Ikeda lift

Not to be confused with Ikeda–Miyawaki 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 Ikeda (2001). It generalized the Saito–Kurokawa lift from modular forms of weight 2k 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 of weight 2k for SL₂(Z) to a Hecke eigenform in the space of Siegel modular forms of weight k+n, degree 2n.

Example

The Ikeda lift takes the Delta function (the weight 12 cusp form for SL₂(Z)) to the Schottky form, a weight 8 Siegel cusp form of degree 4. Here k=6 and n=2.

References

• Duke, W.; Imamoḡlu, Ö. (1996), "A converse theorem and the Saito-Kurokawa lift", International Mathematics Research Notices, 1996 (7): 347–355, doi:10.1155/S1073792896000220, MR 1389957
• Ikeda, Tamotsu (2001), "On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2n", Annals of Mathematics, Second Series, 154 (3): 641–681, doi:10.2307/3062143, JSTOR 3062143, MR 1884618