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... > topology > algebraic topology > manifold > differentiable manifold > complex manifold > Riemann surface > modular curve
... > geometry > topological space > manifold > differentiable manifold > complex manifold > Riemann surface > modular curve
... > geometry > differential geometry > differentiable manifold > complex manifold > Riemann surface > modular curve
... > topology > differential topology > differentiable manifold > complex manifold > Riemann surface > modular curve
geometry > complex geometry > complex manifold > Riemann surface > modular curve
geometry > algebraic geometry > algebraic variety > algebraic curve > modular curve
mathematical analysis > functional analysis > geometry of numbers > arithmetic geometry > modular curve
number > number theory > geometry of numbers > arithmetic geometry > modular curve
geometry > discrete geometry > geometry of numbers > arithmetic geometry > modular curve
... > number > real number > Diophantine approximation > geometry of numbers > arithmetic geometry > modular curve

Terme préférentiel

modular curve  

Définition

  • 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 of rational numbers Q or a cyclotomic field Q(ζn). The latter fact and its generalizations are of fundamental importance in number theory.
    (Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Modular_curve)

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http://data.loterre.fr/ark:/67375/PSR-KFVX9T3D-5

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