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The bilinear forms of Néron and Tate. (Les formes bilinéaires de Néron et Tate.) (French) Zbl 0138.42101
Sémin. Bourbaki 16 (1963/64), Exp. No. 274, 11 p. (1964).
An account of two papers containing fundamental results in the theory of heights of points on algebraic varieties defined over global fields; the first, by John Tate (“non publié, comme d’habitude”), has already been partly published by Yu. I. Manin [Ann. Math. Soc., Transl., II. Ser. 59, 82–110 (1966); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 28, 1363–1390 (1964; Zbl 0192.26801)]; in it is verified the conjecture of Néron [“Valeur asymptotique du nombre des points rationnels de hauteur bornée sur une courbe elliptique”, short communication in Proc. Int. Congr. Math. (August, 1958, Edinburgh), p. xx, Cambridge Univ. Press, Cambridge (1960)] that the height is given, up to a bounded summand, by a quadratic form. The proof depends on an ingenious limiting process, which also gives a bilinear form on the points of an abelian variety and its dual. In the other paper, by Néron, which will doubtless be published in due course in the conventional way, it is shown that the quadratic form and bilinear form are, in fact, sums of summands, one for each local completion of the global field. These local summands are said to be essentially just intersection multiplicities. Results in this direction, but apparently less complete, have already been given by Manin [loc. cit.]. Néron’s proof is said to depend on a reformulation of Weil’s theory of distributions, and an indication of the new theory is given.

MSC:
14G25 Global ground fields in algebraic geometry
14K15 Arithmetic ground fields for abelian varieties
11G10 Abelian varieties of dimension \(> 1\)
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
11G50 Heights