Birch and Swinnerton-Dyer Conjecture: This conjecture links algebraic geometry and number theory by connecting the arithmetic of elliptic curves to their analytic properties. Specifically, it predicts that the rank of the group of rational points on an elliptic curve (a measure of its infinite solutions) equals the order of vanishing of its associated L-function at s = 1. Elliptic curves are central to modern cryptography (e.g., Bitcoin) and played a key role in proving Fermat’s Last Theorem. The conjecture has been verified for many curves but remains unproven in general. Partial results exist for rank 0 and 1, thanks to work by Gross, Zagier, and Kolyvagin. A full proof would unify disparate areas of math and provide powerful tools for solving Diophantine equations. It is one of the seven Clay Millennium Prize problems, reflecting its depth and significance in arithmetic geometry.
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