By Jean-Pierre Serre

ISBN-10: 1461210356

ISBN-13: 9781461210351

ISBN-10: 1461269938

ISBN-13: 9781461269939

Precis of the most Results.- Algebraic Curves.- Maps From a Curve to a Commutative Group.- Singular Algebraic Curves.- Generalized Jacobians.- classification box Theory.- team Extension and Cohomology.- Bibliography.- Supplementary Bibliography.- Index.

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Auxiliary results = If Pi ¢ S, we have WPj(f) 0, and we can thus consider in the right hand side of (6) only the terms corresponding to Pi E S. For these, we have eQj ~ e and WPj(f) < n/(r - 1) in view ofthe choice of n. We thus deduce from (6) the inequality wQ(f' 0 g 0 7r) < en (7) which, together with (4), proves (3). It remains to prove lenuna 8: For that, we go back to writing the composition law on G multiplicatively. We must consider the product (8) ncr being a rational map ofY into the space of matrices N such that nii = 0 for i ~ j.

0 if n:l -e Resu(Tr(tn) du) = { . _ elf n - -e. On the other hand, and we indeed find the same result. Now we pass to the general case. We can write u = t e + Laiti, (**) i>e and conversely such a formula defines a subfield k( (u)) of k( (t)) such that [k«t)) : k«u))] = e; the extension k«t))/k«u)) is separable if and only if u ¢ k«t P )). Formula (**) makes evident the fact that {I, t, t2, ... t i , o~ i ~ e - 1} is a basis of 1, j=O the bn,i,j(U) being formal series in u: bn,i,j(U) = Lbn,i,j,k Uk .

Proposition 20. For every P EX, we have g-l (g( P)) = [F : F'Ji L: 7r 0 (7i(Q), where Q E Y is such that 7r(Q) = P and where the (7i denote representatives in g of the elements of g/~. ) PROOF. Let pi = g(P) and let h = go 7r. First we determine the cycle h-1(P' ). It is a linear combination L:uEg n u (7(Q); as the (7 are automorphisms of Y compatible with h, the nu are equal to the same integer n. The projection formula (cf. [70J, p. 32) shows that h(h-1(P' )) = [L : F'JP ' , whence deg(h-1(P I = [L : F'J which determines the integer n.

### Algebraic Groups and Class Fields by Jean-Pierre Serre

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