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MODULAR FORMS-page46

# MODULAR FORMS-page46 - 42 LECTURE 5 TRANSFORMATIONS OF...

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42 LECTURE 5. TRANSFORMATIONS OF THETA FUNCTIONS Thus for any ϑ Th( k ; Λ τ ) ab , we have ϑ ( z ( γτ + δ )) Th( e λ ; Z + τ Z ) , where τ = ατ + β γτ + δ , e m + ( z ) = e ( m + )( γτ + δ ) ( z ( γτ + δ )) . We have, using (5.1), e 1 ( z ) = e γτ + δ ( z ( γτ + δ )) = e 2 πi ( - ) e - πik (2 γz ( γτ + δ )+ γ 2 τ ) = e - πikγ (( γτ + δ )( z +1) 2 - ( γτ + δ ) z 2 ) e πikγδ e 2 πi ( - ) . This shows that e πikγ ( γτ + δ ) z 2 Th( { e λ ( z ) } ; Z + τ Z ) = Th( { e λ ( z ) } ; Z + τ Z ) , where e 1 ( z ) = e πi [ kγδ +2( - )] . (5.2) Now comes a miracle! Let us compute e τ ( z ). We have e τ ( z ) = e πikγ ( γτ + δ )(( z + τ ) 2 - z 2 ) e τ ( γτ + δ ) ( z ( γτ + δ )) = e πikγ ( γτ + δ )(2 + τ 2 ) e β + ατ ( z ( γτ + δ )) = e πik [ γ ( γτ + δ )(2 + τ 2 ) - (2 αz ( γτ + δ )+ α 2 τ ) e 2 πi ( - + βa )] = e iπikG e 2 πi ( - + βa ) , (5.3) where G = γ ( γτ + δ )(2 + τ 2 ) - 2 αz ( γτ + δ ) - α 2 τ = γ ( γτ + δ )(2 z ( ατ + β γτ + δ ) + ( ατ + β γτ + δ ) 2 ) - 2 αz ( γτ + δ ) - α 2 τ = 2 ( ατ + β ) + γ ( ατ + β ) 2 γτ + δ - 2 αz ( γτ + δ ) - α 2 τ = - 2 z + γ ( ατ + β ) 2 - α 2 τ ( γτ
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