We show, under some natural conditions, that the set of abelian points on the non-anomalous subset of a closed irreducible subvariety intersected with the union of connected algebraic subgroups of codimension at least in a torus is finite, generalising results of Ostafe, Sha, Shparlinski and Zannier (2017). We also generalise their structure theorem for such sets when the algebraic subgroups are not necessarily connected, and obtain a related result in the context of curves and arithmetic dynamics.
Sous certaines conditions naturelles, on montre que l’intersection de l’ensemble de points abéliens d’un sous-ensemble non-atypique d’une sous-variété avec l’union des sous-groupes algébriques connexes de codimension au moins dans un tore est finie, en généralisant les résultats de Ostafe, Sha, Shparlinski and Zannier (2017). Nous généralisons également leur théorème de structure pour de tels ensembles au cas où les sous-groupes algébriques ne sont pas nécesserement connexes et prouvons un résultat connexe pour les courbes dans le contexte de dynamique arithmétique.
Mots-clés : Algebraic torus, subvarieties, abelian closure, height
Jorge Mello 1
CC-BY-ND 4.0
@article{JTNB_2022__34_1_309_0,
author = {Jorge Mello},
title = {On abelian points of varieties intersecting subgroups in a torus},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {309--322},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {34},
number = {1},
year = {2022},
doi = {10.5802/jtnb.1203},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1203/}
}
TY - JOUR AU - Jorge Mello TI - On abelian points of varieties intersecting subgroups in a torus JO - Journal de théorie des nombres de Bordeaux PY - 2022 SP - 309 EP - 322 VL - 34 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1203/ DO - 10.5802/jtnb.1203 LA - en ID - JTNB_2022__34_1_309_0 ER -
%0 Journal Article %A Jorge Mello %T On abelian points of varieties intersecting subgroups in a torus %J Journal de théorie des nombres de Bordeaux %D 2022 %P 309-322 %V 34 %N 1 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1203/ %R 10.5802/jtnb.1203 %G en %F JTNB_2022__34_1_309_0
Jorge Mello. On abelian points of varieties intersecting subgroups in a torus. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 1, pp. 309-322. doi : 10.5802/jtnb.1203. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1203/
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