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- uniform and defined over \mathbb{Q}.
- non-uniform and defined over \mathbb{Q}.
- uniform and defined over a number field k/\mathbb{Q}.
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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? -
Problem 2.2.
Can we classify all non-uniform maximal reflective arithmetic lattices when n=6? How about just those that are congruence? -
Problem 2.3.
Can we classify non-uniform maximal reflective arithmetic lattices in dimensions n \geq 10? How about just those that are congruence? -
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? -
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.