Torsion-free genus zero congruence subgroups of PSL2(ℝ)

Sebbar, Ahmed

doi:10.1215/s0012-7094-01-11028-4

Cited by 23 publications

(48 citation statements)

References 14 publications

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“…We remark that, in the torsionfree genus zero case, all subgroups are congruence when TV = 4, and no subgroups are congruence when TV = 5. For details see Sebbar [6].…”

Section: T H E M a I N R E S U L Tmentioning

confidence: 99%

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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“…We remark that, in the torsionfree genus zero case, all subgroups are congruence when TV = 4, and no subgroups are congruence when TV = 5. For details see Sebbar [6].…”

Section: T H E M a I N R E S U L Tmentioning

confidence: 99%

Existence conditions for a class of modular subgroups of genus zero

Hempel¹

2002

Bull. Austral. Math. Soc.

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“…The list of torsion-free congruence subgroups of genus 0 was completed in 2001 and given in [28] (there are 33 and the levels are all of the form 2 a 3 b 5 c 7 with a 5, b 3, and c 2 with 2 5 being the largest level). Of those only 4 are principal congruence subgroups (of levels 2, 3, 4 and 5).…”

Section: Final Remarksmentioning

confidence: 99%

Principal congruence link complements

Baker¹,

Reid²

2014

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…Beauville's families. We start with certain index 12 genus 0 torsion free congruence subgroups of SL 2 (Z), listed in Table 5 [ Seb01]. Figure 3.1 shows corresponding fundamental domains and generating matrices.…”

Section: 1mentioning

confidence: 99%