Commentary on Sp(6) hypergeometric groups

TL;DR

This paper investigates specific symplectic hypergeometric groups of degree six, establishing the arithmeticity of four examples and providing insights into the broader classification of these groups as either arithmetic or thin.

Contribution

It identifies the arithmeticity of four particular hypergeometric groups of degree six, advancing the classification of these groups within the existing framework.

Findings

Four hypergeometric groups of degree six are proven to be arithmetic.

At least 384 of the 458 known groups are arithmetic.

At least 63 of the groups are thin, with 11 remaining unclassified.

Abstract

We study the examples mentioned in [2,Tables A & C] and establish the arithmeticity of four examples of symplectic hypergeometric groups of degree six. Following [2] we know that there are 458 inequivalent symplectic hypergeometric groups of degree six, and combining the results of this article with the work of [1,2,11], we now know that at least 384 are arithmetic and at least 63 are thin whereas the arithmeticity and thinness of remaining 11 examples are still unknown.