Let $E/\mathbb{Q}$ be an elliptic curve and $P \in E(\mathbb{Q})$. If $P$ is divisible by $p$ in $E(\mathbb{Q}\_p)$ for every prime $p$, does it follow that $P$ is a torsion point?



Local-Global divisibility of points on elliptic curves
Chris Wuthrich has already provided an answer when there are infinitely many anomalous primes, with the correction that the condition to be anomalous is $a\_p=1$, i.e. the number of points in the reduction modulo $p$ is $p$. In the case of non-anomalous prime, the condition that $P$ is $p$-adically divisible by $p$ is equivalent to $p$ being an elliptic Wieferich prime with respect to $E,P$ in the sense of Silverman (J. Number Theory 1988). There it is proved that, assuming the abc conjecture, there will be non-Wieferich primes in some cases if $P$ is of infinite order, which answers your question in the affirmative. There are many papers following up on this. Just as in the classical case of Wieferich primes, one expects very few of them so the answer to your question is almost certainly yes but we can't prove much uncondionally. 


