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MODULAR FORMS-page89

# MODULAR FORMS-page89 - 85 Denition Let X = H A point x = is...

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85 Definition. Let X = H * / Γ. A point x = Γ · τ is called an elliptic point of order 2 (resp, of order 3) if τ Γ(1) · i (resp. τ Γ(1) · ρ ) and Γ τ = 1. Theorem 8.4. The genus of H * / Γ is equal to g = 1 + μ Γ 12 - r 2 4 - r 3 3 - r 2 , where μ Γ is the index of Γ / Γ ( ± 1) in Γ(1) / ( ± 1) , r 2 is the number of elliptic points of Γ of order 2, r 3 is the number of elliptic points of Γ of order 3, and r is the number of cusps of Γ . Proof. Notice first that the number μ Γ is equal to the degree of the meromorphic function X (Γ) X (Γ(1)) = P 1 ( C ) defined by the j -function j : H C . In fact, the number of the points in the pre-image of a general z C is equal to the number of Γ-orbits in H contained in a Γ(1)-orbit. Applying (8.22), we have 2 g - 2 = - 2 μ + X x X ( e x ( j ) - 1) = - 2 μ + X j ( x )= j ( i ) ( e x ( j ) - 1) + X j ( x )= j ( ρ ) ( e x ( j ) - 1) + X j ( x )= ( e x ( j ) - 1) . We have ( μ - r 2 ) / 2 points over j ( i ) with e x ( j ) = 2 and ( μ - r 3 ) / 3 points over j ( ρ ) with e x ( j ) = 3. Also by (8.21), the sum of indices of cusps is equal to μ . This gives 2 g - 2 = - 2 μ + ( μ - r 2 ) / 2 + 2( μ - r 3 ) / 3 + ( μ - r ) , hence g = 1 + μ 12 - r 2 4 - r 3 3 - r 2 . We shall concentrate on the special subgroups Γ of Γ(1) introduced earlier. They
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