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The Ramanujan Journal

, Volume 50, Issue 3, pp 685–715 | Cite as

A study on twisted Koecher–Maass series of Siegel cusp forms via an integral kernel

  • Yves MartinEmail author
Article

Abstract

We find an explicit integral kernel for the twisted Koecher–Maass series of any degree two Siegel cusp form F, where the twist is realized by an arbitrary Maass waveform whose eigenvalue is in the continuum spectrum. We also obtain the analytic properties of such a kernel (functional equations and analytic continuation), as well as a series representation of it in terms of the degree two Siegel Poincare series. From these properties we deduce the analytic properties of the twisted Koecher–Maass series. Moreover, we express the later as a multiple Dirichlet series involving the Dirichlet series associated to the Fourier–Jacobi coefficients of F. Finally, we get the integral kernel of the untwisted Koecher–Maass series (first studied by Kohnen and Sengupta in any degree) as a limit case of our construction.

Keywords

Siegel cusp forms Twisted Koecher–Maass series Multiple Dirichlet series Fourier–Jacobi expansion 

Mathematics Subject Classification

Primary 11F46 Secondary 11F50 11M32 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Matemáticas, Facultad de CienciasUniversidad de ChileSantiagoChile

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