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Algebraic Groups and Class Fields by Jean-Pierre Serre

By Jean-Pierre Serre

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

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If w denotes an inv(lriant differential form on G, then (f + g)*(w) = f*(w) + g*(w). 48 III. Maps From a Curve to a Commutative Group PROOF. Denote by prl and pr2 the two projections of the group G x G to G and put p = prl + pr2. Since G is Abelian these maps are homomorphisms and the differentials p"(w), pri(w) and pr;(w) are invariant differentials on G x G. (w) + pr;(w), this equality is true everywhere. Let (I, g) : X by the pair (I, g). (w) + (I,g)*pr;(w) = f*(w) + g*(w). o 12. Quotient of a variety by a finite group of automorphisms Let V be an algebraic variety and let R be an equivalence relation on V.

If 9 is not constant, (h) (gh - (g) 00 , whence f«h)) 1'(1) - 1'( 00) 0, as was to be shown. 0 = = = = = = 6. Proof in characteristic 0 Let, as before, f : X - S - G be a regular map from X - S to the commutative algebraic group G. Denote by r the dimension of G and let {W1, ... , w r } be a basis of the vector space of differential forms of degree 1 invariant by translation on G (for the properties ofthese forms, cf. no. ll). *(Wi) 1 ~ i ~ r; the O:i are the pull-backs of the Wi by f, and are thus regular on X - S.

It is interesting to note that this proof figures in the work of Chevalley [15] already cited, but not in that of Wei I [88]. As we have seen, the residue formula plays an essential role in identifying differentials with linear forms on repartitions (the "duality" theorem). The first proof of this formula (over a field of any characteristic) is due to Hasse [32]; it is essentially his proof that we have given. The work of Chevalley [15] contains another, rather indirect, but avoiding the difficult lemma 5 (see also Lang [51], chap.

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