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Let $E$ be an elliptic curve over $\mathbb{Q}$ with algebraic rank 0. Is there any way, one can argue that the analytic rank must be even? Of course this would follow from the standard conjectures, such as BSD, Sha is finite, and the parity conjectures, but are there any unconditional results?

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No . – user30035 Mar 15 '13 at 23:16
as long as it's known to be at most two! – Will Sawin Mar 16 '13 at 3:22
In general no. If you know that the p-Selmer rank is 0 for some p, then yes. – Tim Dokchitser Mar 16 '13 at 14:55
@Will Sawin and Tim Dokchitser: "as long as you have a condition, there is an unconditional result" ;-) Tim: are you not tempted to give a reference? ;-) – user30035 Mar 16 '13 at 17:51

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