Lower bounds for the rank of families of abelian varieties under base change

Cecília Salgado
Universidade Federal do Rio de Janeiro
Abstract:
We consider the following question : given a family of abelian varieties
\[\mathcal A\]
over a curve 
\[B\]
defined over a number field
\[k\]
, how does the rank of the Mordell-Weil group of the fibres
\[{\mathcal A}_t(k)\]
vary? A specialization theorem of Silverman guarantees that, for almost all
\[t\]
in
\[C(k)\]
, the rank of the fibre is at least the generic rank, that is the rank of
\[\mathcal A(k(B))\]
. When the base curve
\[B\]
is rational, we show, at least in many cases and under some geometric conditions, that there are infinitely many fibres for which the rank is larger than the generic rank. We will first discuss the problem for families of elliptic curves, treated in my previous work, to then proceed to the general setting of abelian varieties. This is joint work with M. Hindry.