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# Séminaire Théorie des Nombres

Responsables : Eric Balandraud, Dimitrios Chatzakos, Andrea Fanelli.

• Le 27 novembre 2020 à 14:00
• En Visio
Ariyan Javanpeykar (Mayence)
Hilbert's irreducibility theorem for abelian varieties
We will discuss joint work with Corvaja, Demeio, Lombardo, and Zannier in which we extend Hilbert's irreducibility theorem (for rational varieties) to the setting of abelian varieties. Roughly speaking, given an abelian variety $A$ over a number field $k$ and a ramified covering $X$ of $A$, we show that $X$ has 'less' $k$-rational points than $A$.
• Le 4 décembre 2020 à 14:00
• En Visio
Gabriel Dill (Oxford)
Torsion points on isogenous abelian varieties
The Manin-Mumford conjecture, proven by Raynaud, predicted that a subvariety of an abelian variety over a field of characteristic zero contains a Zariski dense set of torsion points if and only if it is a union of torsion cosets, i.e. of translates of abelian subvarieties by torsion points. We study subvarieties of abelian schemes that contain a Zariski dense set of torsion points that lie on pairwise isogenous fibers. If the abelian scheme has maximal variation, conjectures of Zannier and Pink characterize such subvarieties. If everything is defined over the algebraic numbers, we prove one half of the conclusion of these conjectures: the geometric generic fiber of an irreducible such subvariety over its projection to the base is a union of torsion cosets. Our proof is based on a strategy due to Lang, Serre, Tate, and Hindry of using Galois automorphisms that act as homotheties on the torsion points. If the abelian scheme is a fibered power of the Legendre family of elliptic curves, this method yields explicit and uniform results. It also yields uniform Manin-Mumford results within a given isogeny class.
• Le 11 décembre 2020 à 14:00
• En Visio
Jiandi Zou (Versailles)
Titre à préciser

• Le 29 janvier 2021 à 14:00
• Salle de Conférences
Robert Tichy (Graz, CIRM)
Titre à préciser

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