Bounds for canonical Green's function at cusps
In this article, we derive bounds for the canonical Green's function defined on a noncompact hyperbolic Riemann surface, when evaluated at two inequivalent cusps.
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arXiv subjects
Anilatmaja Aryasomayajula
Publications and source records attributed to Anilatmaja Aryasomayajula.
A relation of cusp forms and Maass forms on product of hyperbolic Riemann orbisurfaces of finite volume
In 2006, in a paper published in Compositio, titled "Bounds on canonical Green's functions", J. Jorgenson and J. Kramer 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 product of 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.
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Bounds for Green's functions on noncompact hyperbolic Riemann orbisurfaces of finite volume
In 2006, in a paper published in Compositio titled "Bounds on canonical Green's functions", J. Jorgenson and J. Kramer derived bounds for the canonical Green's function and the hyperbolic Green's function defined on a compact hyperbolic Riemann surface. In this article, we extend these bounds to noncompact hyperbolic Riemann orbisurfaces of finite volume and of genus greater than zero, 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 bounds are optimally derived by following the methods from the above mentioned paper of J. Jorgenson and J. Kramer.
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Extension of a key identity
In this article, we extend a certain key identity proved by J. Jorgenson and J. Kramer for compact hyperbolic Riemann surfaces to noncompact hyperbolic Riemann orbisurfaces of finite volume, which can be realized as the quotient space of the action of a Fuchsian subgroup of the first kind acting on the hyperbolic upper half-plane. The key identity of J. Jorgenson and J. Kramer relates the two natural metrics, namely the hyperbolic metric and the canonical metric defined on a hyperbolic Riemann surface. Via the spectral expansion of the hyperbolic heat kernel, this identity serves as a trace formula relating the weight 2 cusp forms with Maass forms. Our result is an extension of the key identity to elliptic fixed points and cusps at the level of currents acting on a certain space of singular functions. Our result serves as the starting point for extending the work of J. Jorgenson and J. Kramer to non compact hyperbolic Riemann orbisurfaces. In particular, to the problem of deriving bounds for the canonical Green's function defined on a noncompact hyperbolic Riemann orbisurface of finite volume, which is being addressed in an upcoming article by the same author.
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