{"id": "card_joint_weil_conjectures", "kind": "note", "title": "The Weil conjectures (Deligne 1974): the Riemann hypothesis over finite fields, a theorem", "body": "The Weil conjectures are the analogue of the Riemann hypothesis for the zeta functions of varieties over FINITE FIELDS, and they are a theorem (Deligne 1974): every eigenvalue of Frobenius on the i-th cohomology has absolute value q^(i/2). The Riemann hypothesis itself - zeta over the rationals - stays OPEN, and Deligne's proof has not transferred to it. The proof runs on weights - the finite-field shadow of Hodge theory - and the positivity it establishes is what Mulmuley's geometric complexity theory, the one road to P versus NP not excluded by a barrier, leans on. A proven analogue at the centre, three open questions around it.", "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", "weil conjectures", "finite fields", "weights"], "subject": "The Weil conjectures (Deligne 1974): the Riemann hypothesis over finite fields, a theorem", "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_riemann", "relationship": "connects_at", "evidence": "The Weil conjectures (Deligne 1974): the Riemann hypothesis over finite fields, a theorem. deligne 1974; mulmuley 2011. Cited, not sealed: no arithmetic to recompute"}, {"to_card_id": "card_question_hodge", "relationship": "connects_at", "evidence": "The Weil conjectures (Deligne 1974): the Riemann hypothesis over finite fields, a theorem. deligne 1974; mulmuley 2011. Cited, not sealed: no arithmetic to recompute"}, {"to_card_id": "card_question_p_vs_np", "relationship": "connects_at", "evidence": "The Weil conjectures (Deligne 1974): the Riemann hypothesis over finite fields, a theorem. deligne 1974; mulmuley 2011. Cited, not sealed: no arithmetic to recompute"}, {"to_card_id": "card_src_mill_deligne_1974", "relationship": "cites", "evidence": "the record the joint stands on"}, {"to_card_id": "card_src_mill_mulmuley_2011", "relationship": "cites", "evidence": "the record the joint stands on"}, {"to_card_id": "card_chart_weil_rh", "relationship": "charted_by", "evidence": "the stick seals |alpha|^2 = q for a Frobenius eigenvalue over a finite field - the Riemann hypothesis over finite fields, which is exactly this joint"}], "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": ["finite fields", "floor", "millennium", "two trees", "weights", "weil conjectures"]}, "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_riemann", "title": "The Riemann hypothesis", "relationship": "connects at", "why": "The Weil conjectures (Deligne 1974): the Riemann hypothesis over finite fields, a theorem. deligne 1974; mulmuley 2011. Cited, not sealed: no arithmetic to recompute", "href": "/card/card_question_riemann", "resolved": true}, {"id": "card_question_hodge", "title": "The Hodge conjecture", "relationship": "connects at", "why": "The Weil conjectures (Deligne 1974): the Riemann hypothesis over finite fields, a theorem. deligne 1974; mulmuley 2011. Cited, not sealed: no arithmetic to recompute", "href": "/card/card_question_hodge", "resolved": true}, {"id": "card_question_p_vs_np", "title": "P versus NP", "relationship": "connects at", "why": "The Weil conjectures (Deligne 1974): the Riemann hypothesis over finite fields, a theorem. deligne 1974; mulmuley 2011. Cited, not sealed: no arithmetic to recompute", "href": "/card/card_question_p_vs_np", "resolved": true}, {"id": "card_src_mill_deligne_1974", "title": "P. Deligne 1974 — La conjecture de Weil. I", "relationship": "cites", "why": "the record the joint stands on", "href": "/card/card_src_mill_deligne_1974", "resolved": true}, {"id": "card_src_mill_mulmuley_2011", "title": "K. D. Mulmuley 2011 — On P vs. NP and geometric complexity theory", "relationship": "cites", "why": "the record the joint stands on", "href": "/card/card_src_mill_mulmuley_2011", "resolved": true}, {"id": "card_chart_weil_rh", "title": "The finite-field (Weil) Riemann hypothesis", "relationship": "charted by", "why": "the stick seals |alpha|^2 = q for a Frobenius eigenvalue over a finite field - the Riemann hypothesis over finite fields, which is exactly this joint", "href": "/card/card_chart_weil_rh", "resolved": true}], "house": {"door": "FIND", "kind": "card", "trail": "connections", "seal": "/card/card_joint_weil_conjectures", "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_weil_conjectures"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}