# On Orthogonal Hypergeometric Groups of Degree Five

## TL;DR

This paper classifies degree five hypergeometric groups with roots of unity, identifying which have finite, rank one, or rank two Zariski closures, and determines the arithmeticity of many rank two cases.

## Contribution

It provides a comprehensive classification of orthogonal hypergeometric groups of degree five, including new results on the arithmeticity of rank two cases.

## Key findings

- 77 possible polynomial pairs with roots of unity
- 4 finite monodromy groups identified
- 17 groups with rank one Zariski closure

## Abstract

A computation shows that there are 77 (up to scalar shifts) possible pairs of integer coefficient polynomials of degree five, having roots of unity as their roots, and satisfying the conditions of Beukers and Heckman [1], so that the Zariski closures of the associated monodromy groups are either finite or the orthogonal groups of non-degenerate and non-positive quadratic forms. Following the criterion of Beukers and Heckman [1], it is easy to check that only 4 of these pairs correspond to finite monodromy groups and only 17 pairs correspond to monodromy groups, for which, the Zariski closures have real rank one. There are remaining 56 pairs, for which, the Zariski closures of the associated monodromy groups have real rank two. It follows from Venkataramana [16] that 11 of these 56 pairs correspond to arithmetic monodromy groups and the arithmeticity of 2 other cases follows from Singh [11]. In this article, we show that 23 of the remaining 43 rank two cases correspond to arithmetic groups.