{"id": "card_floor_bsd", "kind": "note", "title": "The Birch and Swinnerton-Dyer chain - from the group of rational points to the L-function", "body": "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.", "source": {"label": "Narrow Highway - a chain on the one map (operator seed)", "url": "", "ref": "docs/MILLENNIUM_PREPAREDNESS.md", "authority_tier": "engine_derived"}, "shelf": "codex", "box": "floor", "bands": ["floor", "chain", "two trees", "elliptic curves", "L-functions", "heights", "rank"], "subject": "The Birch and Swinnerton-Dyer chain - from the group of rational points to the L-function", "connections": [{"to_card_id": "card_k_floor_of_discovery", "relationship": "part_of", "evidence": "the bsd chain rests on the one Floor of Discovery"}, {"to_card_id": "card_chain_poincare_1901", "relationship": "has_part", "evidence": "a root of this chain - one of the two trees it began from"}, {"to_card_id": "card_chain_hasse_1936", "relationship": "has_part", "evidence": "a root of this chain - one of the two trees it began from"}, {"to_card_id": "card_question_bsd", "relationship": "has_open_end", "evidence": "the open question this chain reaches"}], "author": "engine", "created_at": 0.0, "updated_at": 0.0, "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "generated": false, "call": "codex.floor", "facets": {"subject": ["L-functions", "chain", "elliptic curves", "heights", "rank", "two trees"]}, "presentation": {"glyph": "•", "kind_label": "floor", "by": "Narrow Highway - a chain on the one map (operator seed)", "authority": "engine_derived", "posted": "", "standing": ""}, "neighbors": [{"id": "card_k_floor_of_discovery", "title": "The Floor of Discovery — one floor, and by its design the fear of God", "relationship": "part of", "why": "the bsd chain rests on the one Floor of Discovery", "href": "/card/card_k_floor_of_discovery", "resolved": true}, {"id": "card_chain_poincare_1901", "title": "Poincaré 1901 — Sur les propriétés arithmétiques des courbes algébriques", "relationship": "has part", "why": "a root of this chain - one of the two trees it began from", "href": "/card/card_chain_poincare_1901", "resolved": true}, {"id": "card_chain_hasse_1936", "title": "Hasse 1936 — Zur Theorie der abstrakten elliptischen Funktionenkörper I–III", "relationship": "has part", "why": "a root of this chain - one of the two trees it began from", "href": "/card/card_chain_hasse_1936", "resolved": true}, {"id": "card_question_bsd", "title": "The Birch and Swinnerton-Dyer conjecture", "relationship": "has open end", "why": "the open question this chain reaches", "href": "/card/card_question_bsd", "resolved": true}], "house": {"door": "FIND", "kind": "card", "trail": "connections", "seal": "/card/card_floor_bsd", "next_step": {"do": "follow a connection: what this card rests on, and what rests on it", "door": "FIND", "tool": "card_connections", "params": {"id": "card_floor_bsd"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}