  This is the third and final installment in our series of papers applying the
method of Atkin and Swinnerton-Dyer to deduce formulas for rank differences.
The study of rank differences was initiated by Atkin and Swinnerton-Dyer in
their proof of Dyson's conjectures concerning Ramanujan's congruences for the
partition function. Since then, other types of rank differences for statistics
associated to partitions have been investigated. In this paper, we prove
explicit formulas for M_2-rank differences for overpartitions. Additionally, we
express a third order mock theta function in terms of rank differences.
