MODULAR FORMS-page51

MODULAR FORMS-page51 - 47 (iv) Show that the expression...

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Unformatted text preview: 47 (iv) Show that the expression Qn−1 ν =1 ν ν ν ϑ00 ( n ; τ )ϑ 1 0 ( n ; τ )ϑ0 1 ( n ; τ ) 2 2 ϑ00 (0; τ )n−1 ϑ 1 0 (0; τ )n−1 ϑ0 1 (0; τ )n−1 2 . 2 does not change when τ is replaced with 2τ . (v) Show that Qn−1 ν =1 ν ν ν ϑ00 ( n ; τ )ϑ 1 0 ( n ; τ )ϑ0 1 ( n ; τ ) 2 2 ϑ00 (0; τ )n−1 ϑ 1 0 (0; τ )n−1 ϑ0 1 (0; τ )n−1 2 (−1) = 2 Q n−1 2 n−1 2 ` ν ν ν ϑ00 ( n ; τ )ϑ 1 0 ( n ; τ )ϑ0 1 ( n ; τ ) ´2 2 2 ϑ00 (0; τ )n−1 ϑ 1 0 (0; τ )n−1 ϑ0 1 (0; τ )n−1 ν =1 2 2 (vi) Prove the formula Q n−1 2 ν =1 ϑ00 ν ν ν ϑ00 ( n ; τ )ϑ 1 0 ( n ; τ )ϑ0 1 ( n ; τ ) 2 2 (0; τ )n−1 ϑ 1 n−1 ϑ 1 (0; τ )n−1 0 (0; τ ) 02 2 z 5.6 Let Λ = Zω1 + Zω2 . Set t(z ; ω1 , ω2 ) = ϑ 1 1 ( ω1 ; 22 =2 1−n 2 . ω2 ). ω1 (i) Show that t(z + ω1 ; ω1 , ω2 ) = −t(z ; ω1 , ω2 ), −πi t(z + ω2 ; ω1 , ω2 ) = −e 2z +ω2 ω1 t(z ; ω1 , ω2 ). (ii) Let ω1 , ω2 be another basis of Λ. Show that t(z ; ω1 , ω2 ) = Ceaz 2 +bz t(z ; ω1 , ω2 ) for some constants C, a, b. (iii) By taking the logarithmic derivative of both sides in (ii) show that a=− t (0; ω1 , ω2 ) t (0; ω1 , ω2 ) + , 6t (0; ω1 , ω2 ) 6t (0; ω1 , ω2 ) b= t (0; ω1 , ω2 ) , 2t (0; ω1 , ω2 ) and t (0; ω1 , ω2 ) ; t (0; ω1 , ω2 ) C= (iv) using (iii) show that ϑ 1 1 (0) a=− 22 2 6ϑ 1 1 (0)ω1 ϑ 1 1 (0) + 22 22 6ϑ 1 1 (0)ω1 2 22 and b = 0; (v) using the Heat equation (see Exercise 3.8) show that ϑ 1 1 (0) 22 ϑ 1 1 (0) 22 where τ = ω2 . ω1 = 12πi d log η (τ ) , dτ ...
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