Download e-book for iPad: 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.- crew Extension and Cohomology.- Bibliography.- Supplementary Bibliography.- Index.
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Then (t, t)p = 1 for P :I 0,00, (t, t)o = -1, (t, t)oo = -1, and the product is indeed equal to 1. b) A:I O. Then (t - A,t)p 1 for P:I O,A,ooj = = = (t - A, t)p = -A (t - A, t)p = 1/ A (t-A,t)p=-1 o and the product is indeed again equal to 1. PROOF OF LEMMA 1. We are going to reduce it to a local result, as we did in chap. II, no. 12 for lemma 4. First of all, we observe that the symbol (f', t)p makes sense when f' and t are any elements of the field K Ep, the completion of the field E with repect to the valuation vp.
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.
The residue formula then shows that its residue at this point is 0 and 1'* (Wj) is a differential of the first kind, thus it is zero because A is a curve of genus zero. This finishes the proof, taking into account what was said above. 0 7. Proof in characteristic p > 0: reduction of the problem Our proof will rely on the structure of commutative algebraic groups. We assume the following two results: Proposition 11 ("Chevalley's theorem"). Every connected algebraic group G contains a normal subgroup R such that: a) R is a connected linear group.
Algebraic Groups and Class Fields by Jean-Pierre Serre