$\ell$-division fields and the  conductor of abelian varieties

Armand Brumer
Fordham University

This is joint work with Kenneth Kramer.
We report on recent progress on criteria for the existence of semistable
abelian varieties defined over Q, with particular emphasis on odd conductor
and dimension two. We obtain necessary conditions sufficient to get lower
bounds compatible with Mestre's conditional bounds in dimension at most
three. This is done by studying carefully the group scheme structure of the
group of $\ell^n$-torsion points and the ramification of the field they
generate.