On the Uniformization of the <i>n</i>
-Punctured Sphere

Hempel, Joachim A.

doi:10.1112/blms/20.2.97

Cited by 39 publications

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“…In [3], the author described functions A in the case of no torsion, but in the general setting of Fuchsian groups. The present paper may be read for its results on rational 518 J.A.…”

Section: Preliminariesmentioning

confidence: 99%

“…ii\P' -E' 3 We could differentiate again and substitute in the previous expression to eliminate P and its derivatives, but refrain from doing so. Instead we examine what happens at points where 5 = 0.…”

Section: Lemma 4 Suppose That T Is a Low Torsion And High Frequencymentioning

confidence: 99%

“…A well known example is that corresponding to the expression for J in terms of the standard Hauptmodul for the subgroup F(2): [3] Modular Subgroups 519…”

Section: K=-oomentioning

confidence: 99%

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Existence conditions for a class of modular subgroups of genus zero

Hempel¹

2002

Bull. Austral. Math. Soc.

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Every subgroup of finite index of the modular group PSL(2, Z) has a signature consisting of conjugacy-invariant integer parameters satisfying certain conditions. In the case of genus zero, these parameters also constitute a prescription for the degree and the orders of the poles of a rational function F with the property:Functions correspond to subgroups, and we use this to establish necessary and sufficient conditions for existence of subgroups with a certain subclass of allowable signatures. PRELIMINARIESWe identify the classical modular group PSL 2 (Z) with the group of self-mappings of the upper half plane %, of the form r -> (ar + b)/(cr + d) where a, b, c, d are integers such that ad -be = 1. It is well known that the field of functions automorphic with respect to PSL2(Z) is the field of rational functions of just one such function J, analytic in H, having multiplicity two at all points equivalent to i, multiplicity three at all points equivalent to p, and normalised by J(i) = 0, J(p) = 1, where p = (1 + i\/3)/2. Suppose now that F is a subgroup of finite index of PSL 2 (%>), and of genus zero. Then again there exists a main function A, referred to in German as the "Hauptmodul", such that the field of functions automorphic with respect to F coincides with the rational functions of A. Since J is such a function, it follows that J = FoX, where F is a rational function. Since we are free to replace A by M~xoA, where M is a Mobius transformation, we are also free to replace F by F o M. In the sequel we shall call such a rational function, determined up to composition with M, the J-defining function for the genus zero subgroup.In [3], the author described functions A in the case of no torsion, but in the general setting of Fuchsian groups. The present paper may be read for its results on rational

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“…In [3], the author described functions A in the case of no torsion, but in the general setting of Fuchsian groups. The present paper may be read for its results on rational 518 J.A.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Lemma 4 Suppose That T Is a Low Torsion And High Frequencymentioning

confidence: 99%

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Existence conditions for a class of modular subgroups of genus zero

Hempel¹

2002

Bull. Austral. Math. Soc.

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“…If λ : ∆ → ∆/G ∼ = X is the universal covering map, the inverse ρ = λ −1 : X → ∆ is a multi-valued function with branching points z j and with branches related by elements of the covering group G ⊂ PSU (1,1). One can show that the Schwarzian derivative of ρ is a holomorphic function on X of the form [2] {ρ, z} = 1 2 It is a well known property of the Schwarzian derivative [2,4] that the map ρ is, to within a Möbius transformation, a quotient of two linearly independent solutions of the Fuchs equation This in particular means that there exists a unique to within SU(1,1) transformation fundamental system {Ψ 1 , Ψ 2 } of normalized (i.e. with the Wronskian equal to 1) solutions for which ρ = Ψ 1 Ψ 2 .…”

Section: Introductionmentioning

confidence: 99%

“…The universal covering map λ : ∆ → ∆/G is however explicitly known only for the thrice-punctured sphere [3] and in few very special, symmetric cases with higher number of punctures [2]. In particular an explicit construction of this map for the four-punctured sphere is a long standing and still open problem.…”

Section: Introductionmentioning

confidence: 99%

Liouville theory and uniformization of four-punctured sphere

Hadasz

Jaskólski

2006

Journal of Mathematical Physics

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Few years ago Zamolodchikov and Zamolodchikov proposed an expression for the 4-point classical Liouville action in terms of the 3-point actions and the classical conformal block [1]. In this paper we develop a method of calculating the uniformizing map and the uniformizing group from the classical Liouville action on n-punctured sphere and discuss the consequences of Zamolodchikovs conjecture for an explicit construction of the uniformizing map and the uniformizing group for the sphere with four punctures.

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Local Minimality Results Related to the Bloch and Landau Constants

BaernsteinII¹,

Vinson²

1998

Quasiconformal Mappings and Analysis

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On the Uniformization of the n -Punctured Sphere

Cited by 39 publications

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Existence conditions for a class of modular subgroups of genus zero

Existence conditions for a class of modular subgroups of genus zero

Liouville theory and uniformization of four-punctured sphere

Local Minimality Results Related to the Bloch and Landau Constants

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