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Eisenstein series for Siegel modular groups. (English) Zbl 0786.11024

Let \(E_ k^{(m)}(z,s)\) be the Siegel-type non-holomorphic Eisenstein series of weight \(k\) (\(\in 2\mathbb{Z}_{\geq 0}\)) for \(\text{Sp}_{2m}(\mathbb{Z})\). Here \(z\) is a variable on the Siegel upper half space of degree \(m\in\mathbb{Z}_{>0}\) and \(s\) is a complex variable. On this Eisenstein series, the author presents:
(I) An elementary proof of the analytic continuation and the functional equation in \(s\) via the Fourier expansion, which is a generalization to all \(m\) of Kaufhold’s result [G. Kaufhold, Math. Ann. 137, 454-476 (1959; Zbl 0086.067)] on degree two Eisenstein series,
(II) a proof that every partial derivative of \(E_ k^{(m)}(z,s)\) in the entries of \(\text{Re}(z)\) and \(\text{Im}(z)\) is slowly increasing (locally uniformly in \(s\)). This fact, for example, enables us to use a wide class of (not necessarily invariant) differential operators in the Rankin-Selberg convolution integral.
Reviewer: S.Mizumoto (Tokyo)

MSC:

11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F30 Fourier coefficients of automorphic forms
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations

Citations:

Zbl 0086.067
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References:

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