# Seminario Dottorato: The l-primary torsion conjecture for abelian varieties and Mordell conjecture

## Wednesday 12 May 2010 - Anna Cadoret

ARGOMENTI: Seminari Dottorato

SEMINARIO DOTTORATO
Wednesday 12 May 2010 h. 15:00, room 1BC/50
Anna Cadoret (Universite' de Bordeaux 1)
"The l-primary torsion conjecture for abelian varieties and Mordell conjecture"

-Abstract
Let k be a field. An abelian variety A over k is a proper group scheme over k. It can be thought of as a functor (with extra properties) from the category of k-schemes to the category of abelian groups. One nice result about such a functor is:
"Theorem (Mordell-Weil): Assume that k is a finitely generated field of characteristic 0: then, for any finitely generated extension K of k, A(K) is a finitely generated group. In particular, the torsion subgroup A(K)_tors of A(K) is finite".
For a prime l, the l-primary torsion conjecture for abelian varieties asserts that the order of the l-Sylow of A(K)_tors should be bounded uniformly only in terms of l, K and the dimension g of A.
For g=1 (elliptic curves), this conjecture was proved by Y. Manin, in 1969. The main ingredient is a special version of Mordell conjecture for modular curves. The general Mordell conjecture was only proved in 1984, by G. Faltings.
For g=2, the l-primary torsion conjecture remains entirely open.
After reviewing the proof of Y. Manin, I would like to explain how the general version of Mordell conjecture can be used to prove - following basically Manin's argument - the l-primary torsion conjecture for 1-dimensional families of abelian varieties (of arbitrary dimension). This result was obtained jointly with Akio Tamagawa (R.I.M.S.), in 2008.

Rif. int. C. Marastoni, T. Vargiolu, M. Dalla Riva