Diamonds in Torsion of Abelian Varieties.

21 May 2009
Moshe Jarden
A theorem of Kuyk says that every Abelian extension of a Hilbertian field is Hilbertian. We conjecture that for an Abelian variety $A$ defined over a Hilbertian field $K$ every extension $L$ of $K$ in $K(A_\tor)$ is Hilbertian. We prove our conjecture when $K$ is a number field. The proofs applies a result of Serre about $l$-torsion of Abelian varieties, information about $l$-adic analytic groups, and Haran's diamond theorem.