In mathematics, the Saito–Kurokawa lift (or lifting) takes
elliptic 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 o ...
s to
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 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 to the space of
Jacobi form
In mathematics, a Jacobi form is an automorphic form on the Jacobi group, which is the semidirect product of the symplectic group Sp(n;R) and the Heisenberg group H^_R. The theory was first systematically studied by .
Definition
A Jacobi form of ...
s of index 1 and weight ''k'', studied by
Eichler and Zagier.
# A map from the space of Jacobi forms of index 1 and weight ''k'' to the Siegel modular forms of degree 2, introduced by Maass.
The Saito–Kurokawa lift can be generalized to forms of higher level.
The image is the Spezialschar (special band), the space of Siegel modular forms whose Fourier coefficients satisfy
:
See also
*
Doi–Naganuma lifting In mathematics, the Doi–Naganuma lifting is a map from elliptic modular forms to Hilbert modular forms of a real quadratic field, introduced by and .
It was a precursor of the base change lifting.
It is named for Japanese mathematicians Kōji D ...
, a similar lift to Hilbert modular forms.
*
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 S ...
, a generalization to Siegel modular forms of higher degree.
References
*
*
*
*
*
*
{{DEFAULTSORT:Saito-Kurokawa lift
Modular forms