The Mordell–Lang question for endomorphisms of semiabelian varieties
Journal de Théorie des Nombres de Bordeaux, Volume 23 (2011) no. 3, pp. 645-666.

The Mordell–Lang conjecture describes the intersection of a finitely generated subgroup with a closed subvariety of a semiabelian variety. Equivalently, this conjecture describes the intersection of closed subvarieties with the set of images of the origin under a finitely generated semigroup of translations. We study the analogous question in which the translations are replaced by algebraic group endomorphisms (and the origin is replaced by another point). We show that the conclusion of the Mordell–Lang conjecture remains true in this setting if either (1) the semiabelian variety is simple, (2) the semiabelian variety is A 2 , where A is a one-dimensional semiabelian variety, (3) the subvariety is a connected one-dimensional algebraic subgroup, or (4) each endomorphism has diagonalizable Jacobian at the origin. We also give examples showing that the conclusion fails if we make slight modifications to any of these hypotheses.

La conjecture de Mordell-Lang décrit l’intersection d’un sous-groupe de type fini avec une variété fermée d’une variété semi-abélienne. De façon équivalente, cette conjecture décrit l’intersection des sous-variétés fermées avec l’ensemble des images de l’origine sous un semigroupe de translations de type fini. Nous étudions la question analogue dans laquelle les translations sont remplacées par les endomorphismes d’un groupe algébrique (et l’origine est remplacée par un autre point). Nous montrons qui la conclusion de la conjecture de Mordell-Lang reste vraie dans cette situation si, ou bien, (1) la variété semi-abélienne est simple, ou, (2) la variété semi-abélienne est A 2 , où A est une variété semi-abéliennne de dimension 1, ou (3) la sous-variété est une sous-variété semi-abéliennne de dimension 1, ou enfin, (4) la matrice jacobienne à l’origine de chaque endomorphisme est diagonalisable. Nous donnons aussi des exemples qui montrent que la conclusion est fausse si nous affaiblissons n’importe laquelle de ces hypothéses.

Received:
Published online:
DOI: 10.5802/jtnb.781
Classification: 14L10,  37P55,  11G20
Keywords: p-adic exponential, Mordell-Lang conjecture, semiabelian varieties
Dragos Ghioca 1; Thomas Tucker 2; Michael E. Zieve 3

1 Department of Mathematics University of British Columbia Vancouver, BC V6T 1Z2 Canada
2 Department of Mathematics University of Rochester Rochester, NY 14627 USA
3 Department of Mathematics University of Michigan 530 Church Street Ann Arbor, MI 48109 USA
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Dragos Ghioca; Thomas Tucker; Michael E. Zieve. The Mordell–Lang question for endomorphisms of semiabelian varieties. Journal de Théorie des Nombres de Bordeaux, Volume 23 (2011) no. 3, pp. 645-666. doi : 10.5802/jtnb.781. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.781/

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