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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
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References:

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