Abstract
and let Λ(N ) be the normalizer of Γ0(N ) in SL(2,R). We shall call a discrete subgroup ∆ of SL(2,R) a congruence group if it contains Γ (N ) for some N . Necessarily the index of Γ (N ) in ∆ is finite and ∆ acts on the extended upper half plane H ∗ = H ∪ Q ∪ {i∞} by fractional linear transformations. The quotient ∆\H ∗ has the structure of a compact Riemann surface and will be denoted by X (∆). Conway and Norton [CN] conjectured relationships between certain congruence groups and M known as the moonshine conjectures. These have been proved by Borcherds:
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