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72 LECTURE 7. THE ALGEBRA OF MODULAR FORMS 7.4 Our goal is to prove an analog of Theorem 7.1 for any subgroup of finite index Γ of Γ(1). Let Γ 0 Γ be two such subgroups. Assume also that Γ 0 is normal in Γ and let G = Γ / Γ 0 be the quotient group. The group G acts on M k 0 ) as follows. Take a representative g of ¯ g G . Then set, for any f ∈ M k 0 ), ¯ g · f = f | k g. Since f | k g 0 = f for any g 0 Γ 0 this definition does not depend on the choice of a representative. The following lemma follows from the definition of elements of M k (Γ). Lemma 7.2. Let Γ 0 be a normal subgroup of Γ and G = Γ / Γ 0 . Then M k (Γ) = M k 0 ) G = { f ∈ M k 0 ) : g · f = f, g G } . It follows from this lemma that the algebra M (Γ) is equal to the subalgebra of M 0 ) which consists of elements invariant with respect to the action of the group Γ / Γ 0 . Let n be the order of the group G = Γ / Γ 0 (recall that we consider only subgroups of finite index of Γ(1)) . For any
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This note was uploaded on 01/08/2012 for the course MATH 300 taught by Professor Ontonkong during the Fall '09 term at SUNY Stony Brook.

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