It is well known that the rational points of an elliptic curve over a number field form a finitely generated group. How does the torsion of this group vary in families?
A theorem of Masser–Zannier says that in a family of elliptic curves over a variety $V$ of characteristic zero, given two non-identically linearly independent sections $s$ and $t$, the set of points $p$ of $V$ such that both $s(p)$ and $t(p)$ are of torsion is finite.
The relative Manin–Mumford conjecture is a broad generalization of this statement, for families of subvarieties of abelian varieties of any dimension.
The ultimate goal of this meeting will be to understand the statement and the proof by Gao-Habegger of the relative Manin-Mumford conjecture. It will be the occasion to learn how to apply in the arithmetic setting the main geometric techniques used in Gao–Habegger's proof: unlikely intersections, mixed Ax–Schaunel and the theory of o-minimality which is behind them.
We plan to revive the tradition of an annual series of conferences in arithmetic geometry based on the model of Oberwolfach's Arbeitsgemeinschaft. They were previously organized by Jean-Benoît Bost and François Loeser from 1995 to 2002 at Luminy. The first editions covered topics such as Euler systems, higher class field theory or modular forms and Galois representations.
The public we have in mind consists mainly of PhD students and early postdocs, with the aim of offering a friendly ambience to learn mathematical subjects that do not necessarily belong to one's own research area. At the end of each edition, the topic for the next one will be voted, and a scientific program will be written in close collaboration with a leading expert. About six months before the conference, participants will apply and the talks will be distributed among them.
Scientific Committee
Philipp Habegger (Universität Basel)