Birch and Swinnerton-Dyer conjecture

The Birch and Swinnerton-Dyer conjecture predicts that an elliptic curve’s algebraic rank equals the order of vanishing of its L-function at 1. Its refined form relates the leading coefficient there to arithmetic invariants of the curve.

Elliptic curve: A smooth projective curve of genus one equipped with a specified point. The conjecture is formulated for elliptic curves over the rational numbers.
Hasse–Weil L-function: The L-function encoding point-counting data of an algebraic variety over finite fields. For an elliptic curve, this is the L-function whose behavior at 1 BSD studies.
Bryan Birch: A British mathematician who helped formulate the Birch and Swinnerton-Dyer conjecture. Birch developed the conjectural link between point growth and L-function behavior.
Modularity theorem: The theorem that every elliptic curve over the rational numbers arises from a modular form. Modularity makes the curve’s L-function analytic and enables rigorous study at 1.
Selmer group: A group defined by local conditions that approximates rational points modulo a fixed integer. Selmer groups provide computable bounds on rank and expose obstacles to proving BSD.
Mordell–Weil theorem: The theorem that an abelian variety over a number field has a finitely generated group of rational points. It makes the curve’s rational points a finitely generated group whose rank the conjecture predicts.
Euler product: A product over primes that expresses a function through local factors. The elliptic-curve L-function is assembled from local factors indexed by primes.
Peter Swinnerton-Dyer: A British mathematician known for work in number theory and the Birch and Swinnerton-Dyer conjecture. Swinnerton-Dyer collaborated with Birch in formulating the conjecture.
Gross–Zagier formula: A formula relating derivatives of certain L-functions to heights of Heegner points. It connects a first-order L-function zero to a non-torsion rational point.
Descent (number theory): A method that studies rational points by mapping them into auxiliary groups with local conditions. Descent can bound rank, but often cannot determine it exactly.

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