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The Birch and Swinnerton-Dyer chain - from the group of rational points to the L-function
floor
Two trees. The arithmetic: Poincaré's group law on the rational points (1901), Mordell's finite generation (1922), Weil's heights (1929), the canonical height (Néron 1965). The analytic: Hasse's Riemann hypothesis for a curve over a finite field (1936), Weil's conjectures and the local factors (1949), modularity (Wiles 1995). They become one in the conjecture itself (Birch and Swinnerton-Dyer 1965): the rank is the order of vanishing of L(E, s). Then Cassels' pairing on Sha (1962), Coates-Wiles (1977), Gross-Zagier (1986), Kolyvagin (1989) - ranks zero and one are theorems - and a majority of curves (2014). The open end is rank two and above.
- part of → The Floor of Discovery — one floor, and by its design the fear of God — the bsd chain rests on the one Floor of Discovery
- has part → Poincaré 1901 — Sur les propriétés arithmétiques des courbes algébriques — a root of this chain - one of the two trees it began from
- has part → Hasse 1936 — Zur Theorie der abstrakten elliptischen Funktionenkörper I–III — a root of this chain - one of the two trees it began from
- has open end → The Birch and Swinnerton-Dyer conjecture — the open question this chain reaches
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