In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a modular form is a (complex)
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
on the
upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...
satisfying a certain kind of
functional equation with respect to the
group action of the
modular group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional ...
, and also satisfying a growth condition. The theory of modular forms therefore belongs to
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
but the main importance of the theory has traditionally been in its connections with
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
. Modular forms appear in other areas, such as
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
,
sphere packing
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three- dimensional Euclidean space. However, sphere pack ...
, and
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
.
A modular function is a function that is invariant with respect to the modular group, but without the condition that be
holomorphic in the upper half-plane (among other requirements). Instead, modular functions are
meromorphic (that is, they are holomorphic on the complement of a set of isolated points, which are poles of the function).
Modular form theory is a special case of the more general theory of
automorphic form
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G o ...
s which are functions defined on
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
s which transform nicely with respect to the action of certain
discrete subgroups, generalizing the example of the modular group
.
General definition of modular forms
In general, given a subgroup
of
finite index, called an
arithmetic group
In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example \mathrm_2(\Z). They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number the ...
, a modular form of level
and weight
is a holomorphic function
from the
upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...
such that the following two conditions are satisfied:
1. (automorphy condition) For any there is the equality[Some authors use different conventions, allowing an additional constant depending only on , see e.g. https://dlmf.nist.gov/23.15#E5]
2. (growth condition) For any the function is bounded for
where
In addition, it is called a cusp form if it satisfies the following growth condition:
3. (cuspidal condition) For any the function as
As sections of a line bundle
Modular forms can also be interpreted as sections of a specific
line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the '' tangent bundle'' is a way of organisi ...
s on
modular varieties. For
a modular form of level
and weight
can be defined as an element of
where
is a canonical line bundle on the
modular curveThe dimensions of these spaces of modular forms can be computed using the
Riemann–Roch theorem. The classical modular forms for
are sections of a line bundle on the
moduli stack of elliptic curves.
Modular forms for SL(2, Z)
Standard definition
A modular form of weight for the
modular group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional ...
:
is a
complex-valued
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 form ...
function on the
upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...
satisfying the following three conditions:
# is a
holomorphic function on .
# For any and any matrix in as above, we have:
#:
# is required to be bounded as .
Remarks:
* The weight is typically a positive integer.
* For odd , only the zero function can satisfy the second condition.
* The third condition is also phrased by saying that is "holomorphic at the cusp", a terminology that is explained below. Explicitly, the condition means that there exist some
such that
, meaning
is bounded above some horizontal line.
* The second condition for
::
:reads
::
:respectively. Since and
generate the modular group , the second condition above is equivalent to these two equations.
* Since , modular forms are
periodic function
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to des ...
s, with period , and thus have a
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
.
Definition in terms of lattices or elliptic curves
A modular form can equivalently be defined as a function ''F'' from the set of
lattices in to the set of
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 fo ...
s which satisfies certain conditions:
# If we consider the lattice generated by a constant and a variable , then is an
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
of .
# If is a non-zero complex number and is the lattice obtained by multiplying each element of by , then where is a constant (typically a positive integer) called the weight of the form.
# The
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
of remains bounded above as long as the absolute value of the smallest non-zero element in is bounded away from 0.
The key idea in proving the equivalence of the two definitions is that such a function is determined, because of the second condition, by its values on lattices of the form , where .
Examples
I. Eisenstein series
The simplest examples from this point of view are the
Eisenstein series. For each even integer , we define to be the sum of over all non-zero vectors of :
:
Then is a modular form of weight . For we have
:
and
:
.
The condition is needed for
convergence; for odd there is cancellation between and , so that such series are identically zero.
II. Theta functions of even unimodular lattices
An
even unimodular lattice in is a lattice generated by vectors forming the columns of a matrix of determinant 1 and satisfying the condition that the square of the length of each vector in is an even integer. The so-called
theta function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field ...
:
converges when Im(z) > 0, and as a consequence of the
Poisson summation formula can be shown to be a modular form of weight . It is not so easy to construct even unimodular lattices, but here is one way: Let be an integer divisible by 8 and consider all vectors in such that has integer coordinates, either all even or all odd, and such that the sum of the coordinates of is an even integer. We call this lattice . When , this is the lattice generated by the roots in the
root system called
E8. Because there is only one modular form of weight 8 up to scalar multiplication,
:
even though the lattices and are not similar.
John Milnor observed that the 16-dimensional
tori obtained by dividing by these two lattices are consequently examples of
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
s which are
isospectral but not
isometric (see
Hearing the shape of a drum.)
III. The modular discriminant
The
Dedekind eta function is defined as
:
where ''q'' is the square of the
nome. Then the
modular discriminant is a modular form of weight 12. The presence of 24 is related to the fact that the
Leech lattice has 24 dimensions.
A celebrated conjecture of
Ramanujan asserted that when is expanded as a power series in q, the coefficient of for any prime has absolute value . This was confirmed by the work of
Eichler,
Shimura,
Kuga,
Ihara, and
Pierre Deligne as a result of Deligne's proof of the
Weil conjectures, which were shown to imply Ramanujan's conjecture.
The second and third examples give some hint of the connection between modular forms and classical questions in number theory, such as representation of integers by
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
s and the
partition function. The crucial conceptual link between modular forms and number theory is furnished by the theory of
Hecke operators, which also gives the link between the theory of modular forms and
representation theory.
Modular functions
When the weight ''k'' is zero, it can be shown using
Liouville's theorem that the only modular forms are constant functions. However, relaxing the requirement that ''f'' be holomorphic leads to the notion of ''modular functions''. A function ''f'' : H → C is called modular
iff it satisfies the following properties:
# ''f'' is
meromorphic in the open
upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...
''H''.
# For every integer
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
in the
modular group ,
.
# As pointed out above, the second condition implies that ''f'' is periodic, and therefore has a
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
. The third condition is that this series is of the form
::
It is often written in terms of
(the square of the
nome), as:
::
This is also referred to as the ''q''-expansion of ''f''. The coefficients
are known as the Fourier coefficients of ''f'', and the number ''m'' is called the order of the pole of ''f'' at i∞. This condition is called "meromorphic at the cusp", meaning that only finitely many negative-''n'' coefficients are non-zero, so the ''q''-expansion is bounded below, guaranteeing that it is meromorphic at ''q'' = 0.
Sometimes a weaker definition of modular functions is used – under the alternative definition, it is sufficient that ''f'' be meromorphic in the open upper half-plane and that ''f'' be invariant with respect to a sub-group of the modular group of finite index. This is not adhered to in this article.
Another way to phrase the definition of modular functions is to use
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. I ...
s: every lattice Λ determines an
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. I ...
C/Λ over C; two lattices determine
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
elliptic curves if and only if one is obtained from the other by multiplying by some non-zero complex number . Thus, a modular function can also be regarded as a meromorphic function on the set of isomorphism classes of elliptic curves. For example, the
j-invariant ''j''(''z'') of an elliptic curve, regarded as a function on the set of all elliptic curves, is a modular function. More conceptually, modular functions can be thought of as functions on the
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such sp ...
of isomorphism classes of complex elliptic curves.
A modular form ''f'' that vanishes at (equivalently, , also paraphrased as ) is called a ''
cusp form'' (''Spitzenform'' in
German
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ge ...
). The smallest ''n'' such that is the order of the zero of ''f'' at .
A ''
modular unit In mathematics, modular units are certain units of rings of integers of fields of modular functions, introduced by . They are functions whose zeroes and poles are confined to the cusps (images of infinity).
See also
*Cyclotomic unit In mathemati ...
'' is a modular function whose poles and zeroes are confined to the cusps.
Modular forms for more general groups
The functional equation, i.e., the behavior of ''f'' with respect to
can be relaxed by requiring it only for matrices in smaller groups.
The Riemann surface ''G''\H∗
Let be a subgroup of that is of finite
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
. Such a group
acts on H in the same way as . The
quotient topological space ''G''\H can be shown to be a
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...
. Typically it is not compact, but can be compactified by adding a finite number of points called ''cusps''. These are points at the boundary of H, i.e. in
Q∪, such that there is a parabolic element of (a matrix with
trace ±2) fixing the point. This yields a compact topological space ''G''\H
∗. What is more, it can be endowed with the structure of 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 ver ...
, which allows one to speak of holo- and meromorphic functions.
Important examples are, for any positive integer ''N'', either one of the
congruence subgroups
:
For ''G'' = Γ
0(''N'') or , the spaces ''G''\H and ''G''\H
∗ are denoted ''Y''
0(''N'') and ''X''
0(''N'') and ''Y''(''N''), ''X''(''N''), respectively.
The geometry of ''G''\H
∗ can be understood by studying
fundamental domain
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
s for ''G'', i.e. subsets ''D'' ⊂ H such that ''D'' intersects each orbit of the -action on H exactly once and such that the closure of ''D'' meets all orbits. For example, the
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
of ''G''\H
∗ can be computed.
Definition
A modular form for of weight ''k'' is a function on H satisfying the above functional equation for all matrices in , that is holomorphic on H and at all cusps of . Again, modular forms that vanish at all cusps are called cusp forms for . The C-vector spaces of modular and cusp forms of weight ''k'' are denoted and , respectively. Similarly, a meromorphic function on ''G''\H
∗ is called a modular function for . In case ''G'' = Γ
0(''N''), they are also referred to as modular/cusp forms and functions of ''level'' ''N''. For , this gives back the afore-mentioned definitions.
Consequences
The theory of Riemann surfaces can be applied to ''G''\H
∗ to obtain further information about modular forms and functions. For example, the spaces and are finite-dimensional, and their dimensions can be computed thanks to the
Riemann–Roch theorem in terms of the geometry of the -action on H. For example,
:
where
denotes the
floor function and
is even.
The modular functions constitute the
field of functions of the Riemann surface, and hence form a field of
transcendence degree one (over C). If a modular function ''f'' is not identically 0, then it can be shown that the number of zeroes of ''f'' is equal to the number of
poles of ''f'' in the
closure of the
fundamental region ''R''
Γ.It can be shown that the field of modular function of level ''N'' (''N'' ≥ 1) is generated by the functions ''j''(''z'') and ''j''(''Nz'').
Line bundles
The situation can be profitably compared to that which arises in the search for functions on the
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
P(''V''): in that setting, one would ideally like functions ''F'' on the vector space ''V'' which are polynomial in the coordinates of ''v'' ≠ 0 in ''V'' and satisfy the equation ''F''(''cv'') = ''F''(''v'') for all non-zero ''c''. Unfortunately, the only such functions are constants. If we allow denominators (rational functions instead of polynomials), we can let ''F'' be the ratio of two
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
polynomials of the same degree. Alternatively, we can stick with polynomials and loosen the dependence on ''c'', letting ''F''(''cv'') = ''c''
''k''''F''(''v''). The solutions are then the homogeneous polynomials of degree . On the one hand, these form a finite dimensional vector space for each ''k'', and on the other, if we let ''k'' vary, we can find the numerators and denominators for constructing all the rational functions which are really functions on the underlying projective space P(''V'').
One might ask, since the homogeneous polynomials are not really functions on P(''V''), what are they, geometrically speaking? The
algebro-geometric answer is that they are ''sections'' of a
sheaf (one could also say a
line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the '' tangent bundle'' is a way of organisi ...
in this case). The situation with modular forms is precisely analogous.
Modular forms can also be profitably approached from this geometric direction, as sections of line bundles on the moduli space of elliptic curves.
Rings of modular forms
For a subgroup of the , the ring of modular forms is the
graded ring generated by the modular forms of . In other words, if be the ring of modular forms of weight , then the ring of modular forms of is the graded ring
.
Rings of modular forms of congruence subgroups of are finitely generated due to a result of
Pierre Deligne and
Michael Rapoport
Michael Rapoport (born 2 October 1948) is an Austrian mathematician.
Career
Rapoport received his PhD from Paris-Sud 11 University in 1976, under the supervision of Pierre Deligne. He held a chair for arithmetic algebraic geometry at the Unive ...
. Such rings of modular forms are generated in weight at most 6 and the relations are generated in weight at most 12 when the congruence subgroup has nonzero odd weight modular forms, and the corresponding bounds are 5 and 10 when there are no nonzero odd weight modular forms.
More generally, there are formulas for bounds on the weights of generators of the ring of modular forms and its relations for arbitrary
Fuchsian groups.
Types
Entire forms
If ''f'' is
holomorphic at the cusp (has no pole at ''q'' = 0), it is called an entire modular form.
If ''f'' is meromorphic but not holomorphic at the cusp, it is called a non-entire modular form. For example, the
j-invariant is a non-entire modular form of weight 0, and has a simple pole at i∞.
New forms
New forms are a subspace of modular forms
of a fixed weight
which cannot be constructed from modular forms of lower weights
dividing
. The other forms are called old forms. These old forms can be constructed using the following observations: if
then
giving a reverse inclusion of modular forms
.
Cusp forms
A
cusp form is a modular form with a zero constant coefficient in its Fourier series. It is called a cusp form because the form vanishes at all cusps.
Generalizations
There are a number of other usages of the term "modular function", apart from this classical one; for example, in the theory of
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
This measure was introduced by Alfréd Haar in 1933, thou ...
s, it is a function determined by the conjugation action.
Maass forms are
real-analytic eigenfunctions of the
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
but need not be
holomorphic. The holomorphic parts of certain weak Maass wave forms turn out to be essentially Ramanujan's
mock theta function
In mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock theta function is essentially a mock modular form of weight . The first examples of mock theta functions were described by Srinivasa Ramanuj ...
s. Groups which are not subgroups of can be considered.
Hilbert modular form
In mathematics, a Hilbert modular form is a generalization of modular forms to functions of two or more variables. It is a (complex) analytic function on the ''m''-fold product of upper half-planes \mathcal satisfying a certain kind of functional ...
s are functions in ''n'' variables, each a complex number in the upper half-plane, satisfying a modular relation for 2×2 matrices with entries in a
totally real number field.
Siegel modular form
In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional ''elliptic'' modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular for ...
s are associated to larger
symplectic groups in the same way in which classical modular forms are associated to ; in other words, they are related to
abelian varieties in the same sense that classical modular forms (which are sometimes called ''elliptic modular forms'' to emphasize the point) are related to elliptic curves.
Jacobi forms are a mixture of modular forms and elliptic functions. Examples of such functions are very classical - the Jacobi theta functions and the Fourier coefficients of Siegel modular forms of genus two - but it is a relatively recent observation that the Jacobi forms have an arithmetic theory very analogous to the usual theory of modular forms.
Automorphic form
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G o ...
s extend the notion of modular forms to general
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
s.
Modular integrals of weight are meromorphic functions on the upper half plane of moderate growth at infinity which ''fail to be modular of weight '' by a rational function.
Automorphic factors are functions of the form
which are used to generalise the modularity relation defining modular forms, so that
:
The function
is called the nebentypus of the modular form. Functions such as the
Dedekind eta function, a modular form of weight 1/2, may be encompassed by the theory by allowing automorphic factors.
History
The theory of modular forms was developed in four periods: first in connection with the theory of
elliptic functions, in the first part of the nineteenth century; then by
Felix Klein and others towards the end of the nineteenth century as the automorphic form concept became understood (for one variable); then by
Erich Hecke from about 1925; and then in the 1960s, as the needs of number theory and the formulation of the
modularity theorem in particular made it clear that modular forms are deeply implicated.
The term "modular form", as a systematic description, is usually attributed to Hecke.
Notes
References
*
* ''Leads up to an overview of the proof of the
modularity theorem''.
*. ''Provides an introduction to modular forms from the point of view of representation theory''.
*
*
*
*. ''Chapter VII provides an elementary introduction to the theory of modular forms''.
*. ''Provides a more advanced treatment.''
*
See also
*
Wiles's proof of Fermat's Last Theorem
{{Authority control
Analytic number theory
Special functions