On the local-global principle for divisibility in the cohomology of elliptic curves

Brendan Creutz

Abstract

For every prime power \(p^n\) with \(p = 2\) or 3 and \(n > 1\) we give an example of an elliptic curve over \(\mathbb Q\) containing a rational point which is locally divisible by \(p^n\) but is not divisible by \(n\). For these same prime powers we construct examples showing that the analogous local-global principle for divisibility in the Weil-Châtelet group can also fail.