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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.

Statement

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 Shimura correspondence from level 1 modular forms of weight 2k − 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 forms 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, introduce 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

$${\begin{pmatrix}n&t/2\\t/2&m\end{pmatrix}}=\sum _{{d|t,m,n}}d^{{k-1}}a{\begin{pmatrix}1&t/2d\\t/2d&nm/d^{2}\end{pmatrix}}$$

Doi–Naganuma lifting, a similar lift to Hilbert modular forms.
Ikeda lift, a generalization to Siegel modular forms of higher degree.

References
Andrianov, Anatolii N. (1979), "Modular descent and the Saito-Kurokawa conjecture", Invent. Math., 53 (3): 267–280, doi:10.1007/BF01389767, MR 0549402
Kurokawa, Nobushige (1978), "Examples of eigenvalues of Hecke operators on Siegel cusp forms of degree two", Invent. Math., 49 (2): 149–165, doi:10.1007/bf01403084, MR 0511188
Maass, Hans (1979), "Über eine Spezialschar von Modulformen zweiten Grades", Invent. Math., 52 (1): 95–104, doi:10.1007/bf01389857, MR 0532746
Maass, Hans (1979), "Über eine Spezialschar von Modulformen zweiten Grades. II", Invent. Math., 53 (3): 249–253, doi:10.1007/bf01389765, MR 0549400
Maass, Hans (1979), "Über eine Spezialschar von Modulformen zweiten Grades. III", Invent. Math., 53 (3): 255–265, doi:10.1007/bf01389766, MR 0549401
Zagier, D. (1981), "Sur la conjecture de Saito-Kurokawa (d'après H. Maass)", Seminar on Number Theory, Paris 1979–80, Progr. Math., 12, Boston, Mass.: Birkhäuser, pp. 371–394, MR 0633910