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Additional resources for Arithmetic of elliptic curves with complex multiplication
N=J(z)r"O on U, we have atfih( w)/ aWj = O,j = I, ... , n, hence tfih( w) are holomorphic in WI. . ,Wn. I Corollary 1. Let be a holomorphic map of a domain Dc en into en. If J(z) does not vanish in D, (D) is a domain in en, I Corollary 2. Let
I> s is m-to-one on ~;;' + ~~ + ~;' ,t. 0, and one-to-one on ~;;' + ~~ + (;' = O. Let C be the algebraic curve in pZ defined by ~;;'+~~+~;'=O. Then S is an m-fold branched covering ofpz with C as its branch locus of order (m -1). Then denoting by X(M) the Euler number of a manifold M, we have x(S) = mx(Pz) - (m -l)X( C). Substituting X(pZ) = 3 and X( C) = 2 - 2g = m(3 - m), we obtain x(S) = m(m 2 -4m+6). In general, let M m be a complex submanifold of a complex manifold W = W n • Then for given q EM, we can choose local coordinates Wq: P-+ 42 2.
W;) in a neighbourhood of q E M n D as we have fM(P) = h(w~, ... , w;;', 0, ... ,0). Let f( p) be a meromorphic function on W n • For any q E W n , we can choose a sufficiently small U(q) such thatf(p)=hq(p)/gq(p) on U(q) where hq (p) and gq (p) are relatively prime holomorphic functions. 13) there is a non-vanishing holomorphic function u(p) on U(ql) n U(q2) such that gql(p) = U(P)gq2(P) there. If q EM, the restrictions hqM (p), gqM (p) of hq( p), gq( p), respectively, to Mare holomorphic in M n U(q).