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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 forms are Siegel modular varieties, which are basic models for what a moduli space for abelian varieties (with some extra level structure) should be and are constructed as quotients of the Siegel upper half-space rather than the upper half-plane by discrete groups. Siegel modular forms are holomorphic functions on the set of symmetric ''n'' × ''n'' matrices with positive definite imaginary part; the forms must satisfy an automorphy condition. Siegel modular forms can be thought of as multivariable modular forms, i.e. as special functions of several complex variables. Siegel modular forms were first investigated by for the purpose of studying quadratic forms analytically. These primarily arise in various branches of number theory, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Geometry
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties. In more abstract terms, arithmetic geometry can be defined as the study of schemes of finite type over the spectrum of the ring of integers. Overview The classical objects of interest in arithmetic geometry are rational points: sets of solutions of a system of polynomial equations over number fields, finite fields, p-adic fields, or function fields, i.e. fields that are not algebraically closed excluding the real numbers. Rational points can be directly characterized by height functions which measure their arithmetic complexity. The structure of algebraic varieties defined over non-algebraically closed fields has become a central area of interest that arose with the modern abstract development of algebraic geometry. Over finite fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Miyawaki Lift
The Miyawaki lift or Ikeda–Miyawaki lift or Miyawaki–Ikeda lift, is a mathematical lift that takes two 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 form ...s to another Siegel modular form. Miyawaki conjectured the existence of this lift for the case of degree 3 Siegel modular forms, and Ikeda proved its existence in some cases using the Ikeda lift. Ikeda's construction starts with a Siegel modular form of degree 1 and weight 2''k'', and a Siegel cusp form of degree ''r'' and weight ''k'' + ''n'' + ''r'' and constructs a Siegel form of degree 2''n'' + ''r'' and weight ''k'' + ''n'' + ''r''. The case when ''n'' = ''r'' = 1 was conjectured by Miyawaki. Here ''n'', ''k'', and ''r'' are non-ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ikeda Lift
In mathematics, the Ikeda lift is a lifting of modular forms to Siegel modular forms. The existence of the lifting was conjectured by W. Duke and Ö. Imamoḡlu and also by T. Ibukiyama, and the lifting was constructed by . It generalized the Saito–Kurokawa lift from modular forms of weight 2''k'' to genus 2 Siegel modular forms of weight ''k'' + 1. Statement Suppose that ''k'' and ''n'' are positive integers of the same parity. The Ikeda lift takes a 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 weight 2''k'' for SL2(Z) to a Hecke eigenform in the space of Siegel modular forms of weight ''k''+''n'', degree 2''n''. Example The Ikeda lift takes the Delta function (the weight 12 cusp form for SL2(Z)) to the Schottky form, a weight 8 Siegel c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saito–Kurokawa Lift
In mathematics, the Saito–Kurokawa lift (or lifting) takes elliptic modular forms to Siegel modular forms of degree 2. The existence of this lifting was conjectured in 1977 independently by Hiroshi Saito and . Its existence was almost proved by , and and completed the proof. Statement The Saito–Kurokawa lift ''σ''''k'' takes level 1 modular forms ''f'' of weight 2''k'' − 2 to level 1 Siegel modular forms of degree 2 and weight ''k''. The L-functions (when ''f'' is a Hecke eigenforms) are related by ''L''(''s'',''σ''''k''(''f'')) = ζ(''s'' − ''k'' + 2)ζ(''s'' − ''k'' + 1)''L''(''s'', ''f''). The Saito–Kurokawa lift can be constructed as the composition of the following three mappings: # The Shimura correspondence from level 1 modular forms of weight 2''k'' − 2 to a space of level 4 modular forms of weight ''k'' − 1/2 in the Kohnen plus-space. #A map from the Kohnen plus-space t ... [...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 -invariance. Explicitly if and then :G_ \left( \frac \right) = (c\tau +d)^ G_(\tau) Relation to modular invariants The modular invariants and of an elliptic curve are given by the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Representation
In mathematics, in the representation theory of algebraic groups, a linear representation of an algebraic group is said to be rational if, viewed as a map from the group to the general linear group, it is a rational map In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible. Definition Formal d ... of algebraic varieties. Finite direct sums and products of rational representations are rational. A rational G module is a module that can be expressed as a sum (not necessarily direct) of rational representations. References * Springer Online Reference Works: Rational Representation Representation theory of algebraic groups {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or can be trivially determined by the context. I_1 = \begin 1 \end ,\ I_2 = \begin 1 & 0 \\ 0 & 1 \end ,\ I_3 = \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end ,\ \dots ,\ I_n = \begin 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end. The term unit matrix has also been widely used, but the term ''identity matrix'' is now standard. The term ''unit matrix'' is ambiguous, because it is also used for a matrix of ones and for any unit of the ring of all n\times n matrices. In some fields, such as group theory or quantum mechanics, the identity matrix is sometimes denoted by a boldface one, \mathbf, or called "id" (short for identity) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic group and is also denoted by \mathrm(n). Many authors prefer slightly different notations, usually differing by factors of . The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group is denoted , and is the compact real form of . Note that when we refer to ''the'' (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension . The name "symplectic group" is due to Hermann Weyl as a replacement for the previous confusing names (line) complex group and Abelian linear group, and is the Greek analog of "complex". The metaple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. In string theory, one of the many vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries the gravitational force. Thus, string theory is a theory of quantum gravity. String theory is a broad and varied subject that attempts to address a number of deep questions of fundamental physics. String theory has contributed a number of advances to mathematical physics, which have been applied to a variety of problems in black hole physics, early universe cosmology, nuclear physics, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Black Hole Thermodynamics
In physics, black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons. As the study of the statistical mechanics of black-body radiation led to the development of the theory of quantum mechanics, the effort to understand the statistical mechanics of black holes has had a deep impact upon the understanding of quantum gravity, leading to the formulation of the holographic principle. Overview The second law of thermodynamics requires that black holes have entropy. If black holes carried no entropy, it would be possible to violate the second law by throwing mass into the black hole. The increase of the entropy of the black hole more than compensates for the decrease of the entropy carried by the object that was swallowed. In 1972, Jacob Bekenstein conjectured that black holes should have an entropy, where by the same year, he proposed no-hair theorems. In 1973 Bekenstein suggested \frac\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
