On the local–global divisibility of torsion points on elliptic curves and GL2-type varieties

Resumen

Let p be a prime number and let k   be a number field. Let E be an elliptic curve defined over k. We prove that if p is odd, then the local–global divisibility by any power of p   holds for the torsion points of E. We also show with an example that the hypothesis over p is necessary.

We get a weak generalization of the result on elliptic curves to the larger family of GL₂-type varieties over k  . In the special case of the abelian surfaces A/k with quaternionic multiplication over k we obtain that for all prime numbers p  , except a finite number depending only on the isomorphism class of the ring End_k(A), the local–global divisibility by any power of p   holds for the torsion points of A.

Autores: Gillibert, F., Ranieri, G.

Journal: Journal of Number Theory

Journal Volume: 174

Journal Issue:

Journal Page: 202-220

Tipo de publicación: Scopus

Fecha de publicación: 2017

Topics: Galois cohomology, Local–global, Elliptic curves, GL2-type varieties, Quaternionic multiplication

DOI: http://dx.doi.org/10.1016/j.jnt.2016.10.012

URL de la publicación: http://www.sciencedirect.com/science/article/pii/S0022314X16302943