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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] |
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] |
Lift (mathematics)
In category theory, a branch of mathematics, given a morphism ''f'': ''X'' → ''Y'' and a morphism ''g'': ''Z'' → ''Y'', a lift or lifting of ''f'' to ''Z'' is a morphism ''h'': ''X'' → ''Z'' such that . We say that ''f'' factors through ''h''. A basic example in topology is lifting a path in one topological space to a path in a covering space. For example, consider mapping opposite points on a sphere to the same point, a continuous map from the sphere covering the projective plane. A path in the projective plane is a continuous map from the unit interval ,1 We can lift such a path to the sphere by choosing one of the two sphere points mapping to the first point on the path, then maintain continuity. In this case, each of the two starting points forces a unique path on the sphere, the lift of the path in the projective plane. Thus in the category of topological spaces with continuous maps as morphisms, we have :\begin f\colon\, & ,1\to \mathbb^2 &&\ \text \\ g\colon\, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a 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, sphere packing, and string theory. A modular function is a function that is invariant with respect to the modular group, but without the condition that be holomorphic in the upper half-plane (among other requirements). Instead, modular functions are meromorphic (that is, they are holomorphic on the complement of a set of isolated points, which are poles of the function). Modular form theory is a special case of the more general theory of automorphic forms which are functions defined on Lie groups which transform nicely w ... [...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] |
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] |
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. In second-order cybernetics, eigenforms are an example of a self-referential system.Kauffman, L. H. (2003). Eigenforms: Objects as tokens for eigenbehaviors. Cybernetics and Human Knowing, 10(3/4), 73-90. 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 wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schottky Form
In mathematics, the Schottky form or Schottky's invariant is a Siegel cusp form ''J'' of degree 4 and weight 8, introduced by as a degree 16 polynomial in the Thetanullwerte of genus 4. He showed that it vanished at all Jacobian points (the points of the degree 4 Siegel upper half-space corresponding to 4-dimensional abelian varieties that are the Jacobian varieties of genus 4 curves). showed that it is a multiple of the difference θ4(''E''8 ⊕ ''E''8) − θ4(''E''16) of the two genus 4 theta functions of the two 16-dimensional even unimodular lattices and that its divisor of zeros is irreducible. showed that it generates the 1-dimensional space of level 1 genus 4 weight 8 Siegel cusp forms. Ikeda showed that the Schottky form is the image of the Dedekind Delta function under the 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. Ibukiy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
