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2. Classification of maximal reflective arithmetic lattices

Restricting to dimensions $n \geq 2$, there are only finitely many maximal reflective arithmetic hyperbolic lattices. One driving goal is to find an exhaustive list of such lattices that are
  1. uniform and defined over $\mathbb{Q}$.
  2. non-uniform and defined over $\mathbb{Q}$.
  3. uniform and defined over a number field $k/\mathbb{Q}$.
Of course, in practice, these three types of lattices may require different approaches and techniques.


    1. Add remark

      Problem 2.1.

      Scharlau has recently classified all non-uniform lattices in dimensions $n=3$ and $n=4$ (possibly also $n = 5$?). It is known that such lattices do not exist when $n=20$ or when $n \geq 22$. When $n=21$, there is only one reflective arithmetic lattice. Can we classify all non-uniform lattices the remaining $n$-values?


        • Add remark

          Problem 2.2.

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


            • Add remark

              Problem 2.3.

              Can we classify non-uniform maximal reflective arithmetic lattices in dimensions $n \geq 10$? How about just those that are congruence?


                • Add remark

                  Problem 2.4.

                  For $n=19$, there is only one known example of a non-uniform reflective arithmetic lattice. Is it unique, which is the case for the lone example in dimension $21$?


                    • Add remark

                      Problem 2.5.

                      Create a central repository (similar to the LMFDB) for all known data relating to Coxeter polytopes in low-dimension.

                          Cite this as: AimPL: Arithmetic reflection groups and crystallographic packings, available at http://aimpl.org/reflectiongpv.