Modular curve
Encyclopedia
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 compactification
s 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 curve
s, 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 Q of rational number
s, or a cyclotomic field
. The latter fact and its generalizations are of fundamental importance in number theory.
The minimal such N is called the level of Γ. A complex structure can be put on the quotient Γ\H to obtain a noncompact Riemann surface commonly denoted Y(Γ).
P1(C). The group Γ acts on the subset }, breaking it up into finitely many orbits called the cusps of Γ. 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
. This space is a compactification of Y(Γ).
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. This group is a simple group of order 60 isomorphic to A5 and PSL(2,5).
The modular curve X(7) is the Klein quartic
of genus 3 with 24 cusps. It can be interpreted as the Riemann sphere 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 Galois group of the covering X(7) → X(1) is a simple group of order 168 isomorphic to PSL(2,7)
.
There is an explicit classical model for X0(N), the classical modular curve
; this is sometimes called the 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 reduction modulo
N. Then Γ0(N) is the larger subgroup consisting of matrices
with 
that are upper triangular modulo N and Γ1(N) is the intermediate group represented by matrices with 
These curves have a direct interpretation as moduli space
s for elliptic curve
s with level structure and for this reason they play an important role in arithmetic geometry. The level N modular curve X(N) is the moduli space for elliptic curves with a basis for the N-torsion. For X0(N) and X1(N), the level structure is, respectively, a cyclic subgroup of order N and a point of order N. These curves have been studied in great detail, and in particular, it is known that X0(N) can be defined over Q.
The equations defining modular curves are the best-known examples of modular equation
s. The "best models" can be very different from those taken directly from elliptic function
theory. Hecke operator
s may be studied geometrically, as correspondence
s connecting pairs of modular curves.
Remark: quotients of H that are compact do occur for Fuchsian group
s Γ other than subgroups of the modular group; a class of them constructed from quaternion algebra
s is also of interest in number theory.
, one can calculate the genus of X(N). 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. This results in a formula
Thus X(5) has genus 0, X(7) has genus 3, and X(11) has genus 26. For p equal to 2 or 3, one must additionally take into account the ramification, that is, the presence of order p 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 curve X(N) of any level N that involves divisors of N.
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 and can be appropriately normalized, is a Hauptmodul (main or principal modular function). First several coefficients of q-expansions of these functions were computed already in 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 the normalizer Γ0(p)+ of Γ0(p) in SL(2,R) has genus zero if and only if p is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71, and these are precisely the prime factors of the order of the monster group
. 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.
The relation runs very deep and as demonstrated by Richard Borcherds
, it also involves generalized Kac–Moody algebra
s. 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.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
and algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, a modular curve Y(Γ) is a Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
, or the corresponding algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
, constructed as a quotient of the complex upper half-plane H by the action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of a congruence subgroup
Congruence subgroup
In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2x2 integer matrices of determinant 1, such that the off-diagonal entries are even.An importance class of congruence...
Γ of the modular group
Modular group
In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...
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 compactification
Compactification (mathematics)
In mathematics, compactification is the process or result of making a topological space compact. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".-An...
s 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 curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s, 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
Field of definition
In mathematics, the field of definition of an algebraic variety V is essentially the smallest field to which the coefficients of the polynomials defining V can belong...
either over the field Q of rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s, or a cyclotomic field
Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
. The latter fact and its generalizations are of fundamental importance in number theory.
Analytic definition
The modular group SL(2, Z) acts on the upper half-plane by fractional 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 the principal congruence subgroup of level N Γ(N), for some positive integer N, whereThe minimal such N is called the level of Γ. A complex structure can be put on the quotient Γ\H to obtain a noncompact Riemann surface commonly denoted Y(Γ).
Compactified modular curves
A common compactification of Y(Γ) is obtained by adding finitely many points called the cusps of Γ. Specifically, this is done by considering the action of Γ on the extended complex upper-half plane H* = }, which is a subset of the Riemann sphereRiemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
P1(C). The group Γ acts on the subset }, breaking it up into finitely many orbits called the cusps of Γ. 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
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
. This space is a compactification of Y(Γ).
Examples
The most common examples are the curves X(N), X0(N), and X1(N) associated with the subgroups Γ(N), Γ0(N), and Γ1(N).The modular curve X(5) has genus 0: it is the Riemann sphere with 12 cusps located at the vertices of a regular icosahedron
Icosahedron
In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
. The covering X(5) → X(1) is realized by the action of the icosahedral group
Icosahedral symmetry
A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation...
on the Riemann sphere. This group is a simple group of order 60 isomorphic to A5 and PSL(2,5).
The modular curve X(7) is the Klein quartic
Klein quartic
In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-preserving automorphisms, and 336 automorphisms if orientation may be reversed...
of genus 3 with 24 cusps. It can be interpreted as the Riemann sphere 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 Galois group of the covering X(7) → X(1) is a simple group of order 168 isomorphic to PSL(2,7)
PSL(2,7)
In mathematics, the projective special linear group PSL is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane...
.
There is an explicit classical model for X0(N), the classical modular curve
Classical modular curve
In number theory, the classical modular curve is an irreducible plane algebraic curve given by an equationwhere for the j-invariant j,is a point on the curve. The curve is sometimes called X0, though often that is used for the abstract algebraic curve for which there exist various models...
; this is sometimes called the 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 reduction modulo
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
N. Then Γ0(N) is the larger subgroup consisting of matrices
with that are upper triangular modulo N and Γ1(N) is the intermediate group represented by matrices with These curves have a direct interpretation as moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
s for elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s with level structure and for this reason they play an important role in arithmetic geometry. The level N modular curve X(N) is the moduli space for elliptic curves with a basis for the N-torsion. For X0(N) and X1(N), the level structure is, respectively, a cyclic subgroup of order N and a point of order N. These curves have been studied in great detail, and in particular, it is known that X0(N) can be defined over Q.
The equations defining modular curves are the best-known examples of modular equation
Modular equation
In mathematics, a modular equation is an algebraic equation satisfied by moduli, in the sense of moduli problem. That is, given a number of functions on a moduli space, a modular equation is an equation holding between them, or in other words an identity for moduli.The most frequent use of the term...
s. The "best models" can be very different from those taken directly from elliptic function
Elliptic function
In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...
theory. Hecke operator
Hecke operator
In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , 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....
s may be studied geometrically, as correspondence
Correspondence (mathematics)
In mathematics and mathematical economics, correspondence is a term with several related but not identical meanings.* In general mathematics, correspondence is an alternative term for a relation between two sets...
s connecting pairs of modular curves.
Remark: quotients of H that are compact do occur for Fuchsian group
Fuchsian group
In mathematics, a Fuchsian group is a discrete subgroup of PSL. The group PSL can be regarded as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting...
s Γ other than subgroups of the modular group; a class of them constructed from quaternion algebra
Quaternion algebra
In mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes the matrix algebra by extending scalars , i.e...
s is also of interest in number theory.
Genus
The covering X(N) → X(1) is Galois, with Galois group SL(2,N)/{1,-1}, which is equal to PSL(2,N) if N is prime. Applying the Riemann–Hurwitz formula and Gauss–Bonnet theoremGauss–Bonnet theorem
The Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry to their topology...
, one can calculate the genus of X(N). For a prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
level p ≥ 5,
where χ = 2 − 2g is the Euler characteristic
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
, |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
Defect (geometry)
In geometry, the defect means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the plane would...
of the spherical (2,3,p) triangle. This results in a formula
Thus X(5) has genus 0, X(7) has genus 3, and X(11) has genus 26. For p equal to 2 or 3, one must additionally take into account the ramification, that is, the presence of order p 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 curve X(N) of any level N that involves divisors of N.
Relation with the Monster group
Modular curves of genusGenus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
0, which are quite rare, turned out to be of major importance in relation with the monstrous moonshine
Monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Horton Conway and Simon P. Norton in 1979, used to describe the connection between the monster group M and modular functions .- History :Specifically, Conway and Norton, following an initial observationby John...
conjectures. In general a modular function field is a function field
Function field of an algebraic variety
In algebraic geometry, the function field of an algebraic variety V consists of objects which are interpreted as rational functions on V...
of a modular curve (or, occasionally, of some other moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
that turns out to be an irreducible variety). Genus 0 means such a function field has a single transcendental function
Transcendental function
A transcendental function is a function that does not satisfy a polynomial equation whose coefficients are themselves polynomials, in contrast to an algebraic function, which does satisfy such an equation...
as generator: for example the j-function
J-invariant
In mathematics, Klein's j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half-plane of complex numbers.We haveThe modular discriminant \Delta is defined as \Delta=g_2^3-27g_3^2...
. 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). First several coefficients of q-expansions of these functions were computed already in 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 the normalizer Γ0(p)+ of Γ0(p) in SL(2,R) has genus zero if and only if p is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71, and these are precisely the prime factors of the order of the monster group
Monster group
In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
. The result about Γ0(p)+ is due to Jean-Pierre Serre
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
, Andrew Ogg
Andrew Ogg
Andrew Pollard Ogg is an American mathematician, a professor emeritus of mathematics at the University of California, Berkeley....
and John G. Thompson
John G. Thompson
John Griggs Thompson is a mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970, the Wolf Prize in 1992 and the 2008 Abel Prize....
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
Jack Daniel's
Jack Daniel's is a brand of sour mash Tennessee whiskey that is among the world's best-selling liquors. It is known for its square bottles and black label. As of November, 2007, one blogger was claiming that it was the best-selling whiskey in the world. It is produced in Lynchburg, Tennessee by...
whiskey to anyone who could explain this fact, which was a starting point for the theory of monstrous moonshine.
The relation runs very deep and as demonstrated by Richard Borcherds
Richard Borcherds
Richard Ewen Borcherds is a British mathematician specializing in lattices, number theory, group theory, and infinite-dimensional algebras. He was awarded the Fields Medal in 1998.- Personal life :...
, it also involves generalized Kac–Moody algebra
Generalized Kac–Moody algebra
In mathematics, a generalized Kac–Moody algebra is a Lie algebra that is similar to aKac–Moody algebra, except that it is allowed to have imaginary simple roots.Generalized Kac–Moody algebras are also sometimes called GKM algebras,...
s. 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
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
, that are holomorphic everywhere, including the cusps, and had been the main objects of study for the better part of the 20th century.
See also
- Modularity theorem
- Shimura varietyShimura varietyIn number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. The term "Shimura variety" applies to the higher-dimensional case, in the case of...
, a generalization of modular curves to higher dimensions