division points of abelian varieties after reduction modulo a prime

Question

I do not have much knowledge of this topic, so I would like also if you can give me some basic references regarding this, in addition to a possible answer to my question.

Given an abelian variety $A$ defined over a field $K$ , and a positive integer $m$, we consider the $m$ division points $A[m](\overline{K})$ of $A$. Also, we consider the $m$-division field $K_{A,m}:=K(A[m])$ and the $m$-division Galois group $G_{A,m}:=Gal(K_m/K)$ of $A$.

For an abelian variety $A$ defined over a number field $K$ (we can suppose with some additional conditions), and its reduction $A_\mathfrak{p}$ modulo a prime $\mathfrak{p}\in \mathcal{O}_K$ (we can suppose with some additional conditions, in particular, such that it has good reduction), what are the relations between $$ A[m],A_\mathfrak{p}[m], K_{A,m}, K_{A_\mathfrak{p},m}, G_{A,m} \text{ and } G_{A_\mathfrak{p},m}? $$

Thanks in advance.

Answer 1

I am supposing that $\mathfrak{p}$ is a prime of good reduction, so $A_{\mathfrak{p}}$ is an abelian variety.

Second, I understand that for $A[m]$ you mean the $m$-torsion points, not the "$m$-division points": so the kernel of the multiplication by $m$. This is generally a finite flat group scheme, and not étale if the characteristic of the field divides $m$.

If $m$ is non-zero modulo $\mathfrak{p}$ then multiplication by $m$ is an étale morphism of group schemes, hence its kernel $A_{\mathfrak{p}}[m]$ is étale. Therefore it is classified by the points $A_{\mathfrak{p}}[m](\overline{k_{\mathfrak{p}}})$, where $k_{\mathfrak{p}}$ is the residue field of $K$ (i.e., of the ring of integers) at $\mathfrak{p}$, as a abelian group together with an action of the absolute Galois group $G_{\mathfrak{p}}$ of $k_{\mathfrak{p}}$. The action factorizes though what you call $G_{A_{\mathfrak{p}},m}$, and the group is (as a group) isomorphic to $(\mathbb{Z}/m\mathbb{Z})^{2d}$, where $d=\dim(A)$.

Now, choosing a prime $\mathfrak{P}$ of $K_{A,m}$ above $\mathfrak{p}$ it gives you a decomposition group $D_\mathfrak{P}$ of $G_{A,m}$, and inertia subgroup $I_\mathfrak{P}$ and a residue field $k_\mathfrak{P}$. The hypothesis on $\mathfrak{P}$ imply that the action of the inertia subgroup acts trivially (by the Criterion of Néron-Ogg-Shafarevich).

Then one has that $$G_{A_\mathfrak{p},m}\cong D_\mathfrak{P}/I_\mathfrak{P},$$ the residue field $k_\mathfrak{P}$ is isomorphic to $k(A_{\mathfrak{p}}[m](\overline{k_{\mathfrak{p}}}))$ and $A_{\mathfrak{p}}[m](\overline{k_{\mathfrak{p}}})$ is equal to $A[m](\overline{K})$ with the natural action of $D_\mathfrak{P}/I_\mathfrak{P}$.