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
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
Keywords:
Dragos Ghioca 1 ; Thomas Tucker 2 ; Michael E. Zieve 3
@article{JTNB_2011__23_3_645_0,
author = {Dragos Ghioca and Thomas Tucker and Michael E. Zieve},
title = {The {Mordell{\textendash}Lang} question for endomorphisms of semiabelian varieties},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {645--666},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {23},
number = {3},
year = {2011},
doi = {10.5802/jtnb.781},
mrnumber = {2861079},
zbl = {1256.14046},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.781/}
}
TY - JOUR AU - Dragos Ghioca AU - Thomas Tucker AU - Michael E. Zieve TI - The Mordell–Lang question for endomorphisms of semiabelian varieties JO - Journal de théorie des nombres de Bordeaux PY - 2011 SP - 645 EP - 666 VL - 23 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.781/ DO - 10.5802/jtnb.781 LA - en ID - JTNB_2011__23_3_645_0 ER -
%0 Journal Article %A Dragos Ghioca %A Thomas Tucker %A Michael E. Zieve %T The Mordell–Lang question for endomorphisms of semiabelian varieties %J Journal de théorie des nombres de Bordeaux %D 2011 %P 645-666 %V 23 %N 3 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.781/ %R 10.5802/jtnb.781 %G en %F JTNB_2011__23_3_645_0
Dragos Ghioca; Thomas Tucker; Michael E. Zieve. The Mordell–Lang question for endomorphisms of semiabelian varieties. Journal de théorie des nombres de Bordeaux, Tome 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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