Nagell–Lutz theorem

In mathematics, the Nagell–Lutz theorem is a result in the diophantine geometry of elliptic curves, which describes rational torsion points on elliptic curves over the integers. It is named for Trygve Nagell and Élisabeth Lutz.

Definition of the terms

Suppose that the equation

y^{2}=x^{3}+ax^{2}+bx+c

defines a non-singular cubic curve with integer coefficients a, b, c, and let D be the discriminant of the cubic polynomial on the right side:

D=-4a^{3}c+a^{2}b^{2}+18abc-4b^{3}-27c^{2}.

Statement of the theorem

If P = (x,y) is a rational point of finite order on C, for the elliptic curve group law, then:

1) x and y are integers
2) either y = 0, in which case P has order two, or else y divides D, which immediately implies that y² divides D.

Generalizations

The Nagell–Lutz theorem generalizes to arbitrary number fields and more general cubic equations.^[1] For curves over the rationals, the generalization says that, for a nonsingular cubic curve whose Weierstrass form

y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}

has integer coefficients, any rational point P=(x,y) of finite order must have integer coordinates, or else have order 2 and coordinates of the form x=m/4, y=n/8, for m and n integers.

History

The result is named for its two independent discoverers, the Norwegian Trygve Nagell (1895–1988) who published it in 1935, and Élisabeth Lutz (1937).

References

^ See, for example, Theorem VIII.7.1 of Joseph H. Silverman (1986), "The arithmetic of elliptic curves", Springer, ISBN 0-387-96203-4.

Élisabeth Lutz (1937). "Sur l'équation y² = x³ − Ax − B dans les corps p-adiques". J. Reine Angew. Math. 177: 237–247.
Joseph H. Silverman, John Tate (1994), "Rational Points on Elliptic Curves", Springer, ISBN 0-387-97825-9.

Topics in algebraic curves

Rational curves

Elliptic curves

Analytic theory	Elliptic function Elliptic integral Fundamental pair of periods Modular form
Arithmetic theory	Counting points on elliptic curves Division polynomials Hasse's theorem on elliptic curves Mazur's torsion theorem Modular elliptic curve Modularity theorem Mordell–Weil theorem Nagell–Lutz theorem Supersingular elliptic curve Schoof's algorithm Schoof–Elkies–Atkin algorithm
Applications	Elliptic curve cryptography Elliptic curve primality

Higher genus

Plane curves

Riemann surfaces

Constructions

Structure of curves

Divisors on curves	Abel–Jacobi map Brill–Noether theory Clifford's theorem on special divisors Gonality of an algebraic curve Jacobian variety Riemann–Roch theorem Weierstrass point Weil reciprocity law
Moduli	ELSV formula Gromov–Witten invariant Hodge bundle Moduli of algebraic curves Stable curve
Morphisms	Hasse–Witt matrix Riemann–Hurwitz formula Prym variety Weber's theorem (Algebraic curves)
Singularities	A_k singularity Acnode Crunode Cusp Delta invariant Tacnode
Vector bundles	Birkhoff–Grothendieck theorem Stable vector bundle Vector bundles on algebraic curves

Read other articles:

Sinagoge Musmeah Yeshua

Sinagoge Musmeah YeshuaAgamaAfiliasiYahudi OrtodoksRitusEdot HamizrachKepemimpinanMoses Samuels (lahir tahun 1940; wafat 29 Mei 2015)[1] Sammy Samuels (2015) [2]Diberkati1896LokasiLokasi85, Jalan 26 Street Yangon, MyanmarKoordinat16°46′27″N 96°09′14″E / 16.774255216139117°N 96.15391813846333°E / 16.774255216139117; 96.15391813846333Koordinat: 16°46′27″N 96°09′14″E / 16.774255216139117°N 96.15391813846333°E...

Artyom Fomichev

Artikel ini sebatang kara, artinya tidak ada artikel lain yang memiliki pranala balik ke halaman ini.Bantulah menambah pranala ke artikel ini dari artikel yang berhubungan atau coba peralatan pencari pranala.Tag ini diberikan pada November 2022. Artyom FomichevInformasi pribadiNama lengkap Artyom Valeryevich FomichevTanggal lahir 25 Maret 1993 (umur 30)Tinggi 1,86 m (6 ft 1 in)Posisi bermain BekInformasi klubKlub saat ini FC DolgoprudnyKarier senior*Tahun Tim Tampil (Gol)2...

ألبيون

لمعانٍ أخرى، طالع ألبيون (توضيح). ألبيون الإحداثيات 42°06′42″N 92°59′27″W / 42.111666666667°N 92.990833333333°W / 42.111666666667; -92.990833333333 [1] تقسيم إداري البلد الولايات المتحدة[2] التقسيم الأعلى مقاطعة مارشال خصائص جغرافية المساحة 1.541417 كيلومتر مربع1.5414...

Unione Calcio Sampdoria 1950-1951

Voce principale: Unione Calcio Sampdoria. Unione Calcio SampdoriaStagione 1950-1951La Sampdoria inaugura lo stadio Faraggiana contro la formazione dell'Albissola Sport calcio Squadra Sampdoria Allenatore Giuseppe Galluzzi (fino al 1º aprile 1951) Gipo Poggi (dal 2 aprile 1951) Presidente Aldo Parodi Serie A12º posto Maggiori presenzeCampionato: Gei (38) Miglior marcatoreCampionato: Gei (12) 1949-1950 1951-1952 Si invita a seguire il modello di voce Questa voce raccoglie le informazion...

West Kilbride

Village in North Ayrshire, Scotland Human settlement in ScotlandWest KilbrideWest Kilbride from Law HillWest KilbrideLocation within North AyrshirePopulation4,860 (mid-2020 est.)[1]OS grid referenceNS205485Council areaNorth AyrshireLieutenancy areaAyrshire and ArranCountryScotlandSovereign stateUnited KingdomPost townWEST KILBRIDEPostcode districtKA23Dialling code01294PoliceScotlandFireScottishAmbulanceScottish UK ParliamentNorth Ayrsh...

Christa Williams

Pour les articles homonymes, voir Williams. Christa Williams Données clés Nom de naissance Christa Bojarzin Naissance 5 février 1926Königsberg, Empire allemand Décès 28 juillet 2012 (à 86 ans)Munich, Allemagne Activité principale Chanteuse Activités annexes Actrice Genre musical Schlager Années actives 1956-1968 Labels Decca Records, Ariola Influences Ella Fitzgerald modifier Christa Williams (de son vrai nom Christa Bojarzin, née le 5 février 1926 à Königsberg, morte le 2...

Vectrex

Not to be confused with Vectrix or Vectrix Corporation. Vector display-based home video game console VectrexA Vectrex and its controllerDeveloperSmith EngineeringManufacturerGeneral Consumer Electronics (1982–83)Milton Bradley Company (1983–84)TypeHome video game consoleGenerationSecond generationRelease dateNA: November 1982EU: 1983JP: 1983Introductory priceUS$199 (equivalent to $630 in 2023)DiscontinuedFebruary 1984MediaROM cartridgeCPUMotorola MC68A09 @ 1.5 MHzMemory1 KBD...

Kau Masih Kekasihku

Kau Masih KekasihkuGenre Drama Roman PembuatSinemArtBerdasarkanAt Dolphin BaySutradaraSridhar JettyPemeran Arifin Putra Asmirandah Baim Wong Slamet Rahardjo Marini Burhan Penggubah lagu temaNaffLagu pembukaKau Masih Kekasihku oleh NaffLagu penutupKau Masih Kekasihku oleh NaffPenata musikPongky PrasetyoNegara asalIndonesiaBahasa asliBahasa IndonesiaJmlh. musim1Jmlh. episode29ProduksiProduserLeo SusantoPengaturan kameraMulti-kameraDurasi60 menitRumah produksiSinemArtDistributorSurya Citr...

國際調查記者同盟

此條目可参照英語維基百科相應條目来扩充。 (2021年10月13日)若您熟悉来源语言和主题，请协助参考外语维基百科扩充条目。请勿直接提交机械翻译，也不要翻译不可靠、低品质内容。依版权协议，译文需在编辑摘要注明来源，或于讨论页顶部标记{{Translated page}}标签。国际调查记者同盟International Consortium of Investigative Journalists成立時間1997年總部华盛顿哥伦比亚特区地址�...

2016年美國總統選舉

2016年美國總統選舉 ← 2012 2016年11月8日 2020 → 538個選舉人團席位獲勝需270票民意調查投票率55.7%[1][2] ▲ 0.8 % 获提名人唐納·川普希拉莉·克林頓政党共和黨民主党家鄉州紐約州紐約州竞选搭档迈克·彭斯蒂姆·凱恩选举人票 304[3][4][註 1] 227[5] 胜出州/省 30 + 緬-2 20 + DC 民選得票 62,984,828[6] 65,853,514[6]...

Wisconsin Public Radio

US state radio network Wisconsin Public RadioTypePublic radio networkCountryUnited StatesBroadcast areaWisconsin, eastern Minnesota, northeastern Iowa, northern Illinois, Chicagoland, southern Upper Peninsula (Michigan)HeadquartersMadison, WisconsinProgrammingAffiliationsNational Public Radio, American Public MediaOwnershipOwnerWisconsin Educational Communications Board & University of Wisconsin System, Board of RegentsSister stationsPBS WisconsinHistoryLaunch date1948LinksWebcastIdeas Ne...

2020年夏季奧林匹克運動會波蘭代表團

2020年夏季奥林匹克运动会波兰代表團波兰国旗IOC編碼POLNOC波蘭奧林匹克委員會網站olimpijski.pl（英文）（波兰文）2020年夏季奥林匹克运动会（東京）2021年7月23日至8月8日（受2019冠状病毒病疫情影响推迟，但仍保留原定名称）運動員206參賽項目24个大项旗手开幕式：帕维尔·科热尼奥夫斯基（游泳）和马娅·沃什乔夫斯卡（自行车）[1]闭幕式：卡罗利娜·纳亚（皮划艇）&#...

صدقة

لمعانٍ أخرى، طالع صدقة (توضيح). فتاة صغيرة تتصدق على عجوز فقيرة. لوحة للرسام الهندي رافي فارما. الصدقة مفهوم ديني، وهي العطية للمحتاج على وجه التقرب إلى الخالق والمعبود، ينظر المؤمنون إليها بإعجاب، ولاسيما عندما تمارس بشكل سري من قبل الغرباء «الذين يخافون الله». وللص...

Bryce Pinkham

American stage and screen actor (born 1982) Bryce PinkhamPinkham in 2016BornBryce Allen Pinkham (1982-10-19) October 19, 1982 (age 41)Redding, California, U.S.EducationBoston College (BA)Yale University (MFA)Occupation(s)Actor, singerYears active2010–presentMusical careerGenres Musical theatre Crossover Pop Instrument(s)Vocals Musical artist Bryce Allen Pinkham (born October 19, 1982) is an American actor and singer. He has appeared in the PBS period drama Mercy Street. On Broadwa...

Bromine

Chemical element, symbol Br and atomic number 35Bromine, 35BrLiquid and gas bromine inside transparent cubeBrominePronunciation/ˈbroʊmiːn, -mɪn, -maɪn/ (BROH-meen, -⁠min, -⁠myne)Appearancereddish-brownStandard atomic weight Ar°(Br)[79.901, 79.907][1]79.904±0.003 (abridged)[2] Bromine in the periodic table Hydrogen Helium Lithium Beryllium Boron Carbon Nitrogen Oxygen Fluorine Neon Sodium Magnesium Aluminium Silicon Phosphorus ...

Dominic West

English actor (born 1969) Dominic WestWest at Testament of Youth Premiere 2014BornDominic Gerard Francis Eagleton West (1969-10-15) 15 October 1969 (age 54)Sheffield, EnglandEducationTrinity College Dublin (BA)Guildhall School of Music and Drama (MA)OccupationsActordirectormusicianYears active1991–presentSpouse Catherine FitzGerald (m. 2010)Children5 Dominic West's voice From Front Row Interviews, July 2011[1] Dominic Gerard Francis E...

Mauro del Vecchio

Italian politician This biography of a living person includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Contentious material about living persons that is unsourced or poorly sourced must be removed immediately, especially if potentially libelous or harmful. Please help improve this article by introducing more precise citations. (January 2013) (Learn how and when to remove this message) Mauro del VecchioBorn (194...

Irish Russians

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Irish Russians – news · newspapers · books · scholar · JSTOR (October 2017) (Learn how and when to remove this message) Irish Russians are Russian nationals whose ancestry originates wholly or partly in Ireland. Migration occurred in the context of conflicts in...

Île Mocha

Cet article est une ébauche concernant le monde insulaire et le Chili. Vous pouvez partager vos connaissances en l’améliorant (comment ?) selon les recommandations des projets correspondants. Pour les articles homonymes, voir Mocha. Île MochaIsla Mocha (es) Mocha vue du ciel Géographie Pays Chili Localisation Océan Pacifique Coordonnées 38° 22′ 15″ S, 73° 54′ 51″ O Superficie 48 km2 Point culminant 390 m Administration Régi...

National Swedish Touring Theatre

Swedish touring theatre company This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: National Swedish Touring Theatre – news · newspapers · books · scholar · JSTOR (July 2019) (Learn how and when to remove this message) Playbill for the production of Man and Superman by George Bernard Shaw, Riksteatern, 1960 A 19...

PROFILPELAJAR.COM