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Questions de corps de définition pour les variétés abéliennes en caractéristique positive

Published online by Cambridge University Press:  07 October 2008

Franck Benoist
Affiliation:
Abteilung für mathemaische Logik, Universität Freiburg, Eckerstrasse 1, Freiburg D-79104, Germany ([email protected])
Françoise Delon
Affiliation:
Equipe de Logique, UFR de Mathématiques—Case 7012, Université Paris 7—Denis Diderot, 2, place Jussieu, 75251 Paris cedex 05, France ([email protected])

Abstract

Dichotomies in various conjectures from algebraic geometry are in fact occurrences of the dichotomy among Zariski structures. This is what Hrushovski showed and which enabled him to solve, positively, the geometric Mordell–Lang conjecture in positive characteristic. Are we able now to avoid this use of Zariski structures? Pillay and Ziegler have given a direct proof that works for semi-abelian varieties they called ‘very thin’, which include the ordinary abelian varieties. But it does not apply in all generality: we describe here an abelian variety which is not very thin. More generally, we consider from a model-theoretical point of view several questions about the fields of definition of semi-abelian varieties.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2008

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