an algorithm that combines Brandhorst-style maximal overlattices with local modifications and contrasting it with the Dubey–Holenstein approach, augmented by Hanke’s improvements. It derives explicit running-time bounds by counting genus symbols via -adic analysis and Meinardus-type asymptotics, yielding bounds of the form and refined bounds across divisors using Dirichlet-series methods. The paper provides rigorous complexity analyses for both Brandhorst and Hanke-based maximal overlattice constructions and local modification steps, and includes publicly available SageMath implementations to facilitate practical enumeration. Together, these results advance efficient tabulation of lattice genera and isometry classes, with potential impact on catalogs of quadratic forms and related arithmetic applications.

Abstract

We present an algorithm to enumerate isometry classes of integral quadratic lattices of a given rank and determinant, and analyze its running time by giving bounds on the number of genus symbols for a fixed rank and determinant. We build on previous work of Brandhorst, Hanke, and Dubey and Holenstein. We analyze the running times of their respective algorithms and compare the practical performance of their implementations with our own. Our implementations are publicly available.
Paper Structure (16 sections, 34 theorems, 61 equations, 5 algorithms)

This paper contains 16 sections, 34 theorems, 61 equations, 5 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Results
  4. Background
  5. Representative of a genus
  6. Brandhorst algorithm
  7. Dubey--Holenstein Algorithm
  8. Maximal overlattices
  9. Brandhorst maximal overlattice algorithm
  10. Hanke algorithm
  11. Local modifications
  12. Counting Genus Symbols
  13. p=2 case
  14. Proof of Theorem \ref{['thm:main theorem']}
  15. Proof of Theorem \ref{['thm:main theorem-upto']}.
  16. ...and 1 more sections

Key Result

Theorem 1.1

There exists an algorithm that, given positive integers $n$ and $D$ such that $\log D \ll n$, outputs an integral lattice in every genus of rank $n$ and determinant $D$, with running time $n^4 D^{\alpha + \frac{\pi (1 + o(1)) }{\log \log D}}$, where $\alpha = \log_2(3 + \sqrt{17}) - 1 \approx 1.833$

Theorems & Definitions (75)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 65 more