A relation of two metrics on a noncompact hyperbolic Riemann orbisurface

Aryasomayajula, Anilatmaja

doi:10.1007/s00229-014-0715-5

Cited by 2 publications

(2 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our estimate of the Bergman metric (2) is far from being optimal. As per evidence from the one-dimensional setting from [AM23], we conjecture the following estimate…”

Section: Let µ Volmentioning

confidence: 62%

“…In [AB19], Biswas and the first named author have derived estimates of the Bergman metric associated to high tensor-powers of the cotangent bundle, defined over a compact hyperbolic Riemann surface. In [AM23], the first named author and Mukherjee have extended estimates from [AB19] to the setting of noncompact hyperbolic Riemann surfaces. In [ARS24], the first named author, Roy, and Sadhukhan have extended the estimates from [AB19] to the setting of Picard surfaces.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Estimates of automorphic cusp forms over quaternion algebras

Aryasomayajula

Balasubramanyam

2018

Int. J. Number Theory

View full text Add to dashboard Cite

In this article, using methods from geometric analysis and theory of heat kernels, we derive qualitative estimates of automorphic cusp forms defined over quaternion algebras. Using which, we prove an average version of the holomorphic QUE conjecture. We then derive quantitative estimates of classical Hilbert modular cusp forms. This is a generalization of the results from [3] and [9] to higher dimensions.The Bergman kernel, which is the reproducing kernel for L 2 -holomorphic functions defined on a domain in C n has been extensively studied in complex analysis. The generalization of the Bergman kernel to complex manifolds as the reproducing kernel for the space of global holomorphic sections of a vector bundle carries the information on the algebraic and geometric structures of the underlying manifolds.Automorphic forms defined over quaternion algebras are global sections of a holomorphic vector bundle. In this article, we derive asymptotic estimates of the Bergman kernel associated to cuspidal holomorphic vector-valued modular forms of large weight defined over quaternion algebras. We prove an average version of the holomorphic QUE conjecture, when the associated complex analytic space is compact. We also derive quantitative estimates of the Bergman kernel associated to classical Hilbert modular cusp forms.In the first half of this article, utilizing certain results from geometric analysis, we derive asymptotic estimates of the Bergman kernel associated to automorphic cusp forms defined over quaternion algebras. Using which, we prove an average version of the holomorphic QUE conjecture.Jorgenson and Kramer have derived optimal estimates of heat kernels defined over hyperbolic Riemann surfaces of finite volume, and used the estimates to derive optimal estimates of the Bergman kernel associated to cusp forms. In the second half of the article, we extend the heat kernel estimates from [9] to the setting of Hilbert modular varieties, to derive quantitative estimates of the Bergman kernel associated to classical Hilbert modular cusp forms. Our estimates for the Bergman kernel are optimal, and complement the asymptotic estimates derived in the first half of the article.Some of the results of this article are known to experts, and the others are very much expected. So this article serves also as a comprehensive collection of estimates for the Bergman kernel associated to automorphic forms defined over quaternion algebras.

show abstract

“…Our estimate of the Bergman metric (2) is far from being optimal. As per evidence from the one-dimensional setting from [AM23], we conjecture the following estimate…”

Section: Let µ Volmentioning

confidence: 62%

Section: Introductionmentioning

confidence: 99%

Estimates of automorphic cusp forms over quaternion algebras

Aryasomayajula

Balasubramanyam

2018

Int. J. Number Theory

View full text Add to dashboard Cite

show abstract

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

Aryasomayajula

2015

Math. Z.

Self Cite

View full text Add to dashboard Cite

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 ...

show abstract

A relation of two metrics on a noncompact hyperbolic Riemann orbisurface

Cited by 2 publications

References 13 publications

Estimates of automorphic cusp forms over quaternion algebras

Estimates of automorphic cusp forms over quaternion algebras

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

Contact Info

Product

Resources

About