{"id": "card_joint_tate_conjecture", "kind": "note", "title": "The Tate conjecture: Hodge's arithmetic twin, and BSD over function fields", "body": "Tate's conjecture (1965) is the Hodge conjecture with Galois representations in place of Hodge structures. For an elliptic surface over a finite field, the Tate conjecture is equivalent to the Birch and Swinnerton-Dyer conjecture for its generic fibre (Artin-Tate 1966, Milne 1975), and both to the finiteness of the Brauer group. One conjecture, three faces.", "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", "tate conjecture", "algebraic cycles", "brauer group"], "subject": "The Tate conjecture: Hodge's arithmetic twin, and BSD over function fields", "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_hodge", "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_question_bsd", "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_src_mill_tate_1965", "relationship": "cites", "evidence": "the record the joint stands on"}, {"to_card_id": "card_src_mill_artin_tate_1966", "relationship": "cites", "evidence": "the record the joint stands on"}, {"to_card_id": "card_src_mill_milne_1975", "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": ["algebraic cycles", "brauer group", "floor", "millennium", "tate conjecture", "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_hodge", "title": "The Hodge conjecture", "relationship": "connects at", "why": "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", "href": "/card/card_question_hodge", "resolved": true}, {"id": "card_question_bsd", "title": "The Birch and Swinnerton-Dyer conjecture", "relationship": "connects at", "why": "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", "href": "/card/card_question_bsd", "resolved": true}, {"id": "card_src_mill_tate_1965", "title": "J. Tate 1965 — Algebraic cycles and poles of zeta functions", "relationship": "cites", "why": "the record the joint stands on", "href": "/card/card_src_mill_tate_1965", "resolved": true}, {"id": "card_src_mill_artin_tate_1966", "title": "M. Artin 1966 — On the conjectures of Birch and Swinnerton-Dyer and a geometric analog", "relationship": "cites", "why": "the record the joint stands on", "href": "/card/card_src_mill_artin_tate_1966", "resolved": true}, {"id": "card_src_mill_milne_1975", "title": "J. S. Milne 1975 — On a conjecture of Artin and Tate", "relationship": "cites", "why": "the record the joint stands on", "href": "/card/card_src_mill_milne_1975", "resolved": true}], "house": {"door": "FIND", "kind": "card", "trail": "connections", "seal": "/card/card_joint_tate_conjecture", "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_tate_conjecture"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}