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.