  Given a reduced abelian $p$-group, we give an upper bound on the Scott
complexity of the group in terms of its Ulm invariants. For limit ordinals, we
show that this upper bound is tight. This gives an explicit sequence of such
groups with arbitrarily high Scott complexity below $\omega_1$. Along the way,
we give a largely algebraic characterization of the back-and-forth relations on
reduced abelian $p$-groups, making progress on an open problem from the book by
Ash and Knight.
