MODULAR FORMS-page47

MODULAR FORMS-page47 - 43 where k k , a + b ). 2 2...

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43 where ( a 0 ,b 0 ) = ( δa - γb + kγδ 2 , - βa + αb - kαβ 2 ) . (5.4) Summarizing we obtain that, for any ϑ ( z,τ ) Th( k τ ) ab , e iπkγ ( γτ + δ ) z 2 ϑ (( γτ + δ ) z ; τ ) Th( k, Λ τ 0 ) a 0 b 0 . (5.5) Now let us replace α β γ δ with its inverse ` - δ - β - γ α ´ . We rewrite (5.13) and (5.14) as e - iπkγ ( - γτ + α ) z 2 ϑ (( - γτ + α ) z ; τ ) Th( k, Λ τ 0 ) a 0 b 0 , (5.6) where ( a 0 ,b 0 ) = ( αa + γb - kγα 2 ,βa + δb + kδβ 2 ) . It remains to replace τ with ατ + β γτ + δ in (5.15) to obtain the assertion of the theorem. Substituting z = 0 we get Corollary 5.1. Let ϑ 1 ( z,τ ) ,..., ϑ k ( z ; τ ) be a basis of the space Th( k τ ) ab and ϑ 0 1 ( z,τ ) ,...,ϑ 0 k ( z ; τ ) be a basis of Th( k τ ) a 0 b 0 , where ( a 0 ,b 0 ) are defined in the Theo- rem. Then, for any M = α β γ δ SL(2 , Z ) there exists a matrix A = ( c ij ) GL( k, C ) depending on M and τ only such that ϑ i (0 , ατ + β γτ + δ ) = k X j =1 c ij ϑ 0 j (0 ) . 5.2 Let us take k = 1 and ( a,b ) = ( ±/ 2 ,η/ 2) ,±,η = 0 , 1. Applying the previous Theorem, we get ϑ ab ( z ; τ + 1) = a,b + a + 1 2 ( z ; τ ) for some C depending only on τ and ( a,b ). In particular,
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