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).
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 modular group SL(2, Z) acts on the upper half-plane by fractional linear transformations.
A complex structure can be put on the quotient Γ\H to obtain a noncompact Riemann surface called a modular curve, and commonly denoted Y(Γ).
Specifically, this is done by considering the action of Γ on the extended complex upper-half plane H* = H ∪ Q ∪ {∞}.
Once again, a complex structure can be put on the quotient Γ\H* turning it into a Riemann surface denoted X(Γ) which is now compact.
The modular curve X(5) has genus 0: it is the Riemann sphere with 12 cusps located at the vertices of a regular icosahedron.
The covering X(5) → X(1) is realized by the action of the icosahedral group on the Riemann sphere.
It can be interpreted as a surface with three handles tiled by 24 heptagons, with a cusp at the center of each face.
These tilings can be understood via dessins d'enfants and Belyi functions – the cusps are the points lying over ∞ (red dots), while the vertices and centers of the edges (black and white dots) are the points lying over 0 and 1.
The definition of Γ(N) can be restated as follows: it is the subgroup of the modular group which is the kernel of the reduction modulo N. Then Γ0(N) is the larger subgroup of matrices which are upper triangular modulo N: and Γ1(N) is the intermediate group defined by: These curves have a direct interpretation as moduli spaces for elliptic curves with level structure and for this reason they play an important role in arithmetic geometry.
Hecke operators may be studied geometrically, as correspondences connecting pairs of modular curves.
For a prime level p ≥ 5, where χ = 2 − 2g is the Euler characteristic, |G| = (p+1)p(p−1)/2 is the order of the group PSL(2, p), and D = π − π/2 − π/3 − π/p is the angular defect of the spherical (2,3,p) triangle.
The traditional name for such a generator, which is unique up to a Möbius transformation and can be appropriately normalized, is a Hauptmodul (main or principal modular function, plural Hauptmoduln).
The following table contains the unique reduced, minimal, integral Weierstrass models, which means
[3] The last column of this table refers to the home page of the respective elliptic modular curve
Modular curves of genus 0, which are quite rare, turned out to be of major importance in relation with the monstrous moonshine conjectures.
First several coefficients of q-expansions of their Hauptmoduln were computed already in the 19th century, but it came as a shock that the same large integers show up as dimensions of representations of the largest sporadic simple group Monster.
The result about Γ0(p)+ is due to Jean-Pierre Serre, Andrew Ogg and John G. Thompson in the 1970s, and the subsequent observation relating it to the monster group is due to Ogg, who wrote up a paper offering a bottle of Jack Daniel's whiskey to anyone who could explain this fact, which was a starting point for the theory of monstrous moonshine.
[4] The relation runs very deep and, as demonstrated by Richard Borcherds, it also involves generalized Kac–Moody algebras.
Work in this area underlined the importance of modular functions that are meromorphic and can have poles at the cusps, as opposed to modular forms, that are holomorphic everywhere, including the cusps, and had been the main objects of study for the better part of the 20th century.