Let $K$ be a number field and $\mathfrak{p}$ be a place of good reduction. It is easy to see that the reduction map on prime-to-$p$ torsion $A(K)[p'] \hookrightarrow A\_{\mathfrak{p}}(\kappa(\mathfrak{p}))$ is injective.


But if $p > e(\mathfrak{p}/p) + 1$, the reduction map is even injective on $p$-torsion. This can be seen by showing that the formal group has no $p$-torsion.


Now I ask if one can show this without using the formal group. By a theorem of Raynaud in "Schémas en groupes de type $(p, \ldots, p)$", the inequality for $p$ implies that for a $p^n$-torsion commutative finite flat group scheme the generic fibre can be spread out uniquely over the special fibre and $\mathrm{Hom}(G,H) = \mathrm{Hom}(G(\bar{K}),H(\bar{K}))$. Can this be used somehow?



$p$-torsion in the Mordell-Weil group of Abelian varieties injecting in reduction
One has the finite flat group scheme $\mathbb Z/p$ over $\mathcal O\_{K\_{\mathfrak p}}$
(I write $K\_{\mathfrak p}$ for the $\mathfrak p$-adic completion of $K$, and 
$\mathcal O\_{K\_{\mathfrak p}}$ for its integer ring), 
as well as
the finite flat group scheme $A[p]$. Giving a $p$-torsion point over $K\_{\mathfrak p}$ (and hence in particular over $K$) is the same as giving a closed embedding
on generic fibres:
$(\mathbb Z/p)\\_{/ K\_{\mathfrak p}} \hookrightarrow A[p]\\_{/K\_{\mathfrak p}}.$


Raynaud's results imply that this extends to a closed embedding over $\mathcal O\_K$:
$\mathbb Z/p \hookrightarrow A[p],$
which is another way of saying the that the non-zero $p$-torsion point has non-zero
reduction.


Just to see concretely what can happen in the situation when $e \geq p-1$, suppose
that $K = \mathbb Q$ and $p = 2$. Then we could have a map
$(\mathbb Z/2)\\_{/\mathbb Q\_2} \hookrightarrow A[2]\_{/\mathbb Q\_2}$
which extends to a closed immersion
$\mu\_2 \hookrightarrow A[2].$
This would correspond to having a 2-torsion point in the kernel of the reduction map.
(Note that $\mu\_2$ has a non-trivial point in char. zero, which collapses down to
the identity in char. two.)


[Added in response to unkwown's comment:]


The point is that one can form the scheme-theoretic closure in $A[p]$ of the image
of $(\mathbb Z/p)\\_K,$ which is *some* finite flat subgroup scheme over $\mathcal O\_K$
which is embedding as a closed subgroup scheme of $A[p]$ (by construction: we formed it as a scheme-theoretic closure). And it
has $(\mathbb Z/p)\\_K$ as its generic fibre (again by construction).


Now when $e < p-1$, Raynaud's results show that this finite flat group scheme has no
choice but to be $\mathbb Z/p$, and so we get a copy of $\mathbb Z/p$ embedding into $A[p]$,
extending the original embedding of generic fibres. Thus the order $p$ point in $A[p](K)$
reduces mod $p$ to an order $p$ point.


But if e.g. $p = 2$, then this scheme-theoretic closure could be $\mu\_2$. Now the non-trivial point ($-1$) of $\mu\_2(K)$ specializes to the trivial point in char. 2,
and so when we have a copy of $\mu\_2$ inside $A[2]$, the non-trivial point of $\mu\_2(K)$
lies in the kernel of the reduction mod 2 map.


