MODULAR FORMS-page89

MODULAR FORMS-page89 - 85 Definition. Let X = H∗ /Γ. A...

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Unformatted text preview: 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 Γτ = 1. Theorem 8.4. The genus of H∗ /Γ is equal to g =1+ µΓ r2 r3 r∞ − − − , 12 4 3 2 where µΓ is the index of Γ/Γ ∩ (±1) in Γ(1)/(±1), r2 is the number of elliptic points of Γ of order 2, r3 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 µΓ is equal to the degree of the meromorphic function X (Γ) → X (Γ(1)) ∼ P1 (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 X 2g − 2 = −2µ + (ex (j ) − 1) = x∈X −2µ + X X (ex (j ) − 1) + j (x)=j (i) (ex (j ) − 1) + X (ex (j ) − 1). j (x)=∞ j (x)=j (ρ) We have (µ − r2 )/2 points over j (i) with ex (j ) = 2 and (µ − r3 )/3 points over j (ρ) with ex (j ) = 3. Also by (8.21), the sum of indices of cusps is equal to µ. This gives 2g − 2 = −2µ + (µ − r2 )/2 + 2(µ − r3 )/3 + (µ − r∞ ), hence g =1+ µ r2 r3 r∞ − − − . 12 4 3 2 We shall concentrate on the special subgroups Γ of Γ(1) introduced earlier. They are the principal congruence subgroup Γ(N ) of level N and „ « αβ Γ0 (N ) = { ∈ SL(2, Z) : N |γ }. γδ Obviously Γ(N ) ⊂ Γ0 (N ). Lemma 8.5. ( µ0,N := µΓ0 (N ) 1 N3 2 Q p|N (1 − p−2 ) if N > 2, if N = 2, Y = [Γ(1) : Γ0 (N )] = N (1 + p−1 ), µN := µΓ(N ) = 6 (8.23) p|N where p denotes a prime number. Proof. This easily follows from considering the action of the group SL(2, Z/N ) on the set (Z/N )2 . The isotropy subgroup of the vector (1, 0) is isomorphic to the group „ « a b of Γ0 (N )/Γ(N ) ⊂ SL(2, Z/N ). It consists of matrices of the form . The 0 a−1 number of invertible elements a in the ring Z/N is equal to the value of the Euler ...
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This note was uploaded on 01/08/2012 for the course MATH 300 taught by Professor Ontonkong during the Fall '09 term at SUNY Stony Brook.

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