Bounds for Green’s functions on noncompact hyperbolic Riemann orbisurfaces of finite volume

Abstract

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 ...

Estimates of automorphic cusp forms over quaternion algebras

“…The estimate that we derived for B k X 1 (z) in Theorem 4.5 depends only on the injectivity radius r X 1 , which is bounded from below, as X 1 is compact. Furthermore, following similar arguments as in [2] (Lemma 6.4 in section 6), it is easy to see that the lower bound for r X 1 remains stable in covers, which implies that our estimate for B k X 1 (z) is stable in covers of compact Hilbert modular varieties.…”

Bounds for canonical Green’s function at cusps

“…The above series is invariant under the action Γ, and hence, defines a function on X. Furthermore, from Proposition 4.2.4 in [4] (or from 2.2 in [3]), the above series converges for all z ∈ X, and satisfies the following equation…”

A relation of two metrics on a noncompact hyperbolic Riemann orbisurface

“…Although it is obvious from the differential equation (10) that g hyp (z, w) is log logsingular at the cusps, the exact asymptotics derived in the following proposition come very useful in the upcoming articles (especially in [2]). Proposition 2.1.…”

“…Hence, in the upcoming article [2], Eq. (31) serves as a starting point for the extension of the bounds for the canonical Green's function g can (z; w) from [6] to noncompact hyperbolic Riemann orbisurfaces of finite volume.…”