On the minimal set for counterexamples to the local-global principle

TL;DR

This paper proves that counterexamples to the local-global divisibility principle for elliptic curves over rationals only occur for powers of 2 and 3, refining previous criteria and illustrating their necessity.

Contribution

It establishes the specific prime powers where counterexamples can occur and refines the criteria for the principle's validity, including necessary assumptions.

Findings

Counterexamples only occur for powers of 2 and 3

Refined criterion for local-global divisibility principle

Provided example demonstrating necessity of assumptions

Abstract

We prove that only for powers of 2 and 3 could occur counterexamples to the local-global divisibility principle for elliptic curves defined over the rationals. For we refine our previous criterion for the validity of the principle. We also give an example that shows that the assumptions of our criterion are necessary.