{"query": "Birch 1965 — Notes on elliptic curves. II", "count": 20, "results": [{"id": "card_chain_birch_swinnerton_dyer_1965", "title": "Birch 1965 — Notes on elliptic curves. II", "shelf": "codex", "surface": "secular", "snippet": "B. J. Birch, H. P. F. Swinnerton-Dyer (1965). Notes on elliptic curves. II. J. reine angew. Math. 218 (1965) 79–108. DOI 10.1515/crll.1965.218.79. Canonical: https://doi.org/10.1515/crll.1965.218.79. ", "authority_tier": "reference", "source": "B. J. Birch, H. P. F. Swinnerton-Dyer (1965), J. reine angew. Math. 218 (1965) 79–108", "readable": false, "generated": false}, {"id": "card_theory_non_euclidean_hyperbolic_elliptic_geometry", "title": "Non-Euclidean (hyperbolic / elliptic) geometry", "shelf": "theories", "surface": "secular", "snippet": "Non-Euclidean (hyperbolic / elliptic) geometry — an engine domain that can touch it: geometry. Calibration: partial — curvature relations verify; the choice of axioms is map-only. Deny Euclid's fifth ", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_algebraic_geometry", "title": "Algebraic geometry (solution sets as shapes)", "shelf": "theories", "surface": "secular", "snippet": "Algebraic geometry (solution sets as shapes) — an engine domain that can touch it: mathematics. Calibration: map-only — specific curve computations verify; the theory is abstract. The study of the sha", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_joint_tate_conjecture", "title": "The Tate conjecture: Hodge's arithmetic twin, and BSD over function fields", "shelf": "codex", "surface": "secular", "snippet": "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 a", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "readable": false, "generated": false}, {"id": "card_src_mill_milne_1975", "title": "J. S. Milne 1975 — On a conjecture of Artin and Tate", "shelf": "millennium", "surface": "secular", "snippet": "J. S. Milne (1975). On a conjecture of Artin and Tate. Ann. Math. 102 (1975) 517–533. DOI 10.2307/1971042. Canonical: https://doi.org/10.2307/1971042. No free copy found; cited by its record. License ", "authority_tier": "reference", "source": "J. S. Milne (1975), Ann. Math. 102 (1975) 517–533", "readable": false, "generated": false}, {"id": "card_chain_bhargava_skinner_zhang_2014", "title": "Bhargava 2014 — A majority of elliptic curves over Q satisfy the Birch and Swinnerton-Dyer conjecture", "shelf": "codex", "surface": "secular", "snippet": "M. Bhargava, C. Skinner, W. Zhang (2014). A majority of elliptic curves over Q satisfy the Birch and Swinnerton-Dyer conjecture. arXiv (2014). Canonical: https://arxiv.org/abs/1407.1826. Free copy: ht", "authority_tier": "reference", "source": "M. Bhargava, C. Skinner, W. Zhang (2014), arXiv (2014)", "readable": false, "generated": false}, {"id": "card_src_mill_silverman_1988", "title": "J. H. Silverman 1988 — Computing heights on elliptic curves", "shelf": "millennium", "surface": "secular", "snippet": "J. H. Silverman (1988). Computing heights on elliptic curves. Math. Comp. 51 (1988) 339–358. DOI 10.1090/S0025-5718-1988-0942161-4. Canonical: https://doi.org/10.1090/S0025-5718-1988-0942161-4. Free c", "authority_tier": "reference", "source": "J. H. Silverman (1988), Math. Comp. 51 (1988) 339–358", "readable": false, "generated": false}, {"id": "card_question_bsd", "title": "The Birch and Swinnerton-Dyer conjecture", "shelf": "millennium", "surface": "secular", "snippet": "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 ", "authority_tier": "reference", "source": "Clay Mathematics Institute, the Millennium Prize Problems (2000)", "readable": false, "generated": false}, {"id": "card_floor_bsd", "title": "The Birch and Swinnerton-Dyer chain - from the group of rational points to the L-function", "shelf": "codex", "surface": "secular", "snippet": "Two trees. The arithmetic: Poincaré's group law on the rational points (1901), Mordell's finite generation (1922), Weil's heights (1929), the canonical height (Néron 1965). The analytic: Hasse's Riema", "authority_tier": "engine_derived", "source": "Narrow Highway - a chain on the one map (operator seed)", "readable": false, "generated": false}, {"id": "card_src_mill_cremona_1997_book", "title": "J. E. Cremona 1997 — Algorithms for Modular Elliptic Curves (2nd ed.; chapter 3, the canonical height; Table 1)", "shelf": "millennium", "surface": "secular", "snippet": "J. E. Cremona (1997). Algorithms for Modular Elliptic Curves (2nd ed.; chapter 3, the canonical height; Table 1). Cambridge University Press, 1997; online edition with corrections. Canonical: https://", "authority_tier": "reference", "source": "J. E. Cremona (1997), Cambridge University Press, 1997; online edition with corrections", "readable": false, "generated": false}, {"id": "card_src_mill_cremona_ecdata", "title": "J. E. Cremona 1997 — ecdata — elliptic curves over Q (the allbsd files: conductor, isogeny class, number, a-invariants, rank, |T|, prod c_p, real period Omega, L^(r)(E,1)/r!, regul", "shelf": "millennium", "surface": "secular", "snippet": "J. E. Cremona (1997-2026). ecdata — elliptic curves over Q (the allbsd files: conductor, isogeny class, number, a-invariants, rank, |T|, prod c_p, real period Omega, L^(r)(E,1)/r!, regulator, |Sha|). ", "authority_tier": "reference", "source": "J. E. Cremona (1997-2026), github.com/JohnCremona/ecdata (conductor <= 500000)", "readable": false, "generated": false}, {"id": "card_joint_l_functions", "title": "One machinery: Euler product, functional equation, critical line", "shelf": "codex", "surface": "secular", "snippet": "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, a", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "readable": false, "generated": false}, {"id": "card_instr_elliptic_curves", "title": "Elliptic curves", "shelf": "codex", "surface": "secular", "snippet": "The Elliptic curves verifier (src/concordance/verifiers/elliptic_curves.py). On the one map it instruments: card_question_bsd. Found from the code's own imports, never invented; the module keeps the l", "authority_tier": "engine_derived", "source": "verifier: elliptic_curves", "readable": false, "generated": false}, {"id": "card_src_rfc_5639", "title": "RFC5639 — Elliptic Curve Cryptography (ECC) Brainpool Standard Curves and Curve Generation", "shelf": "rfcs", "surface": "secular", "snippet": "RFC5639: Elliptic Curve Cryptography (ECC) Brainpool Standard Curves and Curve Generation (March 2010). Status: INFORMATIONAL.", "authority_tier": "reference", "source": "The RFC Index (RFC Editor / IETF) — public domain", "readable": false, "generated": false}, {"id": "card_seal_ac34fe1ca064aacfe486e872990e82ceda80e1b2d995d9f6f4c5cb7d83bd1557", "title": "Receipt ac34fe1ca064… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": null,\n    \"confirmed_steps\": 1,\n    \"error_at\": null,\n    \"gap_at\": null,\n    \"steps\": 1,\n    \"verdict\": \"HOLDS\"\n   },\n   \"gat", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_src_rfc_7027", "title": "RFC7027 — Elliptic Curve Cryptography (ECC) Brainpool Curves for Transport Layer Security (TLS)", "shelf": "rfcs", "surface": "secular", "snippet": "RFC7027: Elliptic Curve Cryptography (ECC) Brainpool Curves for Transport Layer Security (TLS) (October 2013). Status: INFORMATIONAL.", "authority_tier": "reference", "source": "The RFC Index (RFC Editor / IETF) — public domain", "readable": false, "generated": false}, {"id": "card_src_rfc_8734", "title": "RFC8734 — Elliptic Curve Cryptography (ECC) Brainpool Curves for Transport Layer Security (TLS) Version 1.3", "shelf": "rfcs", "surface": "secular", "snippet": "RFC8734: Elliptic Curve Cryptography (ECC) Brainpool Curves for Transport Layer Security (TLS) Version 1.3 (February 2020). Status: INFORMATIONAL.", "authority_tier": "reference", "source": "The RFC Index (RFC Editor / IETF) — public domain", "readable": false, "generated": false}, {"id": "card_src_rfc_6954", "title": "RFC6954 — Using the Elliptic Curve Cryptography (ECC) Brainpool Curves for the Internet Key Exchange Protocol Version 2 (IKEv2)", "shelf": "rfcs", "surface": "secular", "snippet": "RFC6954: Using the Elliptic Curve Cryptography (ECC) Brainpool Curves for the Internet Key Exchange Protocol Version 2 (IKEv2) (July 2013). Status: INFORMATIONAL.", "authority_tier": "reference", "source": "The RFC Index (RFC Editor / IETF) — public domain", "readable": false, "generated": false}, {"id": "card_src_book_31076", "title": "Elliptic Functions: An Elementary Text-Book for Students of Mathematics — Arthur Latham Baker", "shelf": "gutenberg", "surface": "secular", "snippet": "Elliptic Functions: An Elementary Text-Book for Students of Mathematics, by Arthur Latham Baker. Subjects: Elliptic functions. Read the full text (public domain): https://www.gutenberg.org/ebooks/3107", "authority_tier": "reference", "source": "Project Gutenberg (public domain)", "readable": true, "generated": false}, {"id": "card_seal_63611e99d8594a2ecf3e7e25cea892db65ee0127d84ee12a360f5e75756613f7", "title": "Receipt 63611e99d859… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": null,\n    \"confirmed_steps\": 1,\n    \"error_at\": null,\n    \"gap_at\": null,\n    \"steps\": 1,\n    \"verdict\": \"HOLDS\"\n   },\n   \"gat", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}], "house": {"door": "FIND", "kind": "cards", "trail": "results", "seal": null, "next_step": {"do": "open the top card", "door": "FIND", "tool": "card_get", "params": {"id": "card_chain_birch_swinnerton_dyer_1965"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}