Modular curve
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In mathematics, a modular curve is a Riemann surface, or corresponding algebraic curve, constructed as
- H/Γ
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.
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 agree with the identity matrix on and below the main diagonal. 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.
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).
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.
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.