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Hecke Eigenform
In mathematics, an eigenform (meaning simultaneous Hecke eigenform with modular group SL(2,Z)) is a modular form which is an eigenvector for all Hecke operators ''Tm'', ''m'' = 1, 2, 3, .... Eigenforms fall into the realm of number theory, but can be found in other areas of math and science such as analysis, combinatorics, and physics. A common example of an eigenform, and the only non-cuspidal eigenforms, are the Eisenstein series. Another example is the Δ function. Normalization There are two different normalizations for an eigenform (or for a modular form in general). Algebraic normalization An eigenform is said to be normalized when scaled so that the ''q''-coefficient in its Fourier series is one: :f = a_0 + q + \sum_^\infty a_i q^i where ''q'' = ''e''2''πiz''. As the function ''f'' is also an eigenvector under each Hecke operator ''Ti'', it has a corresponding eigenvalue. More specifically ''a''''i'', ''i'' ≥ 1 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Form
In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modular forms has origins in complex analysis, with important connections with number theory. Modular forms also appear in other areas, such as algebraic topology, sphere packing, and string theory. Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group \mathrm_2(\mathbb Z) \subset \mathrm_2(\mathbb R). Every modular form is attached to a Galois representation. The term "modular form", as a systematic description, is usually attributed to Erich Hecke. The importance of modular forms across multiple field of mathematics has been humorously re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvector
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a constant factor \lambda when the linear transformation is applied to it: T\mathbf v=\lambda \mathbf v. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor \lambda (possibly a negative or complex number). Geometrically, vectors are multi- dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. A linear transformation's eigenvectors are those vectors that are only stretched or shrunk, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or shrunk. If the eigenvalue is negative, the eigenvector's direction is reversed. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hecke Operator
In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations. History used Hecke operators on modular forms in a paper on the special cusp form of Ramanujan, ahead of the general theory given by . Mordell proved that the Ramanujan tau function, expressing the coefficients of the Ramanujan form, : \Delta(z)=q\left(\prod_^(1-q^n)\right)^= \sum_^ \tau(n)q^n, \quad q=e^, is a multiplicative function: : \tau(mn)=\tau(m)\tau(n) \quad \text (m,n)=1. The idea goes back to earlier work of Adolf Hurwitz, who treated algebraic correspondences between modular curves which realise some individual Hecke operators. Mathematical description Hecke operators can be realized in a number of contexts. The simplest meaning is combinatorial, namely as taking for a given integer some funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analysis
Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384–322 BC), though ''analysis'' as a formal concept is a relatively recent development. The word comes from the Ancient Greek (''analysis'', "a breaking-up" or "an untying" from ''ana-'' "up, throughout" and ''lysis'' "a loosening"). From it also comes the word's plural, ''analyses''. As a formal concept, the method has variously been ascribed to René Descartes ('' Discourse on the Method''), and Galileo Galilei. It has also been ascribed to Isaac Newton, in the form of a practical method of physical discovery (which he did not name). The converse of analysis is synthesis: putting the pieces back together again in a new or different whole. Science and technology Chemistry The field of chemistry uses analysis in three ways: to i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." It is one of the most fundamental scientific disciplines. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eisenstein Series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generalized in the theory of automorphic forms. Eisenstein series for the modular group Let be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series of weight , where is an integer, by the following series: :G_(\tau) = \sum_ \frac. This series absolutely converges to a holomorphic function of in the upper half-plane and its Fourier expansion given below shows that it extends to a holomorphic function at . It is a remarkable fact that the Eisenstein series is a modular form. Indeed, the key property is its -covariance. Explicitly if and then :G_ \left( \frac \right) = (c\tau +d)^ G_(\tau) Note that is necessary such that the series converges absolutely, whereas needs to be even otherwis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass's Elliptic Functions
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions is also referred to as ℘-functions and they are usually denoted by the symbol ℘, a uniquely fancy script ''p''. They play an important role in the theory of elliptic functions, i.e., meromorphic functions that are doubly periodic. A ℘-function together with its derivative can be used to parameterize elliptic curves and they generate the field of elliptic functions with respect to a given period lattice. Symbol for Weierstrass \wp-function Motivation A cubic of the form C_^\mathbb=\ , where g_2,g_3\in\mathbb are complex numbers with g_2^3-27g_3^2\neq0, cannot be rationally parameterized. Yet one still wants to find a way to parameterize it. For the quadric K=\left\; the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Series
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always Convergent series, converge. Well-behaved functions, for example Smoothness, smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Petersson Inner Product
In mathematics the Petersson inner product is an inner product defined on the space of entire modular forms. It was introduced by the German mathematician Hans Petersson. Definition Let \mathbb_k be the space of entire modular forms of weight k and \mathbb_k the space of cusp forms. The mapping \langle \cdot , \cdot \rangle : \mathbb_k \times \mathbb_k \rightarrow \mathbb, :\langle f , g \rangle := \int_\mathrm f(\tau) \overline (\operatorname\tau)^k d\nu (\tau) is called Petersson inner product, where :\mathrm = \left\ is a fundamental region of the modular group \Gamma and for \tau = x + iy :d\nu(\tau) = y^dxdy is the hyperbolic volume form. Properties The integral is absolutely convergent and the Petersson inner product is a positive definite Hermitian form. For the Hecke operators T_n, and for forms f,g of level \Gamma_0, we have: :\langle T_n f , g \rangle = \langle f , T_n g \rangle This can be used to show that the space of cusp forms of level \Gamma_0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 the upper-half of the complex plane by linear fractional transformations. The name "modular group" comes from the relation to moduli spaces, and not from modular arithmetic. Definition The modular group is the group of fractional linear transformations of the complex upper half-plane, which have the form :z\mapsto\frac, where a,b,c,d are integers, and ad-bc=1. The group operation is function composition. This group of transformations is isomorphic to the projective special linear group \operatorname(2,\mathbb Z), which is the quotient of the 2-dimensional special linear group \operatorname(2,\mathbb Z) by its center \. In other words, \operatorname(2,\mathbb Z) consists of all matrices :\begin a & b \\ c & d \end where a,b, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
