MODULAR FORMS-page72

# MODULAR FORMS-page72 - 68 LECTURE 7 THE ALGEBRA OF MODULAR...

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Unformatted text preview: 68 LECTURE 7. THE ALGEBRA OF MODULAR FORMS In particular, this is true for g 2 . For any other f ∈ M 2 (Γ(1)) we have f/g 2 is Γ(1) invariant and also holomorphic at ∞ (since g 2 is not a cusp form). This shows that f/g 2 is constant and M 2 (Γ(1)) = C g 2 . Similar arguments show that M 3 (Γ(1)) = C g 3 , M 4 (Γ(1)) = C g 2 2 , M 5 (Γ(1)) = C g 2 g 3 . This checks the assertion for k < 6. Now for any cuspidal form f ∈ M k (Γ(1)) with k > 6 we have f/ Δ is a modular form of weight k- 6 (because Δ does not vanish on H and has a simple zero at infinity). This shows that for k > 6 M k (Γ(1)) = Δ M k- 6 (Γ(1)) . (7.4) Since M k (Γ(1)) / M k (Γ(1)) ∼ = C (we have only one cusp) we obtain for k > 6 dim M k (Γ(1)) = dim M k- 6 (Γ(1)) + 1 . Now the assertion follows by induction on k . Corollary 7.1. The algebra M (Γ(1)) is generated by the modular forms g 2 and g 3 . The homomorphism of algebras φ : C [ T 1 ,T 2 ] → M (Γ(1)) defined by sending T 1 to g 2 and T 2 to...
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