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The Birch and Swinnerton-Dyer conjectural formula

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The Birch and Swinnerton-Dyer conjecture (BSD) asserts the equality of the rank of the Mordell-Weil group of an elliptic curve to the order of vanishing of its L-function at 1. The conjectural formula posits a relationship between the leading coefficient of the L-function at 1 and several arithmetic and analytic constants associated to the curve, including the order of the Tate-Shafarevich group, which is conjectured to be finite. If the analytic rank of a curve over Q is at most 1, then there is an algorithm to compute this order and hence prove the conjecture. The aim of this talk is to describe this algorithm in detail and report on the progress of proving BSD for specific elliptic curves of this type.

This talk is part of the Number Theory Seminar series.

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