MODULAR FORMS-page45

MODULAR FORMS-page45 - z ) = e t ( z t ) . In fact, for any...

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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 0 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 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 τ ) a 0 b 0 , where ( a 0 ,b 0 ) = ( 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( { e 0 λ 0 } ; t Λ) , where e 0 λ 0 (
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Unformatted text preview: z ) = e t ( z t ) . In fact, for any = t t , ( z + t ) = f ( z + t t ) = f ( z t + ) = e ( z t ) f ( z t ) = e t ( z t ) ( z ) . We have Th( k ; ) ab = Th( { e } ; Z + Z ) , where e m + n ( z ) = e 2 i ( ma-nb ) e-ik (2 nz + n 2 ) . (5.1) For any M = SL(2 , Z ) we have ( + ) Z + ( + ) Z = Z + Z . 41...
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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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