On ground fields of arithmetic hyperbolic reflection groups. III

TL;DR

This paper extends previous work to explicitly identify finite sets of number fields containing all ground fields of arithmetic hyperbolic reflection groups in dimensions three and higher, providing bounds on their degrees and enabling potential classification.

Contribution

It defines explicit finite sets of number fields containing all ground fields of arithmetic hyperbolic reflection groups and establishes bounds on their degrees across all dimensions.

Findings

Explicit bounds on degrees of ground fields in all dimensions

Finite sets of number fields containing all ground fields

Potential for effective classification of hyperbolic reflection groups

Abstract

This paper continues arXiv.org:math.AG/0609256, arXiv:0708.3991 and arXiv:0710.0162 . Using authors's methods of 1980, 1981, some explicit finite sets of number fields containing all ground fields of arithmetic hyperbolic reflection groups in dimension at least 3 are defined, and explicit bounds of their degrees (over Q) are obtained. Thus, now, explicit bound of degree of ground fields of arithmetic hyperbolic reflection groups is known in all dimensions. Thus, now, we can, in principle, obtain effective finite classification of arithmetic hyperbolic reflection groups in all dimensions together.