Local heights on curves. (English) Zbl 0605.14027

Arithmetic geometry, Pap. Conf., Storrs/Conn. 1984, 327-339 (1986).
[For the entire collection see Zbl 0596.00007.]
The results involving local heights are all special cases of the general theory of Néron’s models for abelian varieties [see A. Néron, Publ. Math., Inst. Hautes Étud. Sci. 21 (1964; Zbl 0132.414) or also M. Artin’s survey ”Néron models”, this volume, 213-230 (1986; Zbl 0603.14028)]. However, in the case of curves one can present the theory of local heights very explicitly (and independently of Néron’s theory). This leads to a more elementary presentation, which is quite instructive for those wanting to introduce themselves to this subject. And indeed, this is the purpose of the paper under review. The author also describes the relationship of this theory to the global height pairings on the Jacobians, and discusses some extensions (due to Arakelov and Tate) of the local pairings to divisors of arbitrary degree and which are not necessarily relatively prime. The theory is illustrated by many examples.
Reviewer: L.Bădescu


14H25 Arithmetic ground fields for curves
14H40 Jacobians, Prym varieties
14K15 Arithmetic ground fields for abelian varieties