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- Heegner_point abstract "In mathematics, a Heegner point is a point on a modular curve that is the image of a quadratic imaginary point of the upper half-plane. They were defined by Bryan Birch and named after Kurt Heegner, who used similar ideas to prove Gauss's conjecture on imaginary quadratic fields of class number one.The Gross–Zagier theorem (Gross & Zagier 1986) describes the height of Heegner points in terms of a derivative of the L-function of the elliptic curve at the point s = 1. In particular if the elliptic curve has (analytic) rank 1, then the Heegner points can be used to construct a rational point on the curve of infinite order (so the Mordell–Weil group has rank at least 1). More generally, Gross, Kohnen & Zagier (1987) showed that Heegner points could be used to construct rational points on the curve for each positive integer n, and the heights of these points were the coefficients of a modular form of weight 3/2. Kolyvagin later used Heegner points to construct Euler systems, and used this to prove much of the Birch–Swinnerton-Dyer conjecture for rank 1 elliptic curves. Shouwu Zhang generalized the Gross–Zagier theorem from elliptic curves to the case of modular abelian varieties. Brown proved the Birch–Swinnerton-Dyer conjecture for most rank 1 elliptic curves over global fields of positive characteristic. (Brown 1994)Heegner points can be used to compute very large rational points on rank 1 elliptic curves (see (Watkins 2006) for a survey) that could not be found by naive methods. Implementation of the algorithm is available in Magma and PARI/GP".
- Heegner_point wikiPageExternalLink 0506325.
- Heegner_point wikiPageExternalLink Book49.
- Heegner_point wikiPageID "5828654".
- Heegner_point wikiPageLength "4121".
- Heegner_point wikiPageOutDegree "25".
- Heegner_point wikiPageRevisionID "673451778".
- Heegner_point wikiPageWikiLink Abelian_variety.
- Heegner_point wikiPageWikiLink Birch_and_Swinnerton-Dyer_conjecture.
- Heegner_point wikiPageWikiLink Birch–Swinnerton-Dyer_conjecture.
- Heegner_point wikiPageWikiLink Bryan_Birch.
- Heegner_point wikiPageWikiLink Bryan_John_Birch.
- Heegner_point wikiPageWikiLink Cambridge_University_Press.
- Heegner_point wikiPageWikiLink Category:Algebraic_number_theory.
- Heegner_point wikiPageWikiLink Category:Elliptic_curves.
- Heegner_point wikiPageWikiLink Class_number_problem.
- Heegner_point wikiPageWikiLink Euler_system.
- Heegner_point wikiPageWikiLink Inventiones_Mathematicae.
- Heegner_point wikiPageWikiLink Kolyvagin.
- Heegner_point wikiPageWikiLink Kurt_Heegner.
- Heegner_point wikiPageWikiLink L-function.
- Heegner_point wikiPageWikiLink Magma_(computer_algebra_system).
- Heegner_point wikiPageWikiLink Magma_computer_algebra_system.
- Heegner_point wikiPageWikiLink Mathematics.
- Heegner_point wikiPageWikiLink Mathematische_Annalen.
- Heegner_point wikiPageWikiLink Mathematische_Zeitschrift.
- Heegner_point wikiPageWikiLink Modular_curve.
- Heegner_point wikiPageWikiLink Mordell–Weil_group.
- Heegner_point wikiPageWikiLink Mordell–Weil_theorem.
- Heegner_point wikiPageWikiLink Néron–Tate_height.
- Heegner_point wikiPageWikiLink GP.
- Heegner_point wikiPageWikiLink Quadratic_field.
- Heegner_point wikiPageWikiLink Quadratic_fields.
- Heegner_point wikiPageWikiLink Rational_point.
- Heegner_point wikiPageWikiLink Shou-Wu_Zhang.
- Heegner_point wikiPageWikiLink Shouwu_Zhang.
- Heegner_point wikiPageWikiLink Upper_half-plane.
- Heegner_point wikiPageWikiLink Victor_Kolyvagin.
- Heegner_point wikiPageWikiLinkText "Heegner point".
- Heegner_point hasPhotoCollection Heegner_point.
- Heegner_point wikiPageUsesTemplate Template:Citation.
- Heegner_point wikiPageUsesTemplate Template:Harv.
- Heegner_point wikiPageUsesTemplate Template:Harvtxt.
- Heegner_point subject Category:Algebraic_number_theory.
- Heegner_point subject Category:Elliptic_curves.
- Heegner_point hypernym Point.
- Heegner_point type Place.
- Heegner_point type Function.
- Heegner_point type Variety.
- Heegner_point comment "In mathematics, a Heegner point is a point on a modular curve that is the image of a quadratic imaginary point of the upper half-plane. They were defined by Bryan Birch and named after Kurt Heegner, who used similar ideas to prove Gauss's conjecture on imaginary quadratic fields of class number one.The Gross–Zagier theorem (Gross & Zagier 1986) describes the height of Heegner points in terms of a derivative of the L-function of the elliptic curve at the point s = 1.".
- Heegner_point label "Heegner point".
- Heegner_point sameAs ヒーグナー点.
- Heegner_point sameAs m.0f7trk.
- Heegner_point sameAs Q5697927.
- Heegner_point sameAs Q5697927.
- Heegner_point wasDerivedFrom Heegner_point?oldid=673451778.
- Heegner_point isPrimaryTopicOf Heegner_point.