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MODULAR FORMS-page50

# MODULAR FORMS-page50 - Using Exercise 5.2 show ϑ 1 2(0;2...

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46 LECTURE 5. TRANSFORMATIONS OF THETA FUNCTIONS Exercises 5.1 Show that the constant ζ ( M ) in (5.16) is equal to i δ - 1 2 ( γ | δ | ) when γ is even and δ is odd. If γ is odd and δ is even, it is equal to e - πiγ/ 4 ( δ γ ). Here ( x y ) is the Jacobi-Legendre symbol, where we also set ( 0 1 ) = 1. 5.2 Extend the transformation law for theta functons by considering transformations deﬁned by matrices α β γ δ with determinant n not necessary equal to 1: e - πi nkγz 2 γτ + δ ϑ ( nz γτ + δ ; ατ + β γτ + δ ) Th( nk, Λ τ ) a 0 b 0 , where ϑ ( z ; τ ) Th( k, Λ τ ) ab and ( a 0 ,b 0 ) = ( αa + γb - kγα 2 ,βa + δb + kδβ 2 ) . 5.3 Using the previous exercise show that (i) 1 2 1 2 ( z ; τ/ 2) = ϑ 0 1 2 ( z ; τ ) ϑ 1 2 1 2 ( z ; τ ) for some constant A ; (ii) A 0 ϑ 1 2 0 ( z ; τ/ 2) = ϑ 00 ( z ; τ ) ϑ 1 2 0 ( z ; τ ) for some constant A 0 ; (iii) ( Gauss’ transformation formulas ϑ 1 2 0 (0; τ/ 2) ϑ 1 2 1 2 ( z ; τ/ 2) = 2 ϑ 1 2 0 ( z ; τ ) ϑ 1 2 1 2 ( z ; τ ) , ϑ 0 1 2 (0; τ/ 2) ϑ 1 2 0 ( z ; τ/ 2) = 2 ϑ 00 ( z ; τ ) ϑ 1 2 0 ( z ; τ ) , [Hint: Apply (3.14) to get A = A 0 , then diﬀerentiate (i) and use the Jacobi theorem]. 5.4 ( Landen’s transformation formulas
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Unformatted text preview: ) Using Exercise 5.2 show ϑ 1 2 (0;2 τ ) ϑ 1 2 1 2 (2 z ;2 τ ) = ϑ 1 2 ( z ; τ ) ϑ 1 2 1 2 ( z ; τ ) , ϑ 1 2 (0;2 τ ) ϑ 1 2 (2 z ;2 τ ) = ϑ 00 ( z ; τ ) ϑ 1 2 ( z ; τ ) , 5.5 Let n be an odd integer. (i) Show that, for any integer ν , ϑ 1 2 ( ν n ; τ ) depends only on the residue of ν modulo n . (ii) Show that n-1 Y ν =1 ϑ 1 2 ( ν n ; τ ) = n-1 Y ν =1 ϑ 1 2 ( 2 ν n ; τ ) . (iii) Using Exercises 5.3 and 5.4 show that ϑ 00 ( z ;2 τ ) ϑ 1 2 ( z ;2 τ ) ϑ 1 2 (2 z ;2 τ ) ϑ 00 (0;2 τ ) ϑ 1 2 (0;2 τ ) ϑ 1 2 (0;2 τ ) = ϑ 00 ( z ; τ ) ϑ 1 2 ( z ; τ ) ϑ 1 2 ( z ; τ ) ϑ 00 (0; τ ) ϑ 1 2 (0; τ ) ϑ 1 2 (0; τ ) ....
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