scholarly journals On Rademacher's conjecture: congruence subgroups of genus zero of the modular group

2004 ◽  
Vol 277 (1) ◽  
pp. 408-428 ◽  
Author(s):  
Kok Seng Chua ◽  
Mong Lung Lang ◽  
Yifan Yang
Keyword(s):  
Modular Group ◽  
Genus Zero ◽  
Congruence Subgroups

scholarly journals Existence conditions for a class of modular subgroups of genus zero

2002 ◽  
Vol 66 (3) ◽  
pp. 517-525
Author(s):  
Joachim A. Hempel
Keyword(s):  
Rational Function ◽  
Finite Index ◽  
Modular Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Genus Zero ◽  
Existence Conditions ◽  
Necessary And Sufficient

Every subgroup of finite index of the modular groupPSL(2, ℤ) has asignatureconsisting 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 functionFwith 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.

scholarly journals Macbeath's curve and the modular group

1985 ◽  
Vol 27 ◽  
pp. 239-247 ◽  
Author(s):  
K. Wohlfahrt
Keyword(s):  
Finite Index ◽  
Modular Group ◽  
Algebraic Curves ◽  
The Other ◽  
Congruence Subgroups ◽  
Related Research ◽  
Other Hand

The theory of algebraic curves associated with subgroups of finite index in the modular group Γ is highly developed for such subgroups of Γ as may be defined by means of congruences in the ring ℤ of rational integers. The situation in he case of non-congruence subgroups of Γ, on the other hand, is drastically different. It reduces to a few isolated examples, two of which may be found in [12]. Related research by A. O. L. Atkin and H. P. F. Swinnerton-Dyer began in [1].

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