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

Jorgenson, Jay

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

Cited by 6 publications

(10 citation statements)

References 13 publications

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“…The above relation, which relates the two natural metrics defined on a Riemann orbisurface has been proved for compact hyperbolic Riemann surfaces, as a relation of differential forms by J. Jorgenson and J. Kramer in [6]. The same authors have also extended the key identity to noncompact hyperbolic Riemann surfaces of finite volume in [5]. In this paper, the authors use different methods from [6], and study the behavior of the key identity over a family of degenerating compact hyperbolic Riemann surfaces.…”

Section: Introductionmentioning

confidence: 95%

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Extension of a key identity

Aryasomayajula¹

2013

Preprint

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In this article, we extend a certain key identity proved by J. Jorgenson and J. Kramer in [6] to noncompact hyperbolic Riemann orbisurfaces of finite volume. This identity relates the two natural metrics, namely the hyperbolic metric and the canonical metric defined on a Riemann orbisurface.The extended version of the key identity enables us to extend the work of J. Jorgenson and J. Kramer to noncompact hyperbolic Riemann orbisurfaces of finite volume. In an upcoming article [2], using the key identity, we extend the bounds derived in [6].Acknowledgements This article is part of the PhD thesis of the author, which was completed under the supervision of J. Kramer at Humboldt Universität zu Berlin. The author would like to express his gratitude to J. Kramer for his support and many valuable scientific discussions. The author would also like to extend his gratitude to J. Jorgenson for sharing new scientific ideas, and to R. S. de Jong for many interesting scientific discussions and for pointing out a mistake in the first proof. Background materialLet Γ ⊂ PSL 2 (R) be a Fuchsian subgroup of the first kind acting by fractional linear transformations on the upper half-plane H. Let X be the quotient space Γ\H, and let g denote the genus of X. The quotient space X admits the structure of a Riemann orbisurface. Let E, P be the finite set of elliptic fixed points and cusps of X, respectively; put S = E ∪ P. For e ∈ E, let m e denote the order of e; for p ∈ P, put m p = ∞; for z ∈ X\E, put m z = 1. Let X denote X = X ∪ P. Locally, away from the elliptic fixed points and cusps, we identity X with its universal cover H, and hence, denote the points on X\S by the same letter as the points on H.Structure of X as a Riemann surface The quotient space X admits the structure of a compact Riemann surface. We refer the reader to section 1.8 in [10], for the details regarding the structure of X as a compact Riemann surface. For the convenience of the reader, we recall the coordinate functions for the neighborhoods of elliptic fixed points and cusps. Let w ∈ U r (e) denote a coordinate disk of radius r around an elliptic fixed point e ∈ E. Then, the coordinate function ϑ e (w) for the coordinate disk U r (e) is given by ϑ e (w) =w − e w − e me .

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Section: Introductionmentioning

confidence: 95%

“…Our main theorem can be seen as an extension of their result to elliptic fixed points and cusps at the level of currents acting on the space of singular function C ℓ,ℓℓ (X). Our methods are different from the ones employed in [5], and are organized around the original line of proof in [6].…”

Section: Introductionmentioning

confidence: 99%

Extension of a key identity

Aryasomayajula¹

2013

Preprint

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“…In fact, J. Jorgenson and J. Kramer have derived a Rankin-Selberg L-function relation relating the Fourier coefficients of cusp forms with those of Mäss forms in [5].…”

Section: Arithmetic Significancementioning

confidence: 97%

“…The same authors have also extended the key identity to noncompact hyperbolic Riemann surfaces of finite volume in [5]. In this paper, the authors use different methods from [6], and study the behavior of the key identity over a family of degenerating compact hyperbolic Riemann surfaces.…”

Section: Introductionmentioning

confidence: 96%

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A relation of two metrics on a noncompact hyperbolic Riemann orbisurface

Aryasomayajula

2014

manuscripta math.

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Jorgenson and Kramer (Compos Math 142:679-700, 2006) proved a certain key identity which relates the two natural metrics, namely the hyperbolic metric and the canonical metric defined on a compact hyperbolic Riemann surface. In this article, we extend this identity to noncompact hyperbolic Riemann orbisurfaces of finite volume, which can be realized as a quotient space of the action of a Fuchsian subgroup of first kind on the hyperbolic upper half plane. Our result can be seen as an extension of the key identity to elliptic fixed points and cusps at the level of currents.

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Remarks on the Patterson–Selberg zeta function, black hole vacua and extremal CFT partition functions

Williams¹

2012

J. Phys. A: Math. Theor.

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We consider some applications (old and new) of the Patterson–Selberg zeta function to 3D gravity with a negative cosmological constant. We also consider 2D black hole vacua with a parabolic generator of their holonomy. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to ‘Applications of zeta functions and other spectral functions in mathematics and physics’.

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

Cited by 6 publications

References 13 publications

Extension of a key identity

Extension of a key identity

A relation of two metrics on a noncompact hyperbolic Riemann orbisurface

Remarks on the Patterson–Selberg zeta function, black hole vacua and extremal CFT partition functions

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