The SaitoâKurokawa lift Ïk takes level 1 modular forms f of weight 2k â 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,Ïk(f)) = ζ(s â k + 2)ζ(s â k + 1)L(s, f).
The SaitoâKurokawa lift can be constructed as the composition of the following three mappings:
The Shimura correspondence from level 1 modular forms of weight 2k â 2 to a space of level 4 modular forms of weight k â 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, introduced by Maass.
The SaitoâKurokawa 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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Zagier, D. (1981), "Sur la conjecture de Saito-Kurokawa (d'aprĂšs H. Maass)", Seminar on Number Theory, Paris 1979â80, Progr. Math., vol. 12, Boston, Mass.: BirkhĂ€user, pp. 371â394,
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