# Experimenting with symplectic hypergeometric monodromy groups

## TL;DR

This paper presents new computational results on symplectic monodromy groups of hypergeometric differential equations, including their arithmetic closure and conditions for arithmeticity, using advanced algorithms based on strong approximation.

## Contribution

It introduces extended algorithms for Zariski dense groups to compute the arithmetic closure of symplectic monodromy groups, advancing understanding of their arithmetic properties.

## Key findings

- Computed arithmetic closures of symplectic monodromy groups
- Identified cases of arithmeticity in hypergeometric monodromy groups
- Extended algorithms for Zariski dense groups based on strong approximation

## Abstract

We present new computational results for symplectic monodromy groups of hypergeometric differential equations. In particular, we compute the arithmetic closure of each group, sometimes justifying arithmeticity. The results are obtained by extending our previous algorithms for Zariski dense groups, based on the strong approximation and congruence subgroup properties.