MODULAR FORMS-page69

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

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Unformatted text preview: Lecture 7 The Algebra of Modular Forms 7.1 Let Γ be a subgroup of finite index of Γ(1). We set Mk (Γ) = {modular forms of weight k with respect to Γ}, We also denote by Mk (Γ)0 the subspace of cuspidal modular forms. It is clear that Mk (Γ) is a vector space over C. Also multiplication of functions defines a bilinear map Mk (Γ) × Ml (Γ) → Mk+l (Γ). This allows us to consider the direct space M(Γ) = ∞ M Mk (Γ) (7.1) k=−∞ as a graded commutative algebra over C. Since Mk (Γ) ∩ Ml (Γ) = {0} if k = l, we may view M(Γ) as a graded subalgebra of O(H). Notice that ∞ M Mk (Γ)0 (7.2) M(Γ)0 = k=−∞ is an ideal in M(Γ). We shall see later that there are no modular forms of negative weight. 7.2 Our next goal is to prove that the algebra M(Γ) is finitely generated. In particular each space Mk (Γ) is finite-dimensional. Let f (z ) be a meromorphic function in a neighborhood of a point a ∈ C and let f (z ) = ∞ X cn (z − a)n n=m be its Laurent expansion in a neighborhood of the point a. We assume that cm = 0 and set νa (f ) = m. We shall call the number νa (f ) the order ( of zero if m ≥ 0 or of pole if m < 0) of f at a. If f is meromorphic at ∞ we set ν∞ (f ) = ν0 (f (1/z )). 65 ...
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