On fields of definition of arithmetic Kleinian reflection groups II

TL;DR

This paper establishes new upper bounds on the degrees and discriminants of fields of definition for arithmetic hyperbolic reflection groups, advancing the classification of such groups across all dimensions.

Contribution

It proves a new degree bound of 9 for dimension 3 and a combined bound of 25 for all dimensions, refining previous finiteness results.

Findings

Degree of fields in dimension 3 does not exceed 9.

Overall degree bound for all dimensions is 25.

Provides upper bounds for discriminants of these fields.

Abstract

Following the previous work of Nikulin and Agol, Belolipetsky, Storm, and Whyte it is known that there exist only finitely many (totally real) number fields that can serve as fields of definition of arithmetic hyperbolic reflection groups. We prove a new bound on the degree $n_k$ of these fields in dimension 3: $n_k$ does not exceed 9. Combined with previous results of Maclachlan and Nikulin, this leads to a new bound $n_k \le 25$ which is valid for all dimensions. We also obtain upper bounds for the discriminants of these fields and give some heuristic results which may be useful for the classification of arithmetic hyperbolic reflection groups.