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Construction de hauteurs archimédinnes et p-adiques suivant la méthode de Bloch. (French) Zbl 0531.14016

Théorie des nombres, Sémin. Delange-Pisot-Poitou, Paris 1980-81, Prog. Math. 22, 175-192 (1982).
[For the entire collection see Zbl 0483.00002.]
S. Bloch [Invent. Math. 58, 65-76 (1980; Zbl 0444.14015)] has introduced a height-function for an abelian variety A over a number field. This height-function is derived from the topological group of adelic points of the \({\mathbb{G}}_ m\)-extension of A associated with a divisor on A. The paper gives an elementary proof that Bloch’s height function coincides with the Néron-height (mod 0(1)). This proof is global and avoides the case of Néron-minimal model of A. The methods are then applied for the construction of p-adic heights. In the special case of an elliptic curve with complex multiplications one finds in this way the canonical height introduced by D. Gross.
Reviewer: M.van der Put

MSC:

14K15 Arithmetic ground fields for abelian varieties
14G20 Local ground fields in algebraic geometry
14H45 Special algebraic curves and curves of low genus
14H52 Elliptic curves
14K22 Complex multiplication and abelian varieties