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Abelian Varieties by Serge Lang (auth.) PDF

By Serge Lang (auth.)

ISBN-10: 0387908757

ISBN-13: 9780387908755

ISBN-10: 1441985344

ISBN-13: 9781441985347

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Sample text

Here of course, we have worked birationally and biholomorphically, and we have [II] HISTORICAL NOTE 53 completely lost track of the integral nature of parameters on which the curve may depend. On the other hand, Chow's method gives rise to Igusa's compatibility principle which allows us to study algebraic systems of Jacobian varieties [39J, [40]. For instance, if we are given an algebraic system of curves, together with their canonical mappings into their Jacobians, if (C, j, f) is a generic member, and if C is a specialization of C which is non-singular, then (f, f) has a uniquely determined specialization (f', f') which is the Jacobian variety of C.

Let u be a generic point of f(C) . g-u = I. Then II 2 (t(P i )) i~1 where the Pi are independent generic points of C, uniquely determined (uP to a permutation) by the relation u = If~lf(Pi)' We are not going to give here the full theory of the Pontrjagin products of f(C) and of the relations which one can find between these products and their intersections, especially with respect to numerical equivalence. A number of questions concerning this topic are not yet completely cleared up. We shall limit ourselves to proving a result which is sufficient to show that f(C) is biholomorphic to C.

In order t? have a completely canonical situation, we proceed as follows. We shall say that a rational map f : V X V -J> A of the product V X V into an abelian variety is admissible if we can find a field k' and a rational map f' : V -J> A such that f(P, Q) = f'(P)-f'(Q) for P, Q independent generic points of V over k'. In view of Theorems 2 and 3, this amounts to saying that f vanishes on the diagonal. , f(P, P) = o. Of course, f may be defined over a field k, while f' is not defined over k. We note that f' is uniquely determined by f up to an additive constant.

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Abelian Varieties by Serge Lang (auth.)


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