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OP-10218Status unassessedNumber theory

Classification of maximal reflective arithmetic lattices · Question 2.2

Arithmetic reflection groups and crystallographic packings · AIMPL workshop record

MSC 11 · Number theory

Editorial mapping of AIMPL source category: Number theory. http://aimpl.org/reflectiongpv/ Not an author-supplied MSC code.

Workshop record · status unassessedAs of 2026-09-10 source snapshot; not a current-status certificationSource & evidence ↓
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Problem statement

Can we classify all non-uniform maximal reflective arithmetic lattices when n=6n=6? How about just those that are congruence?

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Background

Section context

Restricting to dimensions n≥2n \geq 2, there are only finitely many maximal reflective arithmetic hyperbolic lattices. One driving goal is to find an exhaustive list of such lattices that are

  • uniform and defined over Q\mathbb{Q}.

  • non-uniform and defined over Q\mathbb{Q}.

  • uniform and defined over a number field k/Qk/\mathbb{Q}.

Of course, in practice, these three types of lattices may require different approaches and techniques.

Progress & source assessment

Dated source record, inspected 2026-09-10. Current status has not been independently reassessed. Source-reported results have not been independently verified.

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References
  1. [1]Arithmetic reflection groups and crystallographic packings · Original question link
  2. [2]AIMPL source bibliography link
  3. [3]AIMPL text license · CC BY-SA 3.0 linkAIM Problem Lists (AIMPL), original contributors; CC BY-SA 3.0. Adaptation: structural TeX converted to Markdown, documented source macros expanded, and source bibliography links added. Mathematical wording retained. These adapted source texts remain under CC BY-SA 3.0.

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