On ground fields of arithmetic hyperbolic reflection groups

TL;DR

This paper identifies finite sets of number fields that contain the ground fields of arithmetic hyperbolic reflection groups, providing bounds on their degrees and proposing a related mirror symmetric conjecture to aid classification.

Contribution

It explicitly defines finite sets of number fields containing ground fields and establishes bounds on their degrees, advancing the classification of hyperbolic reflection groups.

Findings

Degree of ground fields in dimension ≥6 is bounded by 56

Finite sets of number fields containing ground fields are explicitly constructed

A mirror symmetric conjecture related to finiteness is formulated

Abstract

Using authors's methods of 1980, 1981, some explicit finite sets of number fields containing ground fields of arithmetic hyperbolic reflection groups are defined, and good bounds of their degrees (over Q) are obtained. For example, degree of the ground field of any arithmetic hyperbolic reflection group in dimension at least 6 is bounded by 56. These results could be important for further classification. We also formulate a mirror symmetric conjecture to finiteness of the number of arithmetic hyperbolic reflection groups which was established in full generality recently.