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The Birch and Swinnerton-Dyer conjecture
A question
The Birch and Swinnerton-Dyer conjecture. The rank of the group of rational points of an elliptic curve over Q equals the order of vanishing of its L-function at s = 1, and the leading coefficient is the product of its arithmetic invariants. Its chart is the tick stick stick_birch_and_swinnerton_dyer_conjecture: what is sealed, what is cited and what stays open are read live at /stick?id=stick_birch_and_swinnerton_dyer_conjecture. Open for rank two and above; rank zero and one are theorems (Kolyvagin, Gross-Zagier).
source
card id
card_question_bsd
address
SCI.millennium.FCT/the-birch-and-swinnerton-dyer-conjecture/REF.WITNESSED@clay-mathematics-institu
adjoining cards
- on the shelf of → The Millennium sources — the papers behind the seven sticks — one of the seven questions the Millennium shelf is about
- open end of → The Millennium floor - seven open questions, and where they connect — an open question hanging off the floor of what is proven
- connects at → One machinery: Euler product, functional equation, critical line — One machinery: Euler product, functional equation, critical line. wiles 1995. Sealed: xi_f
- connects at → The Tate conjecture: Hodge's arithmetic twin, and BSD over function fields — The Tate conjecture: Hodge's arithmetic twin, and BSD over function fields. tate 1965; art
- connects at → If Sha is finite, the rank is computable — If Sha is finite, the rank is computable. manin 1971. Cited, not sealed: no arithmetic to
- open end of → The logarithm - the instrument the joints share — the open question this chain reaches
- connects at → Néron 1965 — Quasi-fonctions et hauteurs sur les variétés abéliennes — where the logarithm enters: the height of a rational point is the logarithm of its size; t
- builds on → Kolyvagin 1989 — Finiteness of E(Q) and Sha(E, Q) for a subclass of Weil curves — a later work standing on an earlier one
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