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Rankin-Selberg method for Siegel cusp forms. (English) Zbl 0715.11025

Let F and G be Siegel cusp forms of degree n and weight k for the modular group Sp(n,\({\mathbb{Z}})\). For a positive definite symmetric half-integral matrix T of size r(r\(\leq n)\) denote by \(\phi_ T\) and \(\psi_ T\) the \(T^{th}\)-Fourier-Jacobi coefficients of F and G (respectively). The author studies the following Dirichlet series, which is a generalization of the usual Rankin convolution: \[ D_ r(F,G,s):=\sum_{\{T\}}\frac{1}{\epsilon (T)}<\phi_ T,\psi_ T>\quad \det (T)^{-s}, \] where \(\{\) \(T\}\) runs over a set of representatives for the usual action of GL(r,\({\mathbb{Z}})\) on the space of symmetric positive definite half-integral matrices of size r, \(\epsilon\) (T) denotes the number of units of T and \(<, >\) is the Petersson scalar product on the space of Jacobi forms. The author shows that this Dirichlet series has meromorphic continuation to the complex plane and satisfies a nice functional equation. The proof uses an integral representation of Rankin-Selberg type for \(D_ r(F,G,s).\)
The special case \(n=2\), \(r=1\) was previously studied by W. Kohnen and N.-P. Skoruppa [Invent. Math. 95, 541-558 (1989; Zbl 0665.10019)].
Reviewer: S.Böcherer

MSC:

11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms

Citations:

Zbl 0665.10019
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References:

[1] Volume 62 of Lecture Notes in Math. (1968)
[2] Russian Math. Surveys 29 pp 45– (1974)
[3] J. Fac. Sci., Univ. Tokyo Sec. IA 22 pp 25– (1975)
[4] DOI: 10.1070/SM1977v032n04ABEH002399 · Zbl 0397.10021
[5] Volume 216 of Lecture Notes in Math. (1971)
[6] Acta Arithmetica 24 pp 223– (1973)
[7] DOI: 10.1007/BF01393889 · Zbl 0665.10019
[8] Proc. Cambridge Phil. Soc. 36 pp 351– (1939)
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