{"id": "card_joint_l_functions", "kind": "note", "title": "One machinery: Euler product, functional equation, critical line", "body": "Zeta is the simplest L-function; L(E,s) of an elliptic curve is another, given its analytic continuation by modularity (Wiles 1995). Both have an Euler product, a functional equation about a centre, and a critical line; the Grand Riemann Hypothesis names both at once. The engine computes both sides with the same approximate functional equation: xi(s) = xi(1 - s) sealed at s = 0.3, and L(11a1, 1) sealed on the BSD stick.", "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", "L-functions", "modularity", "functional equation"], "subject": "One machinery: Euler product, functional equation, critical line", "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": "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_question_bsd", "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_src_mill_wiles_1995", "relationship": "cites", "evidence": "the record the joint stands on"}, {"to_card_id": "card_chart_langlands", "relationship": "charted_by", "evidence": "the stick seals modularity for 11a1 - the curve side meeting the automorphic-form side - which is the general form of this joint: zeta and L(E,s), one machinery"}], "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": ["L-functions", "floor", "functional equation", "millennium", "modularity", "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_riemann", "title": "The Riemann hypothesis", "relationship": "connects at", "why": "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)", "href": "/card/card_question_riemann", "resolved": true}, {"id": "card_question_bsd", "title": "The Birch and Swinnerton-Dyer conjecture", "relationship": "connects at", "why": "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)", "href": "/card/card_question_bsd", "resolved": true}, {"id": "card_src_mill_wiles_1995", "title": "A. Wiles 1995 — Modular elliptic curves and Fermat's last theorem", "relationship": "cites", "why": "the record the joint stands on", "href": "/card/card_src_mill_wiles_1995", "resolved": true}, {"id": "card_chart_langlands", "title": "The Langlands program", "relationship": "charted by", "why": "the stick seals modularity for 11a1 - the curve side meeting the automorphic-form side - which is the general form of this joint: zeta and L(E,s), one machinery", "href": "/card/card_chart_langlands", "resolved": true}], "house": {"door": "FIND", "kind": "card", "trail": "connections", "seal": "/card/card_joint_l_functions", "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_l_functions"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}