Arithmetic Monodromy in Sp(2n)

TL;DR

This paper adapts a criterion for identifying when certain subgroups of Sp(2n,Z) are lattices and applies it to hypergeometric monodromy groups, revealing that over half of the Sp(6) groups studied are arithmetic.

Contribution

It extends an existing criterion to cover previously excluded cases and applies it to hypergeometric monodromy groups, providing new insights into their arithmetic nature.

Findings

More than half of the 40 maximally unipotent Sp(6) hypergeometric groups are arithmetic.

The adapted criterion successfully identifies arithmetic groups in new contexts.

Answers a long-standing question of Katz negatively regarding hypergeometric groups.

Abstract

Based on a result of Singh--Venkataramana, Bajpai--Dona--Singh--Singh gave a criterion for a discrete Zariski-dense subgroup of Sp(2n,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 Sp(6,Z) and Sp(4,Z) that arise as the monodromy groups of hypergeometric differential equations. In particular, we show that out of the 40 maximally unipotent Sp(6) hypergeometric groups more than half are arithmetic, answering a question of Katz in the negative.