On local-global divisibility by $p^2$ in elliptic curves

TL;DR

This paper investigates the conditions under which local-global divisibility by p^2 fails in elliptic curves over certain number fields, linking counterexamples to the existence of rational points of order p.

Contribution

It proves that counterexamples imply the existence of a rational point of order p and refines the set of primes where such counterexamples can occur, using Merel's theorem.

Findings

Counterexamples imply a rational point of order p in the elliptic curve.

The set of primes with potential counterexamples is reduced.

The results depend on the field not containing certain subfields of cyclotomic fields.

Abstract

Let $ p $ be a prime lager than 3. Let $k$ be a number field, which does not contain the subfield of $\mathbb{Q} (\zeta_{p^2})$ of degree $p$ over $\mathbb{Q}$. Suppose that $\mathcal{E}$ is an elliptic curve defined over $k$. We prove that the existence of a counterexample to the local-global divisibility by $p^2$ in $\mathcal{E}$, assures the existence of a $k$-rational point of exact order $p$ in $\mathcal{E}$. Using the Merel Theorem, we then shrunk the known set of primes for which there could be a counterexample to the local-global divisibility by $p^2$.