In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
and
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, a modular curve ''Y''(Γ) is a
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
, or the corresponding
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...
, constructed as a
quotient
In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...
of the complex
upper half-plane
In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example ...
H by the
action
Action may refer to:
* Action (philosophy), something which is done by a person
* Action principles the heart of fundamental physics
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video gam ...
of a
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 is the subgroup of invertible integer matrices of determinant 1 in which the off-diag ...
Γ of the
modular group
In mathematics, the modular group is the projective special linear group \operatorname(2,\mathbb Z) of 2\times 2 matrices with integer coefficients and determinant 1, such that the matrices A and -A are identified. The modular group acts on ...
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). The points of a modular curve
parametrize isomorphism classes of
elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...
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 number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s, and, moreover, prove that modular curves are
defined either over the field of
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example,
The set of all ...
s Q or a
cyclotomic field
In algebraic number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to \Q, the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory ...
Q(ζ
''n''). 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'' for some positive integer ''N'', which is defined to be
:
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 called a modular curve, and 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* = . We introduce a topology on H* by taking as a basis:
* any open subset of H,
* for all ''r'' > 0, the set
* for all
coprime integers
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiva ...
''a'', ''c'' and all ''r'' > 0, the image of
under the action of
::
:where ''m'', ''n'' are integers such that ''an'' + ''cm'' = 1.
This turns H* into a topological space which is a subset of the
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann,
is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents ...
P
1(C). The group Γ acts on the subset , breaking it up into finitely many
orbits
In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificia ...
called the cusps of Γ. If Γ acts transitively on , the space Γ\H* becomes the
Alexandroff compactification of Γ\H. 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 as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
. This space is a compactification of ''Y''(Γ).
Examples
The most common examples are the curves ''X''(''N''), ''X''
0(''N''), and ''X''
1(''N'') associated with the subgroups Γ(''N''), Γ
0(''N''), and Γ
1(''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
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...
. The covering ''X''(5) → ''X''(1) is realized by the action of the
icosahedral group
In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of th ...
on the Riemann sphere. This group is a simple group of order 60 isomorphic to ''A''
5 and PSL(2, 5).
The modular curve ''X''(7) is the
Klein 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 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 ''X''
0(''N''), the
classical modular curve
In number theory, the classical modular curve is an irreducible plane algebraic curve given by an equation
:,
such that is a point on the curve. Here denotes the -invariant.
The curve is sometimes called , though often that notation is used f ...
; 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
In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation.
Given two positive numbers and , mo ...
''N''. Then Γ
0(''N'') is the larger subgroup of matrices which are upper triangular modulo ''N'':
:
and Γ
1(''N'') is the intermediate group defined by:
:
These curves have a direct interpretation as
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...
s for
elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...
s with ''
level structure
In the mathematical subfield of graph theory a level structure of a rooted graph is a partition of the vertices into subsets that have the same distance from a given root vertex..
Definition and construction
Given a connected graph ''G'' = (''V ...
'' and for this reason they play an important role in
arithmetic geometry
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.
...
. The level ''N'' modular curve ''X''(''N'') is the moduli space for elliptic curves with a basis for the ''N''-
torsion. For ''X''
0(''N'') and ''X''
1(''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 ''X''
0(''N'') can be defined over Q.
The equations defining modular curves are the best-known examples of
modular equation
In mathematics, a modular equation is an algebraic equation satisfied by ''moduli'', in the sense of moduli problems. That is, given a number of functions on a moduli space, a modular equation is an equation holding between them, or in other wor ...
s. The "best models" can be very different from those taken directly from
elliptic function
In the mathematical field of complex analysis, elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Those integrals are ...
theory.
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 rep ...
s may be studied geometrically, as
correspondences connecting pairs of modular curves.
Quotients of H that ''are'' compact do occur for
Fuchsian group
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of orientation-preserving isometries of the hyperbolic plane, or conformal transformations of the unit disc, or co ...
s Γ other than subgroups of the modular group; a class of them constructed from
quaternion algebra
In mathematics, a quaternion algebra over a field (mathematics), field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension (vector space), dimension 4 ove ...
s is also of interest in number theory.
Genus
The covering ''X''(''N'') → ''X''(1) is Galois, with Galois group SL(2, ''N'')/, which is equal to PSL(2, ''N'') if ''N'' is prime. Applying the
Riemann–Hurwitz formula
In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ''ramified covering'' of the other. It therefore connects ramifica ...
and
Gauss–Bonnet theorem
In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a Surface (topology), surface to its underlying topology.
In the simplest applicati ...
, one can calculate the genus of ''X''(''N''). For a
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
level ''p'' ≥ 5,
:
where χ = 2 − 2''g'' is the
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...
, , ''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 In geometry, the angular defect or simply defect (also called deficit or deficiency) is the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would. The opposite notion is the ''exces ...
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'' = 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''.
Genus zero
In general a modular function field is a
function field of a modular curve (or, occasionally, of some other
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...
that turns out to be an
irreducible variety
In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component of an algebraic set is an algebraic subset that is irred ...
).
Genus
Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
zero means such a function field has a single
transcendental function
In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable that can be written using only the basic operations of addition, subtraction ...
as generator: for example the
j-function generates the function field of ''X''(1) = PSL(2, Z)\H*. The traditional name for such a generator, which is unique up to a
Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying .
Geometrically ...
and can be appropriately normalized, is a Hauptmodul (main or principal modular function, plural Hauptmoduln).
The spaces ''X''
1(''n'') have genus zero for ''n'' = 1, ..., 10 and ''n'' = 12. Since each of these curves is defined over Q and has a Q-rational point, it follows that there are infinitely many rational points on each such curve, and hence infinitely many elliptic curves defined over Q with ''n''-torsion for these values of ''n''. The converse statement, that only these values of ''n'' can occur, is
Mazur's torsion theorem.
''X''0(''N'') of genus one
The modular curves
are of genus one if and only if
equals one of the 12 values listed in the following table. As
elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...
s over
, they have minimal, integral Weierstrass models
. This is,
and the absolute value of the discriminant
is minimal among all integral Weierstrass models for the same curve. The following table contains the unique ''reduced'', minimal, integral Weierstrass models, which means
and
. The last column of this table refers to the home page of the respective elliptic modular curve
on ''
The L-functions and modular forms database (LMFDB)''.
Relation with the Monster group
Modular curves of genus 0, which are quite rare, turned out to be of major importance in relation with the
monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular the ''j'' function. The initial numerical observation was made by John McKay in 1978, ...
conjectures. The first several coefficients of the ''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.
Another connection is that the modular curve corresponding to the
normalizer
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of ele ...
Γ
0(''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
supersingular primes in moonshine theory, i.e. the prime factors of the order of the
monster group
In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group; it has order
:
: = 2463205976112133171923293 ...
. The result about Γ
0(''p'')
+ is due to
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the inau ...
,
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
Jack Daniel's is a brand of Tennessee whiskey produced at Jack Daniel Distillery in Lynchburg, Tennessee, which has been owned by the Brown–Forman Corporation since 1956.
Packaged in square bottles, Jack Daniel's "Black Label" Tennessee wh ...
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 Ewen Borcherds (; born 29 November 1959) is a British mathematician currently working in quantum field theory. He is known for his work in lattices, group theory, and infinite-dimensional algebras,James Lepowsky"The Work of Richard Borch ...
, it also involves
generalized Kac–Moody algebra
In mathematics, a generalized Kac–Moody algebra is a Lie algebra that is similar to a Kac–Moody algebra, except that it is allowed to have imaginary simple roots.
Generalized Kac–Moody algebras are also sometimes called GKM algebras, Borcher ...
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.
See also
*
Manin–Drinfeld theorem
*
Moduli stack of elliptic curves In mathematics, the moduli stack of elliptic curves, denoted as \mathcal_ or \mathcal_, is an algebraic stack over \text(\mathbb) classifying elliptic curves. Note that it is a special case of the moduli stack of algebraic curves \mathcal_. In part ...
*
Modularity theorem
In number theory, the modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way. Andrew Wiles and Richard Taylor proved the modularity theorem for semistable elliptic c ...
*
Shimura variety, a generalization of modular curves to higher dimensions
References
* Steven D. Galbraith
Equations For Modular Curves
*
*
*
{{refend
Algebraic curves
Modular forms
Riemann surfaces