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

TL;DR

This paper provides examples demonstrating the failure of the local-global principle for divisibility in elliptic curves and their Weil-Châtelet groups over Q for prime powers p^n with p=2 or 3 and n>1.

Contribution

It constructs explicit counterexamples showing the breakdown of local-global divisibility principles for specific prime powers in elliptic curves over Q.

Findings

Counterexamples for p=2,3 and n>1 in elliptic curves over Q

Failure of local-global divisibility in Weil-Châtelet groups for these primes

Illustrates limitations of local-global principles in arithmetic geometry

Abstract

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