Advanced Topics in the Arithmetic of Elliptic Curves by Joseph H. Silverman

~. C 53 §6. q-Expansions of Elliptic Functions (b) 1 1 { qn } ·)21](1)=27rZ( 12 -1+242:( 1 - qn )2. n2:1 PROOF. Let We proved above that 1 27rZ 1 2 - . ((z; T) = G(z; T) - 27riG1 (q) - -. Now evaluate at z = ~. 2b), ((~;T) = 1](1). Further, z = ~ corresponds to u = e rri = -1, so all of the terms in G( ~; T) cancel except the n = 0 term.

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