MODULAR FORMS-page104

MODULAR FORMS-page104 - 100 LECTURE 10. THE MODULAR...

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100 LECTURE 10. THE MODULAR EQUATION Example 10.1 . Take N = 2 and f = Δ( τ ) ∈ M (Γ(1)) 6 . We see that Δ(2 τ ) / Δ( τ ) belongs to the space M ( X 0 (2)). Observe that q = e 2 πiτ changes to q 2 when we replace τ with 2 τ . So Δ(2 τ ) / Δ( τ ) = q 2 Q m =1 (1 - q 2 m ) 24 q Q m =1 (1 - q m ) 24 = q Y m =1 (1 + q m ) 24 = 2 - 12 f 2 ( τ ) 24 , (10.2) where f 2 ( τ ) is the Weber function defined in (4.13). In particular, we see that f 24 2 = 2 12 Δ(2 τ ) / Δ( τ ) (10.3) is a modular function with respect to Γ 0 (2). It follows from (10.2) that f 24 2 has a simple zero at the cusp . The index of this cusp is equal to 1 since ( 1 1 0 1 ) Γ 0 (2). We know from Lemma 8.5 that μ 0 , 2 = [ ¯ Γ(1) : Γ 0 (2)] = 3. Thus Γ 0 (2) has another cusp of index 2. Since 0 6∈ Γ 0 (2) · ∞ we can represent it by 0. We have f 24 2 ( - 1 ) = 2 12 Δ( - 2 ) / Δ( - 1 ) = 2 12 Δ( - 1 / ( τ/ 2)) / Δ( - 1 ) = 2 12 ( τ/ 2) 12 Δ( τ/ 2) 12 Δ( τ ) = Δ(
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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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