In
mathematics, the Ikeda lift is a
lifting of
modular forms 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. 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 SL
2(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 SL
2(Z)) to the
Schottky form, a weight 8 Siegel cusp form of degree 4. Here ''k''=6 and ''n''=2.
References
*
*{{citation, mr=1884618
, last=Ikeda, first= Tamotsu
, title=On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2n
, journal=Annals of Mathematics , series=Second Series , volume=154 , year=2001, issue= 3, pages= 641–681, jstor=3062143, doi=10.2307/3062143
Modular forms