A study on twisted Koecher–Maass series of Siegel cusp forms via an integral kernel
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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.
KeywordsSiegel cusp forms Twisted Koecher–Maass series Multiple Dirichlet series Fourier–Jacobi expansion
Mathematics Subject ClassificationPrimary 11F46 Secondary 11F50 11M32
- 5.Cohen, H.: Sur certaines sommes de séries liées aux périodes de formes modulaires. C.R. Journées de Théorie Analytique et Élémentaire des Nombres, Publ. 2 Univ. de Limoges, Limoges (1981)Google Scholar