Local–global divisibility by 4 in elliptic curves defined over $${\mathbb{Q}}$$

Abstract

Let \({\mathcal{E}}\) be an elliptic curve defined over \({\mathbb{Q}}\) . Let \({P\in {\mathcal{E}}(\mathbb {Q})}\) and let q be a positive integer. Assume that for almost all valuations \({v\in \mathbb{Q}}\) , there exist points \({D_v\in {\mathcal{E}}(\mathbb {Q}_v)}\) such that P = qDv. Is it possible to conclude that there exists a point \({D\in {\mathcal{E}}(\mathbb {Q})}\) such that P = qD? A full answer to this question is known when q is a power of almost all primes \({p\in \mathbb{N}}\) , but some cases remain open when \({p\in S=\{2,3,5,7,11,13,17,19,37,43,67,163\}}\) . We now give a complete answer in the case when q = 4.