Finiteness properties of torsion fields of abelian varieties

Abstract

Let $A$ be an abelian variety defined over a field $K.$ We study finite generation properties of the profinite group $\mathrm{Gal}(\Omega /K)$ and of certain closed normal subgroups thereof, where $\Omega$ 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 give an independent proof and generalizations of a recent result of Checcoli and Dill about small exponent subfields of $\Omega /K$ in the number field case. We also give an application of our finite generation results in the realm of permanence principles for varieties with the weak Hilbert property.

17 pages. Slightly improved version now treating representations attached to 'etale cohomology as wee