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MODULAR FORMS-page45

# MODULAR FORMS-page45 - z = e Î t z t In fact for any Î = tÎ...

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Lecture 5 Transformations of Theta Functions 5.1 Let us see now that the theta constants ϑ ab and their derivatives ϑ ab satisfy the functional equation similar to (4.2). This will imply that certain powers of theta constants are modular forms. For brevity we denote the lattice Z + τ Z by Λ τ . Theorem 5.1. Let ϑ ( z ; τ ) Th( k ; Λ τ ) ab and M = α β γ δ SL(2 , Z ) . Then e - ( kγz 2 γτ + δ ) ϑ ( z γτ + δ ; ατ + β γτ + δ ) Th( k ; Λ τ ) a b , where ( a , b ) = ( αa + γb - kγα 2 , βa + δb + kβδ 2 ) . Proof. First observe that for any f ( z ) Th( { e λ } ; Λ) and t C * , φ ( z ) = f ( z t ) Th( {
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Unformatted text preview: z ) = e Î» t ( z t ) . In fact, for any Î» = tÎ» âˆˆ t Î›, Ï† ( z + tÎ» ) = f ( z + tÎ» t ) = f ( z t + Î» ) = e Î» ( z t ) f ( z t ) = e Î» t ( z t ) Ï† ( z ) . We have Th( k ;Î› Ï„ ) ab = Th( { e Î» } ; Z + Ï„ Z ) , where e m + nÏ„ ( z ) = e 2 Ï€i ( ma-nb ) e-Ï€ik (2 nz + n 2 Ï„ ) . (5.1) For any M = â€ž Î± Î² Î³ Î´ Â« âˆˆ SL(2 , Z ) we have ( Î³Ï„ + Î´ ) Z + ( Î±Ï„ + Î² ) Z = Z + Ï„ Z . 41...
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