TITLE: Torsion of an abelian variety under reduction.
QUESTION [11 upvotes]: 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?

REPLY [5 votes]: To complete Pete and Milne's answers when A is not an abelian scheme (for example, when it is the Néron model of an abelian variety over ${\mathbb Q}_p$ with not necessary good reduction), then for any $n$ prime to $p$, the kernel $A[n]$ is still étale over ${\mathbb Z}_p$ (because the tangent map at $0$ of the multiplication by $n$ is just multiplication by $n$ for any commutative algebraic group), but not necessarily finite. There is a biggest closed subscheme $H$ of $A[n]$ which is étale and finite over ${\mathbb Z}_p$. The reduction map on $H$ is injective (see Pete's proof). The generic fiber of $H$ corresponds to the points of the generic fiber of $A[n]$ having specialization mod $p$. You may read Bosch, Lütkebohmert and Raynaud ''Néron Models'', § 7.3.