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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] |
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\, &S^2 ... [...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, su ... [...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] |