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MODULAR FORMS-page81 - Lecture 8 The Modular Curve 8.1 In...

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Lecture 8 The Modular Curve 8.1 In this lecture we shall give an explicit formula for the dimension of the spaces M k (Γ), where Γ is any subgroup of finite index in SL(2 , Z ). For this we have to apply some techinique from algebraic geometry. We shall start with equipping H * / Γ with a structure of a compact Riemann surface. Let Γ be a subgroup of SL(2 , R ). We say that Γ is a discrete subgroup if the usual topology in SL(2 , R ) (considered as a subset of R 4 ) induces a discrete topology in Γ. The latter means that any point of Γ is an open subset in the induced topology. Obviously SL(2 , Z ) is a discrete subgroup of SL(2 , R ). We shall consider the natural action of SL(2 , R ) on the upper half-plane H by Moebius transformations. Lemma 8.1. Any discrete subgroup Γ of SL(2 , R ) acts on H properly discontinuously. Proof. Observe that the group SL(2 , R ) acts transitively on H (view the latter as a subset of R 2 of vectors with positive second coordinate). For any point z ∈ H the stabilizer group is conjugate to the stabilizer of say z = i . The latter consists of matrices ( a b c d ) SL(2 , R ) such that a = d, b = - c . It follows that this group is
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