On a Galois property of fields generated by the torsion of an abelian variety

Abstract In this article, we study a certain Galois property of subextensions of , the minimal field of definition of all torsion points of an abelian variety defined over a number field . Concretely, we show that each subfield of that is Galois over (of possibly infinite degree) and whose Galois group has finite exponent is contained in an abelian extension of some finite extension of . As an immediate corollary of this result and a theorem of Bombieri and Zannier, we deduce that each such field has the Northcott property, that is, does not contain any infinite set of algebraic numbers of bounded height.