Thin monodromy in $\mathrm{Sp}(4)$ and $\mathrm{Sp}(6)$

TL;DR

This paper classifies hypergeometric groups in $ ext{Sp}(4)$ and $ ext{Sp}(6)$ as either arithmetic or thin, using a new computer-assisted ping pong method, and provides the first extensive examples of non-arithmetic hypergeometric monodromy groups.

Contribution

It introduces a novel computer-assisted ping pong approach to classify hypergeometric groups in $ ext{Sp}(4)$ and $ ext{Sp}(6)$ as thin or arithmetic, completing the classification for these cases.

Findings

Proved thinness of 17 hypergeometric groups in $ ext{Sp}(6)$ with maximally unipotent monodromy.

Classified all 40 hypergeometric groups in $ ext{Sp}(6)$ into arithmetic and thin cases.

Established thinness of 46 hypergeometric groups in $ ext{Sp}(6)$ and 3 in $ ext{Sp}(4)$.

Abstract

We explore the thinness of hypergeometric groups of type $\mathrm{Sp}(4)$ and $\mathrm{Sp}(6)$ by applying a new approach of computer-assisted ping pong. We prove the thinness of $17$ hypergeometric groups with maximally unipotent monodromy in $\mathrm{Sp}(6)$, completing the classification of all $40$ such groups into arithmetic and thin cases. In addition, we establish the thinness of further $46$ hypergeometric groups in $\mathrm{Sp}(6)$, and of $3$ hypergeometric groups in $\mathrm{Sp}(4)$, completing the classification of all $\mathrm{Sp}(4)$ hypergeometric groups. To the best of our knowledge, this article produces the first $63$ examples in the cyclotomic family of Zariski dense non-arithmetic hypergeometric monodromy groups of real rank three.