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.

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- Last edited by John Cremona on 2020-01-08 04:08:56

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