Bounds for canonical Green’s function at cusps

Aryasomayajula, Anilatmaja

doi:10.1007/s12188-014-0099-1

Cited by 2 publications

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“…The pieces (𝑎) and (𝑐) contain the data relative to the 'self-intersection at infinity' of 𝜔. Their asymptotics have already been estimated, respectively, in [28] and [27,Section 6]. In this paper, we are able to exactly calculate the term (𝑏).…”

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Intersection matrices for the minimal regular model of X0(N)${X}_0(N)$ and applications to the Arakelov canonical sheaf

Dolce,

Mercuri

2024

Journal of London Math Soc

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Section: Overview Of the Papermentioning

confidence: 98%

Intersection matrices for the minimal regular model of X0(N)${X}_0(N)$ and applications to the Arakelov canonical sheaf

Dolce,

Mercuri

2024

Journal of London Math Soc

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

Aryasomayajula

2015

Math. Z.

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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 ...

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Bounds for canonical Green’s function at cusps

Abstract: 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.

Cited by 2 publications

References 12 publications

Intersection matrices for the minimal regular model of X0(N)${X}_0(N)$ and applications to the Arakelov canonical sheaf

Intersection matrices for the minimal regular model of X0(N)${X}_0(N)$ and applications to the Arakelov canonical sheaf

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

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