# The Mordell–Lang question for endomorphisms of 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
• Volume: 23, Issue: 3, page 645-666
• ISSN: 1246-7405

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## Abstract

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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.

## How to cite

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Ghioca, Dragos, Tucker, Thomas, and Zieve, Michael E.. "The Mordell–Lang question for endomorphisms of semiabelian varieties." Journal de Théorie des Nombres de Bordeaux 23.3 (2011): 645-666. <http://eudml.org/doc/219823>.

@article{Ghioca2011,
abstract = {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.},
affiliation = {Department of Mathematics University of British Columbia Vancouver, BC V6T 1Z2 Canada; Department of Mathematics University of Rochester Rochester, NY 14627 USA; Department of Mathematics University of Michigan 530 Church Street Ann Arbor, MI 48109 USA},
author = {Ghioca, Dragos, Tucker, Thomas, Zieve, Michael E.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {$p$-adic exponential; Mordell-Lang conjecture; semiabelian varieties; -adic exponential},
language = {eng},
month = {11},
number = {3},
pages = {645-666},
publisher = {Société Arithmétique de Bordeaux},
title = {The Mordell–Lang question for endomorphisms of semiabelian varieties},
url = {http://eudml.org/doc/219823},
volume = {23},
year = {2011},
}

TY - JOUR
AU - Ghioca, Dragos
AU - Tucker, Thomas
AU - Zieve, Michael E.
TI - The Mordell–Lang question for endomorphisms of semiabelian varieties
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2011/11//
PB - Société Arithmétique de Bordeaux
VL - 23
IS - 3
SP - 645
EP - 666
AB - 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 - eng
KW - $p$-adic exponential; Mordell-Lang conjecture; semiabelian varieties; -adic exponential
UR - http://eudml.org/doc/219823
ER -

## References

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