Birch and Swinnerton-Dyer conjecture

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In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. It is named after mathematicians Bryan John Birch and Sir Peter Swinnerton-Dyer, who formulated the conjecture in the 1960s with the help of machine computation. Only special cases of the conjecture have been proven. The conjecture proposes a link between arithmetic data associated with an elliptic curve E {\displaystyle E} over a number field K {\displaystyle K} and the behaviour of its associated Hasse–Weil L-function L ( E , s ) {\displaystyle L(E,s)} at s = 1...

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