The celebrated BSD conjecture predicts a deep and mysterious relationship between the algebraic rank of an elliptic curve defined over a number field and an arithmetic invariant arising from the L-function of the elliptic curve. More amazingly, it predicts an exact formula for the leading term of the L-function. Iwasawa theory has proven to be an effective tool in trying to explain the philosophy behind such a mysterious relationship and also in establishing the conjecture in certain cases. We shall outline this theory and discuss its applications.

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Last updated: 01 Oct 2023