ABELIAN VARIETIES AND GALOIS EXTENSIONS OF HILBERTIAN FIELDS

In a recent paper Moshe Jarden (Diamonds in torsion of Abelian varieties, T. Inst. Math. Jussieu 9(3) (2010), 477-480) proposed a conjecture, later named the Kuykian conjecture, which states that if A is an Abelian variety defined over a Hilbertian field K, then every intermediate field of K(A(tor))/K is Hilbertian. We prove that the conjecture holds for Galois extensions of K in K(A(tor)).