Jacobian variety
The Manin–Mumford conjecture of Yuri Manin and David Mumford, proved by Michel Raynaud, states that a curve "C" in its Jacobian variety "J" can only contain a finite number of points that are of finite order (a torsion point) in "J", unless "C" = "J". There are other more general versions, such as the Bogomolov conjecture which generalizes the statement to non-torsion points.