"Abelian varieties over large algebraic fields"

Abstract:
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Let $A$ be a non-zero abelian variety over a finitely
generated field $K$. Then the Mordell-Weil group
$A(K)$ is a finitely generated abelian group. The group
$A(K_{sep})$, however, is known to have infinite
rank unless $K$ is algebraic over a finite field.
Interesting questions arise when we study the rank
of $A$ in other infinite algebraic extensions of
$K$. The following question of Frey and Jarden is
of that flavour: Does any non-zero abelian variety $A$ over
a Hilbertian field $K$ acquire infinite rank over
the maximal abelian extension $K_{ab}$?
Another question in this direction was posed by Larsen:
Is $rank(A(K_{sep}(\sigma)))$ infinite for any vector
$\sigma\in G_K^e$? We will discuss results on these questions.
Some of the results are conditional with respect to
the Birch and Swinnerton-Dyer conjecture.