MODULAR FORMS-page69 - Lecture 7 The Algebra of Modular...

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Lecture 7 The Algebra of Modular Forms 7.1 Let Γ be a subgroup of finite index of Γ(1). We set M k (Γ) = { modular forms of weight k with respect to Γ } , We also denote by M k (Γ) 0 the subspace of cuspidal modular forms. It is clear that M k (Γ) is a vector space over C . Also multiplication of functions defines a bilinear map M k (Γ) × M l (Γ) → M k + l (Γ) . This allows us to consider the direct space M (Γ) = M k = -∞ M k (Γ) (7.1) as a graded commutative algebra over C . Since M k (Γ) ∩ M l (Γ) = { 0 } if k = l , we may view M (Γ) as a graded subalgebra of O ( H ). Notice that M (Γ) 0 = M k = -∞ M k (Γ) 0 (7.2) is an ideal in M (Γ) . We shall see later that there are no modular forms of negative weight.
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