In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and This article is about both real and complex analytic functions In Mathematics, the upper half-plane H is the set of Complex numbers \mathbb{H} = \{x + iy \| y > 0 x y \in \mathbb{R} \} In Mathematics or its applications a functional equation is an Equation expressing a relation between the value of a function (or functions at a point with its values The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Modular forms appear in other areas, such as algebraic topology and string theory. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings

A modular function is a modular form of weight 0: it is invariant under the modular group, instead of transforming in a prescribed way, and is thus a function on the modular region (rather than a section of a line bundle). In Mathematics, a line bundle expresses the concept of a line that varies from point to point of a space

Modular form theory is a special case of the more general theory of automorphic forms, and therefore can now be seen as just the most concrete part of a rich theory of discrete groups. In Mathematics, the general notion of automorphic form is the extension to Analytic functions perhaps of Several complex variables, of the theory of In Mathematics, a discrete group is a group G equipped with the Discrete topology.

## As a function on lattices

A modular form can be thought of as a function F from the set of lattices Λ in C to the set of complex numbers which satisfies certain conditions:

(1) If we consider the lattice $\Lambda = \langle \alpha, z\rangle$ generated by a constant α and a variable z, then F(Λ) is an analytic function of z. In Mathematics, a fundamental pair of periods is an Ordered pair of Complex numbers that define a lattice in the Complex plane. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted This article is about both real and complex analytic functions
(2) If α is a non-zero complex number and αΛ is the lattice obtained by multiplying each element of Λ by α, then F(αΛ) = αkF(Λ) where k is a constant (typically a positive integer) called the weight of the form. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted
(3) The absolute value of F(Λ) remains bounded above as long as the absolute value of the smallest non-zero element in Λ is bounded away from 0. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.

When k = 0, condition 2 says that F depends only on the similarity class of the lattice. This is a very important special case, but the only modular forms of weight 0 are the constants. If we eliminate condition 3 and allow the function to have poles, then weight 0 examples exist: they are called modular functions.

The situation can be profitably compared to that which arises in the search for functions on the projective space 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. In Mathematics a projective space is a set of elements constructed from a vector space such that a distinct element of the projective space consists of all non-zero vectors which 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 polynomials of the same degree. In Mathematics, a homogeneous function is a function with multiplicative scaling behaviour if the argument is multiplied by a factor then the result is multiplied by some power Alternatively, we can stick with polynomials and loosen the dependence on c, letting F(cv) = ckF(v). The solutions are then the homogeneous polynomials of degree k. 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 this case). Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. In Mathematics, a vector bundle is a topological construction which makes precise the idea of a family of Vector spaces parameterized by another space The situation with modular forms is precisely analogous.

## As a function on elliptic curves

Every lattice Λ in C determines an elliptic curve C/Λ over C; two lattices determine isomorphic elliptic curves if and only if one is obtained from the other by multiplying by some α. In Mathematics, an elliptic curve is a smooth, projective Algebraic curve of genus one on which there is a specified point O In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective Modular functions can be thought of as functions on the moduli space of isomorphism classes of complex elliptic curves. In Algebraic geometry, a moduli space is a geometric space (usually a scheme or an Algebraic stack) whose points represent algebro-geometric objects of For example, the j-invariant of an elliptic curve, regarded as a function on the set of all elliptic curves, is modular. In Mathematics, Klein's j -invariant, regarded as a function of a complex variable &tau is a Modular function defined on the Modular forms can also be profitably approached from this geometric direction, as sections of line bundles on the moduli space of elliptic curves.

To convert a modular form F into a function of a single complex variable is easy. Let z = x + iy, where y > 0, and let f(z) = F(<1, z>). (We cannot allow y = 0 because then 1 and z will not generate a lattice, so we restrict attention to the case that y is positive. ) Condition 2 on F now becomes the functional equation

$f\left({az+b\over cz+d}\right) = (cz+d)^k f(z)$

for a, b, c, d integers with adbc = 1 (the modular group). In Mathematics or its applications a functional equation is an Equation expressing a relation between the value of a function (or functions at a point with its values In Mathematics, the modular group Γ is a fundamental object of study in Number theory, Geometry, algebra, and many other areas of advanced For example,

$f(-1/z) = F(\langle 1,-1/z\rangle) = z^k F(\langle z,-1\rangle) = z^k F(\langle 1,z\rangle) = z^k f(z).$

Functions which satisfy the modular functional equation for all matrices in a finite index subgroup of SL2(Z) are also counted as modular, usually with a qualifier indicating the group. Thus modular forms of level N (see below) satisfy the functional equation for matrices congruent to the identity matrix modulo N (often in fact for a larger group given by (mod N) conditions on the matrix entries. )

## Modular functions

In mathematics, modular functions are certain kinds of mathematical functions mapping complex numbers to complex numbers. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted There are a number of other uses of the term "modular function" as well; see below for details.

Formally, a function f is called modular or a modular function iff it satisfies the following properties:

1. f is meromorphic in the open upper half-plane H. In Complex analysis, a meromorphic function on an open subset D of the Complex plane is a function that is holomorphic In Mathematics, the upper half-plane H is the set of Complex numbers \mathbb{H} = \{x + iy \| y > 0 x y \in \mathbb{R} \}
2. For every matrix M in the modular group Γ, f(Mτ) = f(τ). In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, the modular group Γ is a fundamental object of study in Number theory, Geometry, algebra, and many other areas of advanced
3. The Fourier series of f has the form
$f(\tau) = \sum_{n=-m}^\infty a(n) e^{2i\pi n\tau}.$

It is bounded below; it is a Laurent polynomial in e2iπτ, so it is meromorphic at the cusp. In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions In Mathematics, a Laurent polynomial in one variable over a ring R is a Linear combination of positive and negative powers of the variable with

It can be shown that every modular function can be expressed as a rational function of Klein's absolute invariant j(τ), and that every rational function of j(τ) is a modular function; furthermore, all analytic modular functions are modular forms, although the converse does not hold. In Mathematics, a rational function is any function which can be written as the Ratio of two Polynomial functions Definitions In In Mathematics, Klein's j -invariant, regarded as a function of a complex variable &tau is a Modular function defined on the This article is about both real and complex analytic functions 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 RΓ. In Complex analysis, a pole of a Meromorphic function is a certain type of singularity that behaves like the singularity at z = 0 In Mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set In Geometry, the fundamental domain of a Symmetry group of an object or pattern is a part of the pattern as small as possible which based on the Symmetry

### Other uses

There are a number of other usages of the term modular function, apart from this classical one; for example, in the theory of Haar measures, it is a function Δ(g) determined by the conjugation action. In Mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of Locally compact topological groups and subsequently define

## General definitions

Let N be a positive integer. The modular group Γ0(N) is defined as

$\Gamma_0(N) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathbf{Z}) :c \equiv 0 \pmod{N} \right\}.$

Let k be a positive integer. In Mathematics, a congruence subgroup of a Matrix group with Integer entries is a Subgroup defined by congruence conditions on the entries An modular form of weight k with level N (or level group Γ0(N)) is a holomorphic function f on the upper half-plane such that for any

$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_0(N)$

and any z in the upper half-plane, we have

$f\left(\frac{az+b}{cz+d}\right) = (cz+d)^k f(z)$

and f is meromorphic at the cusp. Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane In Mathematics, the upper half-plane H is the set of Complex numbers \mathbb{H} = \{x + iy \| y > 0 x y \in \mathbb{R} \} In Mathematics, the upper half-plane H is the set of Complex numbers \mathbb{H} = \{x + iy \| y > 0 x y \in \mathbb{R} \} In Complex analysis, a meromorphic function on an open subset D of the Complex plane is a function that is holomorphic By "meromorphic at the cusp", it is meant that the modular form is meromorphic as $z\rightarrow i\infty$.

Note that $f\left(z+1\right) = f(z)$, so modular forms are periodic, with period 1, and thus have a Fourier series.

### q-expansion

The q-expansion[1] of a modular form is the Laurent series at the cusp. Equivalently, the Fourier series, written as a Laurent series in terms of q = exp(2πiz) (the square of the nome). In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions In Mathematics, the Laurent series of a complex function f ( z) is a representation of that function as a Power series which includes terms In Mathematics, specifically the theory of Elliptic functions, the nome is a Special function and is given by q \exp(-\pi

Since exp is non-vanishing, $q \neq 0$ on the complex plane, but in the limit, $\exp(w) \to 0$ as $w \to -\infty$ (along the negative real axis), so $q \to 0$ as $2\pi iz \to -\infty$, so as $z \to i\infty$ (along the positive imaginary axis) — thus the q-expansion is the Laurent series expansion at the cusp.

"Meromorphic at the cusp" means that only finitely many negative Fourier coefficients are non-zero, so the q-expansion is bounded below, and meromorphic at q = 0:

$f(z)=\sum_{n=-m}^\infty c_n \exp(2\pi inz) = \sum_{n=-m}^\infty c_n q^n$

The coefficients cn are the Fourier coefficients of f, and the number m is the order of the pole of f at $i\infty$.

### Entire forms, cusp forms

If f is holomorphic at the cusp (has no pole at q = 0), it is called an entire modular form. Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane

If f is meromorphic but not holomorphic at the cusp, it is called non-entire modular form. For example, the j-invariant is a non-entire modular form of weight 0, and has a simple pole at $i\infty$. In Mathematics, Klein's j -invariant, regarded as a function of a complex variable &tau is a Modular function defined on the

If f is entire and vanishes at q = 0 (so c0 = 0), the form is called a cusp form (Spitzenform in German). In Number theory, a branch of Mathematics, a cusp form is a particular kind of Modular form, distinguished in the case of modular forms for the Modular The smallest n such that $c_n \ne 0$ is the order of the zero of f at $i\infty$.

### Automorphic factors and other generalizations

Other common generalizations allow the weight k to not be an integer, and allow a multiplier ε(a,b,c,d) with $\left|\epsilon(a,b,c,d)\right|=1$ to appear in the transformation, so that

$f\left(\frac{az+b}{cz+d}\right) = \epsilon(a,b,c,d) (cz+d)^k f(z).$

Functions of the form ε(a,b,c,d)(cz + d)k are known as automorphic factors. In Mathematics, an automorphic factor is a certain type of Analytic function, defined on Subgroups of SL(2R, appearing in the theory of

By allowing automorphic factors, functions such as the Dedekind eta function may be encompassed by the theory, being a modular form of weight 1/2. The Dedekind eta function, named after Richard Dedekind, is a function defined on the Upper half-plane of Complex numbers whose imaginary part is positive Thus, for example, let χ be a Dirichlet character mod N. In Number theory, Dirichlet characters are certain Arithmetic functions which arise from Completely multiplicative characters on the units of A modular form of weight k, level N (or level group Γ0(N)) with nebentypus χ is a holomorphic function f on the upper half-plane such that for any

$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_0(N)$

and any z in the upper half-plane, we have

$f\left(\frac{az+b}{cz+d}\right) = \chi(d)(cz+d)^k f(z)$

and f is holomorphic at the cusp. Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane In Mathematics, the upper half-plane H is the set of Complex numbers \mathbb{H} = \{x + iy \| y > 0 x y \in \mathbb{R} \} In Mathematics, the upper half-plane H is the set of Complex numbers \mathbb{H} = \{x + iy \| y > 0 x y \in \mathbb{R} \} Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane Sometimes the convention

χ − 1(d)(cz + d)kf(z)

is used for the right hand side of the above equation. In Mathematics, LHS is informal shorthand for the left-hand side of an Equation.

## Examples

The simplest examples from this point of view are the Eisenstein series. This article describes holomorphic Eisenstein series for the non-holomorphic case see Real analytic Eisenstein series In Mathematics For each even integer k > 2, we define Ek(Λ) to be the sum of λk over all non-zero vectors λ of Λ:

$E_k(\Lambda) = \sum_{\lambda\in\Lambda-0}\lambda^{-k}.$

The condition k > 2 is needed for convergence; the condition that k is even prevents λk from cancelling with (−λ)k.

An even unimodular lattice L in Rn is a lattice generated by n vectors forming the columns of a matrix of determinant 1 and satisfying the condition that the square of the length of each vector in L is an even integer. In Mathematics, a unimodular lattice is a lattice of Discriminant 1 or &minus1 As a consequence of the Poisson summation formula, the theta function

$\vartheta_L(z) = \sum_{\lambda\in L}e^{\pi i \Vert\lambda\Vert^2 z}$

is a modular form of weight n/2. The Poisson summation formula is an equation relating the coefficients of the Fourier Series of the periodic extension of a function in terms of the values of the In Mathematics, theta functions are Special functions of Several complex variables. It is not so easy to construct even unimodular lattices, but here is one way: Let n be an integer divisible by 8 and consider all vectors v in Rn such that 2v has integer coordinates, either all even or all odd, and such that the sum of the coordinates of v is an even integer. We call this lattice Ln. When n=8, this is the lattice generated by the roots in the root system called E8. This article discusses root systems in mathematics For root systems of Plants see Root. Because there is only one modular form of weight 8 up to scalar multiplication,

$\vartheta_{L_8\times L_8}(z) = \vartheta_{L_{16}}(z),$

even though the lattices L8×L8 and L16 are not similar. John Milnor observed that the 16-dimensional tori obtained by dividing R16 by these two lattices are consequently examples of compact Riemannian manifolds which are isospectral but not isometric (see Hearing the shape of a drum. John Willard Milnor (b February 20, 1931 in Orange New Jersey) is an American Mathematician known for his work in Differential In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Mathematics, two Linear operators are called isospectral if they have the same spectrum. For the Mechanical engineering and Architecture usage see Isometric projection. To hear the shape of a drum is to infer information about the shape of the Drumhead from the sound it makes i )

The Dedekind eta function is defined as

$\eta(z) = q^{1/24}\prod_{n=1}^\infty (1-q^n),\ q = e^{2\pi i z}.$

Then the modular discriminant Δ(z)=η(z)24 is a modular form of weight 12. The Dedekind eta function, named after Richard Dedekind, is a function defined on the Upper half-plane of Complex numbers whose imaginary part is positive In Mathematics, Weierstrass's elliptic functions are Elliptic functions that take a particularly simple form (cf Jacobi's elliptic functions) they are named A celebrated conjecture of Ramanujan asserted that the qp coefficient for any prime p has absolute value ≤2p11/2. This was settled by Pierre Deligne as a result of his work on the Weil conjectures. Pierre René Viscount Deligne (born 3 October 1944 in Brussels) is a Belgian Mathematician. In Mathematics, the Weil conjectures, which had become theorems by 1974 were some highly-influential proposals from the late 1940s by André Weil on the

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 forms and the partition function. In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables In Number theory, a partition of a positive Integer n is a way of writing n as a Sum of positive integers The crucial conceptual link between modular forms and number theory are furnished by the theory of Hecke operators, which also gives the link between the theory of modular forms and representation theory. In Mathematics, in particular in the theory of Modular forms a Hecke operator is a certain kind of 'averaging' operator that plays a significant role in the structure In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of

## Generalizations

There are various notions of modular form more general than the one discussed above. The assumption of complex analyticity can be dropped; Maass forms are real-analytic eigenfunctions of the Laplacian but are not holomorphic. This article is about both real and complex analytic functions In Mathematics, an eigenfunction of a Linear operator, A, defined on some Function space is any non-zero function f in In Mathematics and Physics, the Laplace operator or Laplacian, denoted by \Delta\  or \nabla^2  and named after Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane Groups which are not subgroups of SL2(Z) can be considered. Hilbert modular forms 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. In Mathematics, a Hilbert modular form is a generalization of the Elliptic modular forms to functions of two or more variables In Number theory, a Number field K is called totally real if for each Embedding of K into the Complex numbers Siegel modular forms are associated to larger symplectic groups in the same way in which the forms we have discussed are associated to SL2(R); in other words, they are related to abelian varieties in the same sense that our forms (which are sometimes called elliptic modular forms to emphasize the point) are related to elliptic curves. In Mathematics, Siegel modular forms are a major type of Automorphic form. In Mathematics, the name symplectic group can refer to two different but closely related types of mathematical groups. In Mathematics, particularly in Algebraic geometry, Complex analysis and Number theory, an Abelian variety is a projective algebraic variety Automorphic forms extend the notion of modular forms to general Lie groups. In Mathematics, the general notion of automorphic form is the extension to Analytic functions perhaps of Several complex variables, of the theory of In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group

## History

It was developed, historically speaking, in three or four periods of development: in connection with the theory of elliptic functions, in the first part of the nineteenth century; by Felix Klein and others towards the end of the nineteenth century, as the automorphic form concept was understood (for one variable); by Erich Hecke from about 1925; and 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. In Complex analysis, an elliptic function is a function defined on the Complex plane which is periodic in two directions (a Doubly-periodic The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group Erich Hecke ( September 20, 1887 &ndash February 13, 1947) was a German Mathematician. In Mathematics, the modularity theorem establishes an important connection between Elliptic curves over the field of Rational numbers and Modular forms

The term modular form, as a systematic description, is usually attributed to Hecke. Curiously, G. H. Hardy is said to have banned it in his circle of students; for example, the deep studies made on the particular cusp form highlighted by Srinivasa Ramanujan often do not use the modern term. Godfrey Harold Hardy FRS ( February 7, 1877 Cranleigh, Surrey, England &ndash December 1, 1947 In Number theory, a branch of Mathematics, a cusp form is a particular kind of Modular form, distinguished in the case of modular forms for the Modular

## References

1. ^ Elliptic and Modular Functions
• Jean-Pierre Serre: A Course in Arithmetic. Graduate Texts in Mathematics 7, Springer-Verlag, New York, 1973. Chapter VII provides an elementary introduction to the theory of modular forms.
• Tom M. Apostol, Modular functions and Dirichlet Series in Number Theory (1990), Springer-Verlag, New York. ISBN 0-387-97127-0
• Goro Shimura: Introduction to the arithmetic theory of automorphic functions. Goro Shimura ( Japanese: 志村 五郎 Shimura Gorō; born 1930 in Hamamatsu Japan) is a Japanese Mathematician, and currently Princeton University Press, Princeton, N. J. , 1971. Provides a more advanced treatment.
• Stephen Gelbart: Automorphic forms on adele groups. Annals of Mathematics Studies 83, Princeton University Press, Princeton, N. J. , 1975. Provides an introduction to modular forms from the point of view of representation theory.
• Robert A. Rankin, Modular forms and functions, (1977) Cambridge University Press, Cambridge. ISBN 0-521-21212-X
• Stein's notes on Ribet's course Modular Forms and Hecke Operators

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