Seminar of Algebra

Let A be an abelian variety defined over a field K. We study finite generation properties of the profinite group $Gal(K_{tor}(A)/K)$ and of certain closed normal subgroups thereof, where $K_{tor}(A)$ is the torsion field of A over K. In fact, we establish more general finite generation properties for monodromy groups attached to smooth projective varieties via étale cohomology. We apply this in order to prove Northcott (and Bogomolov) property for small exponent subfields of $K_{tor}(A)$ in the number field case.