Arithmetic monodromy in $\mathrm{Sp}(2n)$

Abstract

Based on a result of Singh–Venkataramana, Bajpai–Dona–Singh–Singh gave a criterion for a discrete Zariski-dense subgroup of $\mathrm{Sp}(2n,\mathbb{Z})$ to be a lattice. We adapt this criterion so that it can be used in some situations that were previously excluded. We apply the adapted method to subgroups of $\mathrm{Sp}(6,\mathbb{Z})$ and $\mathrm{Sp}(4,\mathbb{Z})$ that arise as the monodromy groups of hypergeometric differential equations. In particular, we show that out of the $40$ maximally unipotent $\mathrm{Sp}(6)$ hypergeometric groups, more than half are arithmetic, answering a question of Katz in the negative.