{"query": "The Birch and Swinnerton-Dyer conjecture", "count": 15, "results": [{"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_spine_millennium_sources", "title": "The Millennium sources — the papers behind the seven sticks", "shelf": "spine", "surface": "secular", "snippet": "63 sources located for the seven Millennium sticks (Riemann, Birch and Swinnerton-Dyer, Navier-Stokes, Yang-Mills, P versus NP, Hodge, Poincare), one reference card each: the bibliographic record, DOI", "authority_tier": "reference", "source": "The Millennium sources — a spine of located records", "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_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_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_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_chain_coates_wiles_1977", "title": "Coates 1977 — On the conjecture of Birch and Swinnerton-Dyer", "shelf": "codex", "surface": "secular", "snippet": "J. Coates, A. Wiles (1977). On the conjecture of Birch and Swinnerton-Dyer. Invent. Math. 39 (1977) 223–251. DOI 10.1007/BF01402975. Canonical: https://doi.org/10.1007/BF01402975. Cited by its record.", "authority_tier": "reference", "source": "J. Coates, A. Wiles (1977), Invent. Math. 39 (1977) 223–251", "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_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_artin_tate_1966", "title": "M. Artin 1966 — On the conjectures of Birch and Swinnerton-Dyer and a geometric analog", "shelf": "millennium", "surface": "secular", "snippet": "M. Artin, J. Tate (1966). On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. Séminaire Bourbaki 9 (1964–66), exp. 306, 415–440. Canonical: http://www.numdam.org/item/SB_1964-1966_", "authority_tier": "reference", "source": "M. Artin, J. Tate (1966), Séminaire Bourbaki 9 (1964–66), exp. 306, 415–440", "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_book_72789", "title": "R. L. Stevenson — Frank Swinnerton", "shelf": "gutenberg", "surface": "secular", "snippet": "R. L. Stevenson, by Frank Swinnerton. Subjects: Stevenson, Robert Louis, 1850-1894 -- Criticism and interpretation. Read the full text (public domain): https://www.gutenberg.org/ebooks/72789", "authority_tier": "reference", "source": "Project Gutenberg (public domain)", "readable": true, "generated": false}, {"id": "card_src_book_31005", "title": "Coquette — Frank Swinnerton", "shelf": "gutenberg", "surface": "secular", "snippet": "Coquette, by Frank Swinnerton. Subjects: Young women -- Fiction; Love stories; Dressmakers -- Fiction. Read the full text (public domain): https://www.gutenberg.org/ebooks/31005", "authority_tier": "reference", "source": "Project Gutenberg (public domain)", "readable": true, "generated": false}, {"id": "card_src_book_15177", "title": "Nocturne — Frank Swinnerton", "shelf": "gutenberg", "surface": "secular", "snippet": "Nocturne, by Frank Swinnerton. Subjects: London (England) -- Fiction; Sisters -- Fiction; Fathers and daughters -- Fiction; Man-woman relationships -- Fiction. Read the full text (public domain): http", "authority_tier": "reference", "source": "Project Gutenberg (public domain)", "readable": true, "generated": false}, {"id": "card_src_book_63479", "title": "The Three Lovers — Frank Swinnerton", "shelf": "gutenberg", "surface": "secular", "snippet": "The Three Lovers, by Frank Swinnerton. Subjects: Orphans -- Fiction; Young women -- Fiction; Man-woman relationships -- Fiction; Kensington (London, England) -- Fiction; Bohemianism -- Fiction. Read t", "authority_tier": "reference", "source": "Project Gutenberg (public domain)", "readable": true, "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_question_bsd"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}