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Siegel Eisenstein Series
In mathematics, a Siegel Eisenstein series (sometimes just called an Eisenstein series or a Siegel series) is a generalization of Eisenstein series to Siegel modular forms. gave an explicit formula for their coefficients. Definition The Siegel Eisenstein series of degree ''g'' and weight an even integer ''k'' > 2 is given by the sum :\sum_\frac Sometimes the series is multiplied by a constant so that the constant term of the Fourier expansion is 1. Here ''Z'' is an element of the Siegel upper half space of degree ''d'', and the sum is over equivalence classes of matrices ''C'',''D'' that are the "bottom half" of an element of the Siegel modular group In mathematics, the Siegel upper half-space of degree ''g'' (or genus ''g'') (also called the Siegel upper half-plane) is the set of ''g'' × ''g'' symmetric matrices over the complex numbers whose imaginary part is positive definite. It .... Example See also * Klingen Eisenstein series, a generaliza ... [...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 poin ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Siegel Upper Half Space
In mathematics, the Siegel upper half-space of degree ''g'' (or genus ''g'') (also called the Siegel upper half-plane) is the set of ''g'' × ''g'' symmetric matrices over the complex numbers whose imaginary part is positive-definite matrix, positive definite. It was introduced by . It is the symmetric space associated to the symplectic group . The Siegel upper half-space has properties as a complex manifold that generalize the properties of the upper half-plane, which is the Siegel upper half-space in the special case ''g=1''. The group of automorphisms preserving the complex structure of the manifold is isomorphic to the symplectic group . Just as the Poincaré half-plane model, two-dimensional hyperbolic metric is the unique (up to scaling) metric on the upper half-plane whose isometry group is the complex automorphism group = , the Siegel upper half-space has only one metric up to scaling whose isometry group is . Writing a generic matrix ''Z'' in the Siegel upper hal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Siegel Modular Group
In mathematics, the Siegel upper half-space of degree ''g'' (or genus ''g'') (also called the Siegel upper half-plane) is the set of ''g'' × ''g'' symmetric matrices over the complex numbers whose imaginary part is positive definite. It was introduced by . It is the symmetric space associated to the symplectic group . The Siegel upper half-space has properties as a complex manifold that generalize the properties of the upper half-plane, which is the Siegel upper half-space in the special case ''g=1''. The group of automorphisms preserving the complex structure of the manifold is isomorphic to the symplectic group . Just as the two-dimensional hyperbolic metric is the unique (up to scaling) metric on the upper half-plane whose isometry group is the complex automorphism group = , the Siegel upper half-space has only one metric up to scaling whose isometry group is . Writing a generic matrix ''Z'' in the Siegel upper half-space in terms of its real and imaginary parts as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klingen Eisenstein Series
In mathematics, a Klingen Eisenstein series is a Siegel modular form of weight ''k'' and degree ''g'' depending on another Siegel cusp form ''f'' of weight ''k'' and degree ''r'' ''g'' + ''r'' + 1 an even integer. The Klingen Eisenstein series is : \sum_ f\left(\frac\right)\det(c\tau+d)^. It is a Siegel modular form of weight ''k'' and degree ''g''. Here ''P''''r'' is the integral points of a certain Borel subgroup, parabolic subgroup of the symplectic group, and Γ''r'' is the group of integral points of the degree ''g'' symplectic group. The variable τ is in the Siegel upper half plane of degree ''g''. The function ''f'' is originally defined only for elements of the Siegel upper half plane of degree ''r'', but extended to the Siegel upper half plane of degree ''g'' by projecting this to the smaller Siegel upper half plane. The cusp form ''f'' is the image of the Klingen Eisenstein series under the operator Φ''g''−''r'', where Φ is the Siegel oper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
