MODULAR FORMS-page62

MODULAR FORMS-page62 - 58 LECTURE 6. MODULAR FORMS as a...

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58 LECTURE 6. MODULAR FORMS as a holomorphic map from C \{ e 1 ,e 2 ,e 3 } to ( C / Λ τ ) / ( z → - z ). It can be shown that it extends to a holomorphic isomorphism from the Weiertrass cubic y 2 = 4 x 3 - g 2 x - g 3 onto ( C / Λ τ ) \ { 0 } . This is the inverse of the map given by z ( ( z ) ,℘ ( z ) 0 ). As was first shown by Euler, the elliptic integral (6.23) with special values of g 2 and g 3 over a special path in the real part of the complex plane x gives the value of the length of an arc of an ellipse. This explains the names “elliptic”. 6.5 Next we shall show that, considered as functions of the lattice Λ = Z + Z τ , and hence as functions of τ , the coefficients g 2 and g 3 are modular forms of level 4 and 6, respectively. Set for any positive even integer k : E k ( τ ) = X λ Λ τ \{ 0 } 1 λ k . Assume | τ | > R > 0 and k > 2. Since X λ Z + τ Z \{ 0 } 1 | λ | k < Z Z | x + iy | >R | x + iy | - k dxdy = Z R 2 π Z 0 r - k +1 drdθ = 2 π Z R r 1 - k dr, we see that E k ( τ ) is absolutely convergent on any compact subset of H . Thus E k ( τ ) are holomorphic functions on H for k > 2. From (6.21) we infer g 2 = 60 E 4 , g 3 = 140 E 6 . (6.24) We have E k ( ατ
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This note was uploaded on 01/08/2012 for the course MATH 300 taught by Professor Ontonkong during the Fall '09 term at SUNY Stony Brook.

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