Lecture 8The Modular Curve8.1In this lecture we shall give an explicit formula for the dimension of the spacesMk(Γ), where Γ is any subgroup of finite index in SL(2,Z). For this we have to applysome techinique from algebraic geometry. We shall start with equippingH*/Γ with astructure of a compact Riemann surface.Let Γ be a subgroup of SL(2,R). We say that Γ is adiscrete subgroupif the usualtopology in SL(2,R) (considered as a subset ofR4) induces a discrete topology inΓ. 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 naturalaction of SL(2,R) on the upper half-planeHby Moebius transformations.Lemma 8.1.Any discrete subgroupΓofSL(2,R)acts onHproperly discontinuously.Proof.Observe that the group SL(2,R) acts transitively onH(view the latter as asubset ofR2of vectors with positive second coordinate).For any pointz∈ Hthestabilizer group is conjugate to the stabilizer of sayz=i.The latter consists ofmatrices (a bc d)∈SL(2,R) such thata=d, b=-c.It follows that this group is
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