2017/03/18 by Gillibert, Florence, Ranieri, Gabriele
#11G10 #11R34 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1703.06235
Let k be a number field and let A be a \rm GL2-type variety defined over k of dimension d. We show that for every prime number p satisfying certain conditions (see Theorem 2), if the local-global divisibility principle by a power of p does not hold for A over k, then there exists a cyclic extension \widetildek of k of degree bounded by a constant depending on d such that A is \widetildek-isogenous to a \rm GL2-type variety defined over \widetildek that admits a \widetildek-rational point of order p. Moreover, we explain how our result is related to a question of Cassels on the divisibility of the Tate-Shafarevich group, studied by Ciperiani and Stix and Creutz.