Journal de Theorie des Nombres de Bordeaux | 2019
On class numbers of division fields of abelian varieties
Abstract
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 Kn, obtained by adjoining to K all the coordinates of the p-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 Kn. This lower bound depends on the Mordell– Weil rank of A and the reduction of p-torsion points modulo primes above p.