{"id": "card_question_bsd", "kind": "reference", "title": "The Birch and Swinnerton-Dyer conjecture", "body": "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": {"label": "Clay Mathematics Institute, the Millennium Prize Problems (2000)", "url": "https://www.claymath.org/millennium-problems/", "ref": "stick_birch_and_swinnerton_dyer_conjecture", "domain": "mathematics", "authority_tier": "reference"}, "shelf": "millennium", "box": "question", "bands": ["millennium", "open question", "clay", "bsd", "elliptic curves", "L-functions", "number theory"], "subject": "The Birch and Swinnerton-Dyer conjecture", "connections": [{"to_card_id": "card_spine_millennium_sources", "relationship": "member_of", "evidence": "one of the seven questions the Millennium shelf is about"}, {"to_card_id": "card_floor_millennium", "relationship": "open_end_of", "evidence": "an open question hanging off the floor of what is proven"}, {"to_card_id": "card_joint_l_functions", "relationship": "connects_at", "evidence": "One machinery: Euler product, functional equation, critical line. wiles 1995. Sealed: xi_functional_equation_at_s_0_3 (Riemann stick) and the 11a1 instance (BSD stick)"}, {"to_card_id": "card_joint_tate_conjecture", "relationship": "connects_at", "evidence": "The Tate conjecture: Hodge's arithmetic twin, and BSD over function fields. tate 1965; artin tate 1966; milne 1975. Cited, not sealed: no arithmetic to recompute"}, {"to_card_id": "card_joint_manin_algorithm", "relationship": "connects_at", "evidence": "If Sha is finite, the rank is computable. manin 1971. Cited, not sealed: no arithmetic to recompute"}, {"to_card_id": "card_floor_logarithm", "relationship": "open_end_of", "evidence": "the open question this chain reaches"}, {"to_card_id": "card_chain_neron_1965", "relationship": "connects_at", "evidence": "where the logarithm enters: the height of a rational point is the logarithm of its size; the regulator sealed on 37a1 is a determinant of those logs"}, {"to_card_id": "card_chain_kolyvagin_1989", "relationship": "builds_on", "evidence": "a later work standing on an earlier one"}, {"to_card_id": "card_chain_bhargava_skinner_zhang_2014", "relationship": "builds_on", "evidence": "a later work standing on an earlier one"}, {"to_card_id": "card_chain_cassels_1962", "relationship": "builds_on", "evidence": "a later work standing on an earlier one"}, {"to_card_id": "card_floor_bsd", "relationship": "open_end_of", "evidence": "the open question this chain reaches"}, {"to_card_id": "card_instr_elliptic_curves", "relationship": "connects_at", "evidence": "instruments the map: elliptic_curves computes L(E,1), the root number and the analytic rank, and the full BSD formula - the BSD stick's instances are its verdicts, sealed"}], "author": "engine", "created_at": 0.0, "updated_at": 0.0, "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "generated": false, "call": "millennium.question", "facets": {"subject": ["L-functions", "bsd", "clay", "elliptic curves", "millennium", "number theory", "open question"]}, "presentation": {"glyph": "?", "kind_label": "A question", "by": "Clay Mathematics Institute, the Millennium Prize Problems (2000)", "authority": "reference", "posted": "", "standing": "", "link": {"url": "https://www.claymath.org/millennium-problems/", "host": "claymath.org", "provider": "", "titled": "The Birch and Swinnerton-Dyer conjecture", "waybill_line": "", "reach": "", "embed": ""}}, "neighbors": [{"id": "card_spine_millennium_sources", "title": "The Millennium sources — the papers behind the seven sticks", "relationship": "on the shelf of", "why": "one of the seven questions the Millennium shelf is about", "href": "/card/card_spine_millennium_sources", "resolved": true}, {"id": "card_floor_millennium", "title": "The Millennium floor - seven open questions, and where they connect", "relationship": "open end of", "why": "an open question hanging off the floor of what is proven", "href": "/card/card_floor_millennium", "resolved": true}, {"id": "card_joint_l_functions", "title": "One machinery: Euler product, functional equation, critical line", "relationship": "connects at", "why": "One machinery: Euler product, functional equation, critical line. wiles 1995. 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Cited, not sealed: no arithmetic to recompute", "href": "/card/card_joint_manin_algorithm", "resolved": true}, {"id": "card_floor_logarithm", "title": "The logarithm - the instrument the joints share", "relationship": "open end of", "why": "the open question this chain reaches", "href": "/card/card_floor_logarithm", "resolved": true}, {"id": "card_chain_neron_1965", "title": "Néron 1965 — Quasi-fonctions et hauteurs sur les variétés abéliennes", "relationship": "connects at", "why": "where the logarithm enters: the height of a rational point is the logarithm of its size; the regulator sealed on 37a1 is a determinant of those logs", "href": "/card/card_chain_neron_1965", "resolved": true}, {"id": "card_chain_kolyvagin_1989", "title": "Kolyvagin 1989 — Finiteness of E(Q) and Sha(E, Q) for a subclass of Weil curves", "relationship": "builds on", "why": "a later work standing on an earlier one", "href": "/card/card_chain_kolyvagin_1989", "resolved": true}], "house": {"door": "FIND", "kind": "card", "trail": "connections", "seal": "/card/card_question_bsd", "next_step": {"do": "read the source itself", "door": "FIND", "web": "/reader.html?card=card_question_bsd"}, "ends": "a verdict or a card · the trail · a seal · one next step"}}