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About Gamma0(n) and its
Normalizer in the Rational cases
Abstract
Let Gamma0(n), n in N, be the group of transformations z -> (az+b)/(cz+d), a,b,c,d in Z, ad-bc = 1, n|c, of the upper half of the complex plane with corresponding field K(n) of invariant (modular) functions. A thorough investigation of these groups is presented including a formula for the length of a minimal generating system and a method to construct "canonical" fundamental regions. In the cases of genus zero the generating function of K(n) is explicitly determined. The same investigation is done for some groups between Gamma0(n) and its normalizer in PSL2(R) with special attention to Fricke-Involutions and the "translation part" of the normalizer. For the latter a formula for the genus is presented.
Reference
Markus Püschel (Diploma Thesis Mathematics 1994, ref.: Prof. Dr. H.-W. Leopoldt, 144 pages)
Über Gamma0(n) und seinen Normalisator in den rationalen Fällen (On Gamma0(n) and its Normalizer in the Rational Cases)
dipl.ps (1361 KB, German)
Example: Fundamental Region and some Data for Gamma0(16)
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