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 ((1)) we have f/g 2 is (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 ((1)) = C g 2 . Similar arguments show that M 3 ((1)) = C g 3 , M 4 ((1)) = C g 2 2 , M 5 ((1)) = C g 2 g 3 . This checks the assertion for k < 6. Now for any cuspidal form f M k ((1)) with k > 6 we have f/ is a modular form of weight k- 6 (because does not vanish on H and has a simple zero at infinity). This shows that for k > 6 M k ((1)) = M k- 6 ((1)) . (7.4) Since M k ((1)) / M k ((1)) = C (we have only one cusp) we obtain for k > 6 dim M k ((1)) = dim M k- 6 ((1)) + 1 . Now the assertion follows by induction on k . Corollary 7.1. The algebra M ((1)) is generated by the modular forms g 2 and g 3 . The homomorphism of algebras : C [ T 1 ,T 2 ] M ((1)) defined by sending T 1 to g 2 and T 2 to...
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