Bounded Height in Pencils of Finitely Generated Subgroups

F Amoroso (1) , D Masser (2) , U Zannier (3)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme
2 Unibas - Université de Bâle = University of Basel = Basel Universität
3 SNS - Scuola Normale Superiore di Pisa

We prove height bounds concerning intersections of finitely generated subgroups in a torus with algebraic subvarieties, all varying in a pencil. This vastly extends the previously treated constant case and involves entirely different, and more delicate, techniques. The paper complements and sharpens Mordell-Lang by replacing finiteness by emptyness, for the intersection of varieties and subgroups, all moving in a pencil, except for bounded height values of the parameters (and excluding identical relations).

Domaines: Théorie des nombres [math.NT]

F Amoroso, D Masser, U Zannier. Bounded Height in Pencils of Finitely Generated Subgroups. 2016. ⟨hal-01200626v2⟩