In mathematics, a modular curve is a Riemann surface, or corresponding algebraic curve, constructed as Mathematics is the study of quantity, structure, space and change. ... In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ...

HΓ

where H is the upper half-plane in the complex numbers, and Γ is a Fuchsian group acting on H, with Γ a subgroup of the modular group of integral 2×2 matrices. This quotient will not be compact, requiring the addition of one or more cusps for its compactification. It is therefore an open Riemann surface; it is the corresponding compact Riemann surface and algebraic curve that is usually, more strictly, meant by a modular curve. In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... In mathematics, a Fuchsian group is a particular type of group of isometries of the hyperbolic plane. ... In mathematics, groups are often used to describe symmetries of objects. ... In mathematics, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation... In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ... Several specialized usages of the terms compact and compactness exist. ... A cusp is a sharp point or apex, such as occurs in two dimensions at the end of a crescent, or in three dimensions at the tip of a cone or horn. ... In mathematics, compactification is applied to topological spaces to make them compact spaces. ...

The most common examples are the curves X(N) and X₀(N), associated with the groups Γ(N) and Γ₀(N). Here for any N ≥ 1 Γ(N) is the subgroup of the modular group of matrices that are in the kernel of reduction modulo N, and Γ₀(N) is the larger subgroup, of matrices that modulo N are upper triangular. These curves have a direct interpretation as moduli spaces for elliptic curves, with 'markings'; for example X(N) is (the compactified) moduli space for elliptic curves with a given basis for the N-torsion. These curves have been studied in great detail. The word modulo is the Latin ablative of modulus. ... In algebraic geometry, the moduli problem is to describe the parameters on which algebraic varieties depend. ... In mathematics, an elliptic curve is a non-singular projective algebraic curve of genus 1 over a field K, together with a distinguished point defined over K. A more accessible (though less accurate) definition is that an elliptic curve is a plane curve defined by an equation of the form...

Modular curves of genus 0, which are quite rare, turned out to be of major importance in relation with the monstrous moonshine conjectures. In general a modular function field is a function field of a modular curve (or, occasionally, of some other moduli space that turns out to be an irreducible variety). Genus 0 means such a function field has a single transcendental function as generator: for example the j-function. The traditional name for such a generator, which is unique up to a Möbius transformation, is a Hauptmodul (main or principal modular function). In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ... In mathematics, monstrous moonshine is a term devised by John Horton Conway and Simon P. Norton in 1979, used to describe the (then totally unexpected) connection between the monster group M and modular functions (particularly, the j function). ... In algebraic geometry, the function field of an irreducible algebraic variety is the field of fractions of the ring of regular functions. ... In algebraic geometry, the moduli problem is to describe the parameters on which algebraic varieties depend. ... A transcendental function is a function which does not satisfy a polynomial equation whose coefficients are themselves polynomials. ... In mathematics, the j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half plane of complex numbers with positive imaginary part. ... In mathematics, a Möbius transformation, also called a homographic function, is a conformal mapping that is a bijection on the extended complex plane (that is, the complex plane augmented by the point at infinity, written ∞.) It is named in honor of August Ferdinand Möbius. ...

The equations defining modular curves are the best-known examples of modular equations. The best models can be very different from those taken directly from elliptic function theory. Hecke operators may be studied geometrically, as correspondences connecting pairs of modular curves. In mathematics, a modular equation is an algebraic equation satisfied by moduli, in the sense of moduli problem. ... In complex analysis, an elliptic function is, roughly speaking , a function defined on the complex plane which is periodic in two directions. ... In mathematics, in particular in the theory of modular forms, a Hecke operator is a certain kind of averaging operator that plays a significant role in the structure of vector spaces of modular forms (and more general automorphic representations). ... Correspondence may refer to: In the theology of Emanuel Swedenborg, correspondence is the relationship between spiritual and physical realities. ...

Quotients

H/Γ

that are compact do occur with Fuchsian groups Γ other than subgroups of the modular group; these are of interest in number theory, also, in cases where they are constructed from quaternion algebras.