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MODULAR FORMS-page67 - 63(iii Show that L is almost modular...

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63 (iii) Show that θ L ( τ ) is “almost” modular form for the group Γ 0 ( N ) = { α β γ δ « SL(2 , Z ) : N | c } , i.e. θ L ( ατ + β γτ + δ ) = ( γτ + δ ) k/ 2 χ ( d ) θ L ( τ ) , α β γ δ « Γ 0 ( N ) , where χ ( d ) = ( ( - 1) k 2 D d ) is the quadratic residue symbol. (iv) Prove that θ L ( τ ) is a modular form for Γ 0 (2) whenever D = 1 and k 0 mod 4. 6.11 Let Φ( z ; τ ) be a function in z and τ satisfying the assumptions of Theorem 6.2 (such a function is called a Jacobi form of weight m and index 0 with respect to the group Γ). Show that (i) ( z ; τ ) is a Jacobi form of weight 2 and index 0 with respect to Γ(1); (ii) σ ( z ; 1 , τ ) is a Jacobi form of weight 1 with respect to Γ(1). 6.12 Let n be a positive integer greater than 2. Consider the map of a complex torus E τ \ { 0 } → C n given by the formula z (1 , ℘ ( z ) , . . . , ℘ ( z ) n - 1 2 , ℘ ( z ) , ℘ ( z ) ( z ) , . . . , ℘ ( z ) n - 3 2 ( z ) ) if n is odd and z (1 , ℘ ( z ) , . . . , ℘ ( z ) n 2 , ℘
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