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Modular form
From Wikipedia
In mathematics, a modular form is an 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 with number theory. Modular forms appear in other areas, such as algebraic topology and string theory. fish is good Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, an analytic function is one that is locally given by a convergent power series. ... In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ... In mathematics or its applications, a functional equation is an equation in terms of independent variables, and also unknown functions, which are to be solved for. ... Complex analysis is the branch of mathematics investigating holomorphic functions, i. ... Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... String theory is a physical model whose fundamental building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles) that were the basis of most earlier physics. ...
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. 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 Taniyama-Shimura conjecture in particular made it clear that modular forms are deeply implicated. In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms. ...
In mathematics, a discrete group is a group G equipped with the discrete topology. ...
In complex analysis, an elliptic function is, roughly speaking , a function defined on the complex plane which is periodic in two directions. ...
Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ...
Felix Christian Klein (April 25, 1849 – June 22, 1925) was a German mathematician. ...
Erich Hecke (September 20, 1887 – February 13, 1947) was a German mathematician. ...
The Taniyama-Shimura theorem establishes an important connection between elliptic curves, which are objects from algebraic geometry, and modular forms, which are certain periodic holomorphic functions investigated in number theory. ...
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; the deep studies made on the particular cusp form highlighted by Srinivasa Ramanujan often do not use the modern term. A modular function is in practical terms a modular form of weight 0; but to be strictly accurate modular functions are meromorphic functions rather than analytic. G. H. Hardy Godfrey Harold Hardy (February 7, 1877 – December 1, 1947) was a prominent British mathematician, known for his achievements in number theory and mathematical analysis. ...
In number theory, a cusp form is a particular kind of modular form, distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion of the constant coefficient a0. ...
Ramanujan Srinivasa Aiyangar Ramanujan (Tamil: ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (December 22, 1887 – April 26, 1920) was a groundbreaking Indian mathematician. ...
In mathematics, modular functions are certain kinds of mathematical functions mapping complex numbers to complex numbers. ...
A meromorphic function is a function that is holomorphic on an open subset of the complex number plane C (or on some other connected Riemann surface) except at points in a set of isolated poles, which are certain well-behaved singularities. ...
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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 functions which satisfies certain conditions: In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. ... Complex analysis is the branch of mathematics investigating holomorphic functions, i. ...
- (1) If we consider the lattice Λ = <α, z> generated by a constant α and a variable z, then F(Λ) is an analytic function of z.
- (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) = α−kF(Λ) where k is a constant (typically a positive integer) called the weight of the form.
- (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.
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 graph of the absolute value function In mathematics, the absolute value (or modulus) of a real number is its numerical value without regard to its sign. ... Several equivalence relations in mathematics are called similarity. ...
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. 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. 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). In mathematics, a projective space is a fundamental construction from any vector space. ...
Homogeneous is an adjective that has several meanings. ...
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). The situation with modular forms is precisely analogous. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...
In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain...
In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ...
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. Modular functions can be thought of as functions on the moduli space of isomorphism classes of complex elliptic curves. For example, the j-invariant of an elliptic curve, regarded as a function on the set of all elliptic curves, is modular. Modular forms can also be profitably approached from this geometric direction, as sections of line bundles on the moduli space of elliptic curves. In mathematics, an elliptic curve is a non-singular projective algebraic curve of genus 1 over a field K, together with a distinguished point defined over K. A more accessible (though less accurate) definition is that an elliptic curve is a plane curve defined by an equation of the form... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In algebraic geometry, the moduli problem is to describe the parameters on which algebraic varieties depend. ... In mathematics, Kleins j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half plane of complex numbers with positive imaginary part. ...
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 In mathematics or its applications, a functional equation is an equation in terms of independent variables, and also unknown functions, which are to be solved for. ...
for a, b, c, d integers with ad − bc = 1 (the modular group). For example, In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ...
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.)
General definitions
Let N be a positive integer. The modular group Γ0(N) is defined as In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. ...
Let k be a positive integer. A 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 Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ...
and any z in the upper half plane, we have In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ...
and f is holomorphic at the cusp. By "holomorphic at the cusp", it is meant that the modular form is holomorphic as , or equivalently, has a Fourier series Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In common parlance, a cusp is an important moment usually regarded as a decision point upon which consequent events are determined. ... In mathematics, a Fourier series, named in honor of Joseph Fourier (1768-1830), is a representation of a periodic function with period 2π as a sum of periodic functions of the form which are the harmonics of ei x. ...
where x = exp(2πiz) is the square of the nome. Such a form, having no pole at x=0, is sometimes called an entire modular form. If c(0)=0, then the form is called a cusp form (Spitzenform in German). The smallest n such that is called the order of the zero of f at . More general treatments allow poles at x=0; thus, for example, the j-invariant is a non-entire modular form of weight 0, because it has a simple pole at . Nome refers to several things: The town of Nome, Norway The town of Nome, Alaska, USA Nome Census Area, Alaska, USA A subnational division (see Nome (subnational division): in Greece (see Nome (Greece)) in Ancient Egypt (see Nome (Egypt)) In mathematics, the Nome (mathematics). ... In number theory, a cusp form is a particular kind of modular form, distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion of the constant coefficient a0. ... In mathematics, Kleins j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half plane of complex numbers with positive imaginary part. ...
Other common generalizations allow the weight k to not be an integer, and allow a multiplier ε(a,b,c,d) with to appear in the transformation
This allows functions such as the Dedekind eta function to be encompassed by the theory, being a modular form of weight 1/2. Thus, for example, let χ be a Dirichlet character mod N. 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 The Dedekind eta function is a function defined on the upper half plane of complex numbers whose imaginary part is positive. ... In number theory, a Dirichlet character is a function χ from the positive integers to the complex numbers which has the following properties: There exists a positive integer k such that χ(n) = χ(n + k) for all n. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ...
and any z in the upper half plane, we have In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ...
and f is holomorphic at the cusp. Sometimes the convention Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In common parlance, a cusp is an important moment usually regarded as a decision point upon which consequent events are determined. ...
- 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: For each even integer k > 2 we define Ek(Λ) to be the sum of λ−k over all non-zero vectors λ of Λ (the condition k > 2 is needed for convergence and the condition k is even to prevent λ−k from cancelling with (−λ)−k and producing the 0 form.) In mathematics, Eisenstein series are particular modular forms with infinite series expansions that may be written down directly. ...
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. As a consequence of the Poisson summation formula, the theta function In mathematics, a unimodular lattice is a lattice of discriminant 1 or −1. ...
In mathematics, the Poisson summation formula is a relation holding between a sum of a function F over all integers, and a corresponding sum for the Fourier transform G. If the normalization of the Fourier transform is correctly adjusted, it takes the form Σ F(n) = Σ G(n). ...
In mathematics, theta functions are special functions of several complex variables. ...
is a modular form of weight n/2. 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. Because both sides of the equation are modular forms of weight 8, and because there is only one modular form of weight 8 up to scalar multiplication, See also Simple Lie group. ... In mathematics, E8 is the name of a Lie group and also its Lie algebra . ...
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. John Willard Milnor (b. ... Geometry In geometry, a torus (pl. ... Several specialized usages of the terms compact and compactness exist. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In mathematics, two linear operators are called isospectral if they have the same spectrum. ... The term isometric comes from the Greek for having equal measurement and is a descriptive word associated with several topics: Isometric projection, a method for the visual representation of three-dimensional objects in two dimensions, is a form of orthographic projection, or more specifically, an axonometric projection. ...
The Dedekind eta function is defined as The Dedekind eta function is a function defined on the upper half plane of complex numbers whose imaginary part is positive. ...
Then the modular discriminant Δ(z)=η(z)24 is a modular form of weight 12. 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. In number theory, a cusp form is a particular kind of modular form, distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion of the constant coefficient a0. ... Ramanujan Srinivasa Aiyangar Ramanujan (Tamil: ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (December 22, 1887 – April 26, 1920) was a groundbreaking Indian mathematician. ... Pierre Deligne (born 3 October 1944) is a Belgian mathematician. ... In mathematics, the Weil conjectures, which had become theorems by 1975, were some highly-influential proposals from the late 1940s by Andre Weil on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields. ...
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. 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, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...
In number theory, the partition function p(n) represents the number of possible partitions of a natural number n, which is to say the number of distinct (and order independent) ways of representing n as a sum of natural numbers. ...
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 of vector spaces of modular forms (and more general automorphic representations). ...
In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ...
Generalizations
There are various notions of modular form more general than the one discussed above. The assumption of analyticity can be dropped; Maass forms are eigenfunctions of the Laplacian but are not analytic. 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. 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. Automorphic forms extend the notion of modular forms to general Lie groups. In mathematics, an eigenfunction f of a linear operator A on a function space is an eigenvector of the linear operator; it satisfies for some scalar λ, the corresponding eigenvalue. ... In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ... In number theory , a number field K is called totally real if for each embedding of K into the complex numbers the image lies inside the real numbers. ... 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, abelian variety is a term used to denote a complex torus that can be embedded into projective space as a projective variety. ... In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...
References
- 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. 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
