Conditional Mordell-Weil group (reviewed)

Under the (weak) BSD conjecture, the Mordell-Weil rank of an abelian variety $A$ over $\Q$ is equal to its analytic rank. If the torsion subgroup of $A$ and its analytic rank are known (or if one simply assumes a known upper bound on the analytic rank is tight), then the Mordell-Weil group of $A$ is conditionally determined up to isomorphism,

In cases were one knows a lower bound on the Mordell-Weil group that is one less than a known upper bound on its analytic rank, it is only necessary to assume the parity conjecture.