{"id": "card_joint_manin_algorithm", "kind": "note", "title": "If Sha is finite, the rank is computable", "body": "Manin (1971): if the Tate-Shafarevich group is finite - part of what BSD asserts - then the rank of an elliptic curve over Q is effectively computable by descent. Without it, no algorithm is known. A conjecture in arithmetic that would decide a question in computation.", "source": {"label": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "url": "", "domain": "mathematics", "authority_tier": "engine_derived"}, "shelf": "codex", "box": "joint", "bands": ["joint", "two trees", "millennium", "floor", "manin", "descent", "computability"], "subject": "If Sha is finite, the rank is computable", "connections": [{"to_card_id": "card_floor_millennium", "relationship": "part_of", "evidence": "a joint that is proven or observed - a part of the floor"}, {"to_card_id": "card_question_bsd", "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_question_p_vs_np", "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_src_mill_manin_1971", "relationship": "cites", "evidence": "the record the joint stands on"}], "author": "engine", "created_at": 0.0, "updated_at": 0.0, "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "generated": false, "call": "codex.joint", "facets": {"subject": ["computability", "descent", "floor", "manin", "millennium", "two trees"]}, "presentation": {"glyph": "•", "kind_label": "joint", "by": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "authority": "engine_derived", "posted": "", "standing": ""}, "neighbors": [{"id": "card_floor_millennium", "title": "The Millennium floor - seven open questions, and where they connect", "relationship": "part of", "why": "a joint that is proven or observed - a part of the floor", "href": "/card/card_floor_millennium", "resolved": true}, {"id": "card_question_bsd", "title": "The Birch and Swinnerton-Dyer conjecture", "relationship": "connects at", "why": "If Sha is finite, the rank is computable. manin 1971. Cited, not sealed: no arithmetic to recompute", "href": "/card/card_question_bsd", "resolved": true}, {"id": "card_question_p_vs_np", "title": "P versus NP", "relationship": "connects at", "why": "If Sha is finite, the rank is computable. manin 1971. Cited, not sealed: no arithmetic to recompute", "href": "/card/card_question_p_vs_np", "resolved": true}, {"id": "card_src_mill_manin_1971", "title": "Yu. I. Manin 1971 — Cyclotomic fields and modular curves", "relationship": "cites", "why": "the record the joint stands on", "href": "/card/card_src_mill_manin_1971", "resolved": true}], "house": {"door": "FIND", "kind": "card", "trail": "connections", "seal": "/card/card_joint_manin_algorithm", "next_step": {"do": "follow a connection: what this card rests on, and what rests on it", "door": "FIND", "tool": "card_connections", "params": {"id": "card_joint_manin_algorithm"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}