2-Selmer rank (reviewed)

The 2-Selmer rank of an abelian variety is the rank of its 2-Selmer group.

The difference between the 2-Selmer rank and the 2-rank of the torsion subgroup is an upper bound on the rank of the Mordell-Weil group. This bound is tight if and only if the 2-part of the Tate-Shafarevich group is trivial