In 2006, 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.Mathematics Subject Classification (2010): 14G40, 30F10, 11F72, 30C40.Keywords: Green's functions, Arakelov theory, modular curves, hyperbolic heat kernels. IntroductionNotation Let X be a noncompact hyperbolic Riemann orbisurface of finite volume vol hyp (X) with genus g X ≥ 1, and can be realized as the quotient space Γ X \H, where Γ X ⊂ PSL 2 (R) is a Fuchsian subgroup of the first kind acting on the hyperbolic upper half-plane H, via fractional linear transformations. Let P X and E X denote the set of cusps and the set of elliptic fixed points of Γ X , respectively. Put X = X ∪ P X . Then, X admits the structure of a Riemann surface.Let µ hyp (z) denote the (1,1)-form associated to hyperbolic metric, which is the natural metric on X, and of constant negative curvature minus one. Let µ shyp (z) denote the rescaled hyperbolic metric µ hyp (z)/ vol hyp (X), which measures the volume of X to be one.The Riemann surface X is embedded in its Jacobian variety Jac(X) via the Abel-Jacobi map. Then, the pull back of the flat Euclidean metric by the Abel-Jacobi map is called the canonical metric, and the (1,1)-form associated to it is denoted by µ can (z). We denote its restriction to X by µ can (z).For µ = µ shyp (z) or µ can (z), let g X,µ (z, w) defined on X ×X denote the Green's function associated to the metric µ. The Green's function g X,µ (z, w) is uniquely determined by the differential equation (which is to be interpreted in terms of currents)with the normalization conditionThe Green's function g X ,can (z, w) associated to the canonical metric µ can (z) is called the canonical Green's function. Similarly the Green's function g X ,hyp (z, w) associated to the (rescaled) hyperbolic metric µ shyp (z) is called the hyperbolic Green's function.From differential equation (1), we can deduce that for a fixed w ∈ X, as a function in the variable z, both the Green's functions g X ,can (z, w) and g X ,hyp (z, w) are log-singular at z = w. Recall that µ hyp (z) is singular at the cusps and at the elliptic fixed points, and µ can (z) the pull back of the smooth and flat Euclidean metric is smooth on X. Hence, from the elliptic regularity of the d z d c z operator, it follows that g X ,hyp (z, w) is log log-singular at the cusps, and g X ,can (z, w) remains smooth at the cusps.1 From a geometric perspective, it is very interesting to compare the two metrics µ hyp (z) and µ can (z), and study the difference of the two Green's functionson compact subsets of X.In [10], J. Jorgenson and J. Kramer have already established these tasks, when X is a compact Riemann surface devoid of elliptic fixed ...
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 .
In this article, we derive off-diagonal estimates of the Bergman kernel associated to tensorproducts of the cotangent line bundle defined over a hyperbolic Riemann surface of finite volume.
In this article, using the heat kernel approach from [2], we derive sup-norm bounds for cusp forms of integral and half-integral weight. Let Γ ⊂ PSL 2 (R) be a cocompact Fuchsian subgroup of first kind. For k ∈ 1 2 Z (or k ∈ 2Z), let S k (Γ) denote the complex vector space of weight-k cusp forms. Let {f 1 , . . . , f j k } denote an orthonormal basis of S k (Γ). In this article, we show that as k → ∞, the sup-norm forwhere the implied constant is independent on Γ. Furthermore, using results from [1], we extend these results to the case when Γ is cofinite.Mathematics Subject Classification (2010): 30F30, 30F35, 30F45.
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