The reduction map on the $\ell$-primary torsion of abelian variety

Question

Let $A/\mathbb{Q}$ be an abelian variety with good reduction at a prime $p$. Assume $\mathcal{A}/\mathbb{Z}_{(p)}$ is an integral model at $p$(hence proper smooth).

For any number field $K$ and any prime ideal $\mathfrak{p}$ of $K$ over $p$ with residue field $\kappa$, by valuation criterion on $\mathcal{O}_{K,\mathfrak{p}}$, any $K$-point of $A$ extends uniquely to an $\kappa$-point of $\mathcal{A}$'s special fiber.

By fixing a prime ideal of $\overline{\mathbb{Z}}$ lying over $p$ and taking injective limit, we get a well defined reduction map\begin{equation} A(\overline{\mathbb{Q}})\rightarrow \mathcal{A}_{\mathbb{F}_p}(\overline{\mathbb{F}_p}). \end{equation}

Does the above map induce isomprhisms on $\ell$-primary parts of both groups? Notice that both of their $\ell$-primary subgroups are isomorphic to $(\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell})^{\oplus 2g}$.

Answer 1

This is true. One reference for this is Theorem C.1.4 in Hindry-Silverman's "Diophantine Geometry: An Introduction" which says that for an abelian variety $A$, number field $K$ and any prime $\mathfrak p$ above $p$, the reduction map $A(K)\to A(O_K/\mathfrak p)$ is injective on prime-to-$p$-torsion. The result you ask for follows since $\overline{\mathbb Q}$ is a union of number fields.

One way to prove the required injectivity follows from the description of the kernel of the reduction map over the completion $K_{\mathfrak p}$, which can be identified with the formal group associated to $A$. Formal groups over $p$-adic fields have no prime-to-$p$ torsion.