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MODULAR FORMS-page52 - 48 LECTURE 5. TRANSFORMATIONS OF...

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Unformatted text preview: 48 LECTURE 5. TRANSFORMATIONS OF THETA FUNCTIONS 5.7 Define the Weierstrass σ -function by ω2 z 2 −z 2 (ϑ 1 1 /6ω1 ϑ 1 1 ) ϑ 1 1 ( ω ; ω ) 1 1 22 σ (z ; ω1 , ω2 ) = ω1 e 22 22 ϑ 1 1 (0) . 22 Show that (i) σ (z ; ω1 , ω2 ) does not depend on the basis ω1 , ω2 of the lattice Λ; (ii) σ (−z ) = −σ (z ); (iii) σ (z + ω1 ) = −eη1 (z+ω1 2) σ (z ), where η1 = σ (ω1 2)/σ (ω1 2); σ (z + ω2 ) = −eη2 (z+ω2 2) σ (z ), η2 = σ (ω2 2)/σ (ω2 2). (iv) (Legendre-Weierstrass relation) η1 ω2 − η2 ω1 = 2πi. [Hint: integrate the function σ along the fundamental parallelogram using (iii)]; (v) η1 = − πi d log η (τ ) , 2 ω1 dτ η2 = − πiω2 d log η (τ ) π − , 2 ω1 dτ 2ω1 where τ = ω2 /ω1 . 5.8 Using formulas from Lecture 4 prove the following infinite product expansion of σ (z ; ω1 , ω2 ): σ (z ; ω1 , ω2 ) = ω 2πi ω2 where q = e 1 ∞ 2 Y (1 − q m v −2 )(1 − q m v 2 ) ω1 η1 z , e 2ω1 (v − v −1 ) (1 − q m )2 2πi m=1 , v = eπiz/ω1 . ...
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