Saito–Kurokawa conjecture
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 Nobushige Kurokawa (1978). Its existence was almost proved by Maass (1979), and Andrianov (1979) and Zagier (1981) completed the proof.
The Saito–Kurokawa lift σk takes level 1 modular forms f of weight 2k − 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,σk(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 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
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