In number theory and algebraic geometry, a modular curveY(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 curvesX(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.
number theory and algebraic geometry, a modularcurve Y(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex...
A modular elliptic curve is an elliptic curve E that admits a parametrisation X0(N) → E by a modularcurve. This is not the same as a modularcurve that...
classical modularcurve is an irreducible plane algebraic curve given by an equation Φn(x, y) = 0, such that (x, y) = (j(nτ), j(τ)) is a point on the curve. Here...
The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Shimura-Weil conjecture or modularity conjecture for elliptic curves) states...
Classical modularcurve Fuchsian group J-invariant Kleinian group Mapping class group Minkowski's question-mark function Möbius transformation Modularcurve Modular...
rational functions F and G, in the function field of the modularcurve, will satisfy a modular equation P(F,G) = 0 with P a non-zero polynomial of two...
)} where ω {\displaystyle \omega } is a canonical line bundle on the modularcurve X Γ = Γ ∖ ( H ∪ P 1 ( Q ) ) {\displaystyle X_{\Gamma }=\Gamma \backslash...
asserts that every elliptic curve over Q is a modularcurve, which implies that its L-function is the L-function of a modular form whose analytic continuation...
number theory, a Shimura variety is a higher-dimensional analogue of a modularcurve that arises as a quotient variety of a Hermitian symmetric space by...
field of the corresponding quotient, i.e., it is a Hauptmodul for the modularcurve X(2). Over any point τ, its value can be described as a cross ratio...
In mathematics, a Heegner point is a point on a modularcurve that is the image of a quadratic imaginary point of the upper half-plane. They were defined...
varieties Shimura variety Modularcurve Elliptic cohomology Silverman, Joseph H. (2009). The arithmetic of elliptic curves (2nd ed.). New York: Springer-Verlag...
Enrique; Gonzalez, Josep; Poonen, Bjorn (2005), "Finiteness results for modularcurves of genus at least 2", American Journal of Mathematics, 127 (6): 1325–1387...
this curve has a complicated form, it is natural and conceptually significant in the number theory of elliptic curves. The equation describes a modular curve...
Taniyama–Shimura conjecture (now known as the modularity theorem) relating elliptic curves to modular forms. This connection would ultimately lead to...
function Weil conjectures Modular form modular group Congruence subgroup Hecke operator Cusp form Eisenstein series Modularcurve Ramanujan–Petersson conjecture...
face) is the modularcurve X(5); this explains the relevance for number theory. More subtly, the (projective) Klein quartic is a Shimura curve (as are the...
Modular synthesizers are synthesizers composed of separate modules for different functions. The modules can be connected together by the user to create...
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus...
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