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- Modular_curve abstract "In number theory and algebraic geometry, a modular curve Y(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z). The term modular curve can also be used to refer to the compactified modular curves X(Γ) which are compactifications obtained by adding finitely many points (called the cusps of Γ) to this quotient (via an action on the extended complex upper-half plane). The points of a modular curve parametrize isomorphism classes of elliptic curves, together with some additional structure depending on the group Γ. This interpretation allows one to give a purely algebraic definition of modular curves, without reference to complex numbers, and, moreover, prove that modular curves are defined either over the field Q of rational numbers, or a cyclotomic field. The latter fact and its generalizations are of fundamental importance in number theory.".
- Modular_curve wikiPageExternalLink m064410.htm.
- Modular_curve wikiPageID "669552".
- Modular_curve wikiPageRevisionID "594441243".
- Modular_curve hasPhotoCollection Modular_curve.
- Modular_curve subject Category:Algebraic_curves.
- Modular_curve subject Category:Modular_forms.
- Modular_curve subject Category:Riemann_surfaces.
- Modular_curve type Abstraction100002137.
- Modular_curve type AlgebraicCurves.
- Modular_curve type Attribute100024264.
- Modular_curve type Curve113867641.
- Modular_curve type Form106290637.
- Modular_curve type LanguageUnit106284225.
- Modular_curve type Line113863771.
- Modular_curve type ModularForms.
- Modular_curve type Part113809207.
- Modular_curve type Relation100031921.
- Modular_curve type Shape100027807.
- Modular_curve type Word106286395.
- Modular_curve comment "In number theory and algebraic geometry, a modular curve Y(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z).".
- Modular_curve label "Courbe modulaire".
- Modular_curve label "Modular curve".
- Modular_curve label "モジュラー曲線".
- Modular_curve label "模曲線".
- Modular_curve sameAs Courbe_modulaire.
- Modular_curve sameAs モジュラー曲線.
- Modular_curve sameAs 모듈러_곡선.
- Modular_curve sameAs m.031dxv.
- Modular_curve sameAs Q3001220.
- Modular_curve sameAs Q3001220.
- Modular_curve sameAs Modular_curve.
- Modular_curve wasDerivedFrom Modular_curve?oldid=594441243.
- Modular_curve isPrimaryTopicOf Modular_curve.