On the local-global divisibility over GL2-type varieties

Resumen

Let k be a number field and let A be a 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 k˜ of k of degree bounded by a constant depending on d such that A is k˜-isogenous to a GL2-type variety defined over k˜ that admits a k˜-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