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A certain Dirichlet series attached to Siegel modular forms of degree two. (English) Zbl 0665.10019

The authors introduce a new type of Dirichlet series attached to a pair \((F,G)\) of two Siegel modular forms of degree two (and same weight): Let \(\sum_{m}\phi_ m(z_ 1,z_ 2)e^{2\pi imz_ 4}\), \(\sum_{m}\psi_ m(z_ 1,z_ 2)e^{2\pi imz_ 4}\) with \(Z=\left( \begin{matrix} z_ 1\\ z_ 2\end{matrix} \begin{matrix} z_ 2\\ z_ 4\end{matrix} \right)\) be the Fourier-Jacobi expansions of \(F\) and G and consider \[ D_{F,G}(s)=\sum_{m}<\phi_ m,\psi_ m>/m^ s \] where \(<\,,\,>\) is the Petersson scalar product of the Jacobi forms \(\phi_ m,vf_ m\). By a variant of the Rankin-Selberg method the authors show that this series has a meromorphic continuation to the whole complex plane and satisfies a functional equation.
Surprisingly this type of Dirichlet series may (in some cases) be related to the degree 4 \(L\)-function \(Z_ F(s)\) attached to a Hecke eigenform; this “spinor” \(L\)-function was first investigated by A. N. Andrianov [Usp. Mat. Nauk 29, No. 3(117), 43–110 (1974; Zbl 0304.10020)]. The authors prove that \[ D_{F,G}(s)=\text{const} \times Z_ F(s), \] if \(F\) is a Hecke eigenform and \(G\) is in the “Maaß Spezialschar”.

MSC:

11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F50 Jacobi forms

Citations:

Zbl 0304.10020
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References:

[1] Andrianov, A.N.: Euler products corresponding to Siegel modular forms of genus 2. Russ. Math. Surv.29, 45-116 (1974) · Zbl 0304.10021
[2] Eichler, M., Zagier, D.: The theory of Jacobi forms (Progress in Maths. Vol. 55). Boston: Birkhäuser 1985 · Zbl 0554.10018
[3] Evdokimov, S.A.: A characterization of the Maass space of Siegel cusp forms of degree 2 (in Russian). Mat. USSR Sb. (154)112, 133-142 (1980) · Zbl 0433.10014
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[5] Klingen, H.: Zum Darstellungssatz für Siegelsche Modulformen. Math. Z.102, 30-43 (1967) · Zbl 0155.40401
[6] Kohnen, W.: On the Petersson norm of a Siegel-Hecke eigenform of degree two in the Maass space. J. Reine Angew. Math.357, 96-100 (1985) · Zbl 0547.10026
[7] Kurokawa, N.: Examples of eigenvalues of Hecke operators on Siegel cusp forms of degree two. Invent. Math.49, 149-165 (1978) · Zbl 0397.10018
[8] Oda, T.: On the poles of AndrianovL-functions. Math. Ann.256, 323-340 (1981) · Zbl 0465.10021
[9] Zagier, D.: Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields. In: Serre, J.-P., Zagier, D. (eds.). Modular functions of one variable VI. (Lect. Notes Maths. Vol. 627, pp. 105-169). Berlin Heidelberg New York: Springer 1977 · Zbl 0372.10017
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