~. 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 = ~.

4bJ, which says that deg(Kx) = 2g - 2. 0 The next proposition gives the precise relationship between a modular function J of weight 2k and the corresponding k-form J(7} (d7)k. 7. Let J be a non-zero modular function of weight 2k. (a) The k-form J(7) (d7)k on H descends to give a meromorphic k-form wf on the Riemann surface X(l). In other words, there is a k-form Wf E n~(1) such that ¢*(wf) = J(7) (d7)k, ! where ¢ : H -> X(l) is the usual projection. (b) Let x E X(l), and let 7x E H* with ¢(7x ) = x.