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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.- classification box Theory.- workforce Extension and Cohomology.- Bibliography.- Supplementary Bibliography.- Index.

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Maps From a Curve to a Commutative Group If I : X -+ Y is a regular map from the variety X to the variety Y (X and Y being supposed non-singular), I defines a homomorphism Oy -+ Ox, thus, passing to differentials, a homomorphism £ly -+ £lx ' But, in general, if Ex and Ey are vector bundles over X and Y respectively, every Oy-linear homomorphism from S(Ey) to S(Ex) corresponds to a homomorphism of the fibre space Ex to Ey compatible with I, and conversely; this can be immediately checked by using local coordinates.

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. Denote by VIR the quotient set of V by R and let 0 : V --+ VIR be the canonical projection from V to VIR. We give VIR the quotient topology, where V has the Zariski topology.

G) = Ep .... g) = 1I'((g)). Thus suppose pi E SI, and choose a function h such that g/h == 1 mod m at the points P mapping to pi and h == 1 mod m at the points PES not mapping to pl. h == 1 mod ml on SI - pl. S = = L L p .... P' (110 1I',h)p (110 1I',g)p. o 33 §1. Local symbols 3. Example of a local symbol: additive group case From now on, we limit ourselves to the case where the commutative group G is a connected algebraic group, the map I : X - S -+ G being a regular map. We can then consider I as a rational map from X to G, regular away from S.

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