## Saito-Kurokawa lifts of square-free level.(English)Zbl 1326.11019

In this article, the authors survey the classical construction of a Saito-Kurokawa lift $$F_f$$ from a Hecke eigen cuspform $$f\in S_{2k-2}(\Gamma_0(M))$$ for $$M\geq 1$$ an odd square-free integer and $$k\geq 2$$ an even integer. Here $$F_f$$ is a Siegel modular form of weight $$k$$ with respect to a congruence subgroup $$\Gamma_0^{(2)}(M)$$ of the group $$\text{Sp}(4,\mathbb Z)$$ of level $$M$$. They show that, up to a constant, the Fourier coefficients of $$F_f$$ are in the ring generated by those of $$f$$. Further, they correct the result for the norm of $$F_f$$ in a previous paper of the second author [Ramanujan J. 14, No. 1, 89–105 (2007; Zbl 1197.11057)], based on the correct definition of the Maass lifting from Jacobi forms to Siegel forms given in [T. Ibukiyama, Kyoto J. Math. 52, No. 1, 141–178 (2012; Zbl 1284.11085)].

### MSC:

 11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11F32 Modular correspondences, etc. 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations

### Citations:

Zbl 1197.11057; Zbl 1284.11085
Full Text:

### References:

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