On abelian points of varieties intersecting subgroups in a torus

Mello, Jorge

Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 1, pp. 309-322.

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 $X$ avec l’union des sous-groupes algébriques connexes de codimension au moins $dimX$ 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.

We show, under some natural conditions, that the set of abelian points on the non-anomalous subset of a closed irreducible subvariety $X$ intersected with the union of connected algebraic subgroups of codimension at least $dimX$ 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.

Classification : 14G40, 14G25
Mots clés : Algebraic torus, subvarieties, abelian closure, height

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