zbMATH — the first resource for mathematics

Periods and isogenies of abelian varieties of number fields [after D. Masser and G. Wüstholz]. (Périodes et isogénies des variétés abéliennes sur les corps de nombres [d’après D. Masser et G. Wüstholz].) (French) Zbl 0936.11042
Séminaire Bourbaki. Volume 1994/95. Exposés 790-804. Paris: Société Mathématique de France, Astérisque. 237, 115-161, Exp. No. 795 (1996).
This paper surveys results of D. Masser and G. Wüstholz published in a series of papers between 1990 and 1995, concerning abelian varieties defined over a number field \(K\). Among others, these results include estimates for isogenies between such abelian varieties, an effective version of the theorem of complete reducibility of Poincaré for these abelian varieties and upper bounds for the discriminant of their ring of endomorphisms.
The point of view of this survey is new and different from that of the original papers. It uses Arakelov theory and should prove useful in the applications. In particular, the author announces in §2.3.3 an explicit form for the constant appearing in the theorem of periods of D. Masser and G. Wüstholz.
The paper is completed by an appendix which introduces the necessary concepts from Arakelov theory.
For the entire collection see [Zbl 0851.00039].
Reviewer: D.Roy (Ottawa)

11J81 Transcendence (general theory)
14G05 Rational points
11G10 Abelian varieties of dimension \(> 1\)
Full Text: Numdam EuDML