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

Jay Jorgenson () and Jürg Kramer ()
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Jay Jorgenson: City College of New York, Department of Mathematics
Jürg Kramer: Humboldt-Universität zu Berlin, Institut für Mathematik

A chapter in Representation Theory, Complex Analysis, and Integral Geometry, 2012, pp 9-40 from Springer

Abstract: 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.

Keywords: Automorphic forms; Eisenstein series; L-functions; Rankin– Selberg transform; Heat kernel (search for similar items in EconPapers)
Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-0-8176-4817-6_2

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DOI: 10.1007/978-0-8176-4817-6_2

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