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

Let $p$ be a prime lager than 3. Let $k$ be a number field, which does not contain the subfield of $\mathbb{Q} (ζ_{p^2})$ 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 $p^2$.

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