Commensurability and arithmetic equivalence for orthogonal hypergeometric monodromy groups

TL;DR

This paper investigates the properties of orthogonal hypergeometric monodromy groups of degree five by computing invariants of associated quadratic forms, revealing their commensurability relations and conditions for non-conjugacy.

Contribution

It introduces a method to compute invariants of quadratic forms for these groups, enabling the classification of their commensurability and conjugacy properties.

Findings

Identified conditions for commensurability between groups.

Determined when thin groups are not conjugate.

Provided a framework for analyzing hypergeometric monodromy groups.

Abstract

We compute invariants of quadratic forms associated to orthogonal hypergeometric groups of degree five. This allows us to determine some commensurabilities between these groups, as well as to say when some thin groups cannot be conjugate to each other.