Let $p$ be a prime. Suppose you have an Abelian scheme $A$ over $Spec\ \mathbb{Z}\_p$. How do you prove that if $q$ is another prime, then the $q$-torsion of $A$ injects into the torsion of $A\_p$, under the reduction map?



Torsion of an abelian variety under reduction.
Let me try again at an alternate answer. If $A\_{\mathbb{Z}\_p}$ is an abelian scheme of dimension $g$ and $\ell \neq p$ is a prime, then for any positive integer $n$, the isogeny $[\ell^n]: A \rightarrow A$ is an etale map. [If I am not mistaken, the proof of this does not require formal groups!] Since the special fiber has $\ell^{2gn}$ points over the algebraic closure, by Hensel's Lemma all of the $\ell^n$-torsion on $A$ is defined over the maximal unramified extension, and it follows that the reduction map over the maximal unramified extension is an isomorphism on the $\ell^n$-torsion, hence an injection over $\mathbb{Q}\_p$. 


