On class numbers of division fields of abelian varieties

Let A be an abelian variety defined over a number field K. Fix a prime p and a natural number n and consider the field K-n, obtained by adjoining to K all the coordinates of the p(n)-torsion points of A. We give a lower bound on the p-part of the class group of Kn for large n, by finding a large unramified extension of K-n. This lower bound depends on the Mordell-Weil rank of A and the reduction of p-torsion points modulo primes above p.