  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.
