Abstract
We define a twisted two complex variables Rankin-Selberg convolution of Siegel cusp forms of degree 2. We find its group of functional equations and prove its analytic continuation to \({\mathbb{C}^2}\) . As an application we obtain a non-vanishing result for special values of the Fourier Jacobi coefficients. We also prove the analytic properties for the characteristic twists of convolutions of Jacobi cusp forms.
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Andrianov, A.N., Zhuravlev, V.G.: Modular forms and Hecke operators. Math. Monographs, vol. 145. AMS, Providence (1995)
Bocherer S., Bruinier J.H., Kohnen W.: Non-vanishing of scalar products of Fourier–Jacobi coefficients of Siegel cusp forms. Math. Ann. 313, 1–13 (1999)
Bump D.: Automorphic forms and representations. Cambridge University Press, Cambridge (1997)
Christian, U.: Selberg’s Zeta-, L-, and Eisensteinseries. Lect. Notes Math., vol. 1030. Springer, Berlin (1983)
Eichler M., Zagier D.: The theory of Jacobi forms. Progress in Math., vol. 55. Birkhauser, Basel (1985)
Hormander L.: An Introduction to Complex Analysis in Several Variables. Van Nostrand Co., Princeton (1966)
Imamoglu Ö, Martin Y.: On a Rankin-Selberg convolution of two variables for Siegel modular forms. Forum Math. 15, 565–589 (2003)
Iwaniec, H.: Topics in Classical Automorphic forms. GSM, vol. 17. AMS, Providence (1997)
Klingen H.: Introductory Lectures on Siegel Modular Forms. Cambridge Univ. Press, Cambridge (1990)
Kohnen W., Skoruppa N.-P.: A certain Dirichlet series attached to Siegel modular forms of degree two. Invent. Math. 95, 541–558 (1989)
Kohnen W.: On characteristic twists of certain Dirichlet series. Mem. Fac. Sci. Kyushu Univ. Ser. A 47, 103–117 (1993)
Kohnen W., Krieg A., Sengupta J.: Characteristic twists of a Dirichlet series for Siegel cusp forms. Manusc. Math. 87, 489–499 (1995)
Maass, H.: Siegel’s modular forms and Dirichlet series. Lect. Notes Math., vol. 216. Springer, Berlin (1971)
Miyake T.: Modular Forms. Springer, Berlin (1989)
Siegel C.: Lectures on Advanced Analytic Number Theory. Tata Inst. Fund. Research, Bombay (1961)
Terras A.: Harmonic Analysis on Symmetric Spaces and Application I and II. Springer, Berlin (1998)
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Research Supported by Fondecyt grants 1061147, 7060241.
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Imamōlu, Ö., Martin, Y. On characteristic twists of multiple Dirichlet series associated to Siegel cusp forms. Math. Z. 263, 345–368 (2009). https://doi.org/10.1007/s00209-008-0421-7
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DOI: https://doi.org/10.1007/s00209-008-0421-7