'l/Jj(Ui n Uj ) is holomorphic. The map 'l/Ji is called a local parameter for the points in Ui .

1. Each of the vector spaces has dimension 1. ) Similarly, GlO = t1 G 4 G 6 and G14 = 134°3 G~G6. More generally, M2k has a basis consisting of functions of the form G 4Gg. 1) of the G 2k 'S. 34 I. 2. 10b), the spaces also have dimension 1. In particular, up to multiplication by a constant, there is only one cusp form of weight 12, namely ~(7). §4. 1]. This theorem says that every elliptic curve over IC is parametrized by Weierstrass elliptic functions. 7(a), which says in particular that every modular function of weight 0 defines a meromorphic function on the Riemann surface X(l).

TEr(l)\H* T,pi,p,oo Equating these two expressions for deg( div W f) gives the desired formula. 8, we can give a good description of the space of all modular forms of a given weight. We set the notation M2k = {modular forms Mgk = of weight 2k for r(1)}, {cusp forms of weight 2k for reI)}. Note that both M2k and Mgk are C-vector spaces. 9. For all k ~ 2, the Eisenstein series G 2k(r) is in M2k but is not in Mgk • The modular discriminant 6(r) is in MP2. 3). 10. (a) For all integers k ~ 2, M2k ~ Mgk +CG 2k· (b) For all integers k, the map is an isomorphism of C- vector spaces.

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

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