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MODULAR FORMS-page90

MODULAR FORMS-page90 - 86 LECTURE 8 THE MODULAR CURVE...

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86 LECTURE 8. THE MODULAR CURVE function φ ( N ). The number of elements b is N . This gives the index of Γ( N ) in Γ 0 ( N ). The index of Γ 0 ( N ) in Γ(1) is equal to the number of elements in the orbit of (1 , 0). It is the set of pairs ( a,b ) Z /N which are coprime modulo N . This is easy to compute. Lemma 8.6. There are no elliptic points for Γ( N ) if N > 1 . The number of cusps is equal to μ N /N . Each of them is of order N . Proof. The subgroup Γ = Γ( N ) is normal in Γ(1). If Γ τ 6 = 1 } , then g Γ g - 1 = Γ i for any g Γ(1) which sends τ to i . Similarly for elliptic points of order 3 we get a subgroup of Γ fixing e 2 πi/ 3 . It is easy to see that only the matrices 1 or - 1 , if N = 2, from Γ( N ) satisfy this property. We leave to the reader to prove the assertion about the cusps. Next computation will be given without proof. The reader is referred to [Shimura] . Lemma 8.7.
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