This preview shows page 1. Sign up to view the full content.
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
This is the end of the preview. Sign up to access the rest of the document.