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A Relation Involving Rankin–Selberg L-Functions of Cusp Forms and Maass Forms

In: Representation Theory, Complex Analysis, and Integral Geometry

Author

Listed:
  • Jay Jorgenson

    (City College of New York, Department of Mathematics)

  • Jürg Kramer

    (Humboldt-Universität zu Berlin, Institut für Mathematik)

Abstract

In previous articles, an identity relating the canonical metric to the hyperbolic metric associated with any compact Riemann surface of genus at least two has been derived and studied. In this article, this identity is extended to any hyperbolic Riemann surface of finite volume. The method of proof is to study the identity given in the compact case through degeneration and to understand the limiting behavior of all quantities involved. In the second part of the paper, the Rankin–Selberg transform of the noncompact identity is studied, meaning that both sides of the relation after multiplication by a nonholomorphic, parabolic Eisenstein series are being integrated over the Riemann surface in question. The resulting formula yields an asymptotic relation involving the Rankin–Selberg L-functions of weight two holomorphic cusp forms, of weight zero Maass forms, and of nonholomorphic weight zero parabolic Eisenstein series.

Suggested Citation

  • Jay Jorgenson & Jürg Kramer, 2012. "A Relation Involving Rankin–Selberg L-Functions of Cusp Forms and Maass Forms," Springer Books, in: Bernhard Krötz & Omer Offen & Eitan Sayag (ed.), Representation Theory, Complex Analysis, and Integral Geometry, pages 9-40, Springer.
    • Handle: RePEc:spr:sprchp:978-0-8176-4817-6_2
    DOI: 10.1007/978-0-8176-4817-6_2
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