Innumber theory andalgebraic geometry, amodular curveY(Γ) is aRiemann surface, or the correspondingalgebraic curve, constructed as aquotient of the complexupper half-planeH by theaction of acongruence subgroup Γ of themodular group of integral 2×2 matrices SL(2, Z). The term modular curve can also be used to refer to thecompactified modular curvesX(Γ) which arecompactifications obtained by adding finitely many points (called thecusps of Γ) to this quotient (via an action on theextended complex upper-half plane). The points of a modular curveparametrize isomorphism classes ofelliptic 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 tocomplex numbers, and, moreover, prove that modular curves aredefined either over the field ofrational numbersQ or acyclotomic fieldQ(ζ_n). The latter fact and its generalizations are of fundamental importance in number theory.

The modular group SL(2, Z) acts on the upper half-plane byfractional linear transformations. The analytic definition of a modular curve involves a choice of a congruence subgroup Γ of SL(2, Z), i.e. a subgroup containing theprincipal congruence subgroup of levelN for some positive integerN, which is defined to be

The minimal suchN is called thelevel of Γ. Acomplex structure can be put on the quotient Γ\H to obtain anoncompact Riemann surface called amodular curve, and commonly denotedY(Γ).

A common compactification ofY(Γ) is obtained by adding finitely many points called the cusps of Γ. Specifically, this is done by considering the action of Γ on theextended complex upper-half planeH* = H ∪Q ∪ {∞}. We introduce a topology onH* by taking as a basis:

• any open subset ofH,
• for allr > 0, the set${\displaystyle \{\mathrm{\infty }\}\cup \{\tau \in \mathbf{H}\mid \text{Im}(\tau )>r\}}$
• for allcoprime integersa,c and allr > 0, the image of${\displaystyle \{\mathrm{\infty }\}\cup \{\tau \in \mathbf{H}\mid \text{Im}(\tau )>r\}}$ under the action of

wherem,n are integers such thatan +cm = 1.

This turnsH* into a topological space which is a subset of theRiemann sphereP¹(C). The group Γ acts on the subsetQ ∪ {∞}, breaking it up into finitely manyorbits called thecusps of Γ. If Γ acts transitively onQ ∪ {∞}, the space Γ\H* becomes theAlexandroff compactification of Γ\H. Once again, a complex structure can be put on the quotient Γ\H* turning it into a Riemann surface denotedX(Γ) which is nowcompact. This space is a compactification ofY(Γ).^[1]

The most common examples are the curvesX(N),X₀(N), andX₁(N) associated with the subgroups Γ(N), Γ₀(N), and Γ₁(N).

The modular curveX(5) has genus 0: it is the Riemann sphere with 12 cusps located at the vertices of a regularicosahedron. The coveringX(5) →X(1) is realized by the action of theicosahedral group on the Riemann sphere. This group is a simple group of order 60 isomorphic toA₅ and PSL(2, 5).

The modular curveX(7) is theKlein quartic of genus 3 with 24 cusps. 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 viadessins d'enfants andBelyi 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 Galois group of the coveringX(7) → X(1) is a simple group of order 168 isomorphic toPSL(2, 7).

There is an explicit classical model forX₀(N), theclassical modular curve; this is sometimes calledthe modular curve. The definition of Γ(N) can be restated as follows: it is the subgroup of the modular group which is the kernel of the reductionmoduloN. Then Γ₀(N) is the larger subgroup of matrices which are upper triangular moduloN:

and Γ₁(N) is the intermediate group defined by:

These curves have a direct interpretation asmoduli spaces forelliptic curves withlevel structure and for this reason they play an important role inarithmetic geometry. The levelN modular curveX(N) is the moduli space for elliptic curves with a basis for theN-torsion. ForX₀(N) andX₁(N), the level structure is, respectively, a cyclic subgroup of orderN and a point of orderN. These curves have been studied in great detail, and in particular, it is known thatX₀(N) can be defined overQ.

The equations defining modular curves are the best-known examples ofmodular equations. The "best models" can be very different from those taken directly fromelliptic function theory.Hecke operators may be studied geometrically, ascorrespondences connecting pairs of modular curves.

Quotients ofH thatare compact do occur forFuchsian groups Γ other than subgroups of the modular group; a class of them constructed fromquaternion algebras is also of interest in number theory.

The coveringX(N) →X(1) is Galois, with Galois group SL(2,N)/{1, −1}, which is equal to PSL(2, N) ifN is prime. Applying theRiemann–Hurwitz formula andGauss–Bonnet theorem, one can calculate the genus ofX(N). For aprime levelp ≥ 5,

${\displaystyle -\pi \chi (X(p))=|G|\cdot D,}$

where χ = 2 − 2g is theEuler characteristic, |G| = (p+1)p(p−1)/2 is the order of the group PSL(2,p), andD = π − π/2 − π/3 − π/p is theangular defect of the spherical (2,3,p) triangle. This results in a formula

${\displaystyle g=\frac{1}{24}(p+2)(p-3)(p-5).}$

ThusX(5) has genus 0,X(7) has genus 3, andX(11) has genus 26. Forp = 2 or 3, one must additionally take into account the ramification, that is, the presence of orderp elements in PSL(2,Z), and the fact that PSL(2, 2) has order 6, rather than 3. There is a more complicated formula for the genus of the modular curveX(N) of any levelN that involves divisors ofN.

In general amodular function field is afunction field of a modular curve (or, occasionally, of some othermoduli space that turns out to be anirreducible variety).Genus zero means such a function field has a singletranscendental function as generator: for example thej-function generates the function field ofX(1) = PSL(2,Z)\H*. The traditional name for such a generator, which is unique up to aMöbius transformation and can be appropriately normalized, is aHauptmodul (main orprincipal modular function, pluralHauptmoduln).

The spacesX₁(n) have genus zero forn = 1, ..., 10 andn = 12. Since each of these curves is defined overQ and has aQ-rational point, it follows that there are infinitely many rational points on each such curve, and hence infinitely many elliptic curves defined overQ withn-torsion for these values ofn. The converse statement, that only these values ofn can occur, isMazur's torsion theorem.

The modular curves${\displaystyle X_0(N)}$ are of genus one if and only if${\displaystyle N}$ equals one of the 12 values listed in the following table.^[2] Aselliptic curves over${\displaystyle \mathbb{Q}}$, they have minimal, integral Weierstrass models${\displaystyle y^2+a_{1}xy+a_{3}y=x^3+a_2{x}^2+a_{4}x+a_6}$. This is,${\displaystyle a_j\in \mathbb{Z}}$ and the absolute value of the discriminant${\displaystyle \mathrm{\Delta }}$ is minimal among all integral Weierstrass models for the same curve. The following table contains the uniquereduced, minimal, integral Weierstrass models, which means${\displaystyle a_1,a_3\in \{0,1\}}$ and${\displaystyle a_2\in \{-1,0,1\}}$.^[3] The last column of this table refers to the home page of the respective elliptic modular curve${\displaystyle X_0(N)}$ onThe L-functions and modular forms database (LMFDB).

${\displaystyle X_0(N)}$ of genus 1

${\displaystyle y^2+a_{1}xy+a_{3}y=x^3+a_2{x}^2+a_{4}x+a_6}$
${\displaystyle N}$	${\displaystyle [a_1,a_2,a_3,a_4,a_6]}$	${\displaystyle \mathrm{\Delta }}$	LMFDB
11	[0, -1, 1, -10, -20]	${\displaystyle -{11}^5}$	link
14	[1, 0, 1, 4, -6]	${\displaystyle -2^6\cdot 7^3}$	link
15	[1, 1, 1, -10, -10]	${\displaystyle 3^4\cdot 5^4}$	link
17	[1, -1, 1, -1, -14]	${\displaystyle -{17}^4}$	link
19	[0, 1, 1, -9, -15]	${\displaystyle -{19}^3}$	link
20	[0, 1, 0, 4, 4]	${\displaystyle -2^8\cdot 5^2}$	link
21	[1, 0, 0, -4, -1]	${\displaystyle 3^4\cdot 7^2}$	link
24	[0, -1, 0, -4, 4]	${\displaystyle 2^8\cdot 3^2}$	link
27	[0, 0, 1, 0, -7]	${\displaystyle -3^9}$	link
32	[0, 0, 0, 4, 0]	${\displaystyle -2^{12}}$	link
36	[0, 0, 0, 0, 1]	${\displaystyle -2^4\cdot 3^3}$	link
49	[1, -1, 0, -2, -1]	${\displaystyle -7^3}$	link

Modular curves of genus 0, which are quite rare, turned out to be of major importance in relation with themonstrous moonshine conjectures. The first several coefficients of theq-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.

Another connection is that the modular curve corresponding to thenormalizer Γ₀(p)⁺ ofΓ₀(p) in SL(2,R) has genus zero if and only ifp is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71, and these are preciselysupersingular primes in moonshine theory, i.e. the prime factors of the order of themonster group. The result about Γ₀(p)⁺ is due toJean-Pierre Serre,Andrew Ogg andJohn 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 ofJack 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 byRichard Borcherds, it also involvesgeneralized Kac–Moody algebras. Work in this area underlined the importance ofmodularfunctions that are meromorphic and can have poles at the cusps, as opposed tomodularforms, that are holomorphic everywhere, including the cusps, and had been the main objects of study for the better part of the 20th century.

• Manin–Drinfeld theorem
• Moduli stack of elliptic curves
• Modularity theorem
• Shimura variety, a generalization of modular curves to higher dimensions

• Steven D. Galbraith -Equations For Modular Curves

• Algebraic curves
• Modular forms
• Riemann surfaces

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