[proofplan] We localise the global statement at the finite prime $\mathfrak{p}$ and compare good reduction with the inertia action on the rational Tate modules. The main input is the local Néron-Ogg-Shafarevich criterion: for an abelian variety over a non-archimedean local field, good reduction is equivalent to the inertia subgroup acting as the identity on one, equivalently every, rational Tate module of residue characteristic different from $l$. Applying this local theorem to the base change of $A$ to $K_{\mathfrak{p}}$ proves both directions and the independence of $l$. [/proofplan]

[step:Localise the abelian variety and the Galois representation at $\mathfrak{p}$] Let $K_{\mathfrak{p}}$ denote the completion of $K$ at $\mathfrak{p}$, let $\overline{K_{\mathfrak{p}}}$ be a separable [algebraic closure](/page/Algebraic%20Closure), and let \begin{align*} G_{K_{\mathfrak{p}}} := \operatorname{Gal}(\overline{K_{\mathfrak{p}}}/K_{\mathfrak{p}}) \end{align*} be the absolute [Galois group](/page/Galois%20Group). Let \begin{align*} I_{\mathfrak{p}} \subset G_{K_{\mathfrak{p}}} \end{align*} denote the inertia subgroup. For a rational prime $l \ne p$, define the rational Tate module \begin{align*} V_lA : G_K &\to \operatorname{Aut}_{\mathbb{Q}_l}(T_lA \otimes_{\mathbb{Z}_l}\mathbb{Q}_l) \end{align*} from the natural action of $G_K$ on the $l$-power torsion of $A(\overline{K})$. Choose an embedding $\overline{K} \hookrightarrow \overline{K_{\mathfrak{p}}}$ extending $K \hookrightarrow K_{\mathfrak{p}}$. It determines a decomposition subgroup $D_{\mathfrak{p}} \subset G_K$ identified with $G_{K_{\mathfrak{p}}}$ up to conjugacy, and the inertia subgroup of $D_{\mathfrak{p}}$ is identified with $I_{\mathfrak{p}}$. Therefore the statement that $V_lA$ is unramified at $\mathfrak{p}$ means precisely that $I_{\mathfrak{p}}$ acts as the identity on $V_lA$ after restricting the representation from $G_K$ to $G_{K_{\mathfrak{p}}}$. [/step]

[step:Apply the local Néron Ogg Shafarevich criterion]Let \begin{align*} A_{\mathfrak{p}} := A \times_{\operatorname{Spec} K} \operatorname{Spec} K_{\mathfrak{p}} \end{align*} be the base change of $A$ to the local field $K_{\mathfrak{p}}$. The field $K_{\mathfrak{p}}$ is a non-archimedean local field with residue characteristic $p$, and $A_{\mathfrak{p}}/K_{\mathfrak{p}}$ is an abelian variety. Hence the [local Néron-Ogg-Shafarevich criterion](/theorems/???) applies to $A_{\mathfrak{p}}$ and gives, for each rational prime $l \ne p$, the equivalence \begin{align*} A_{\mathfrak{p}} \text{ has good reduction over } K_{\mathfrak{p}} \iff I_{\mathfrak{p}} \text{ acts as the identity on } V_lA. \end{align*} By the definition of good reduction at a finite prime, $A$ has good reduction at $\mathfrak{p}$ exactly when $A_{\mathfrak{p}}$ has good reduction over $K_{\mathfrak{p}}$. By the definition of unramifiedness from the preceding step, the right-hand condition is exactly that $V_lA$ is unramified at $\mathfrak{p}$.[/step]

[guided]We now use the local form of the theorem, so we first verify its hypotheses. The base field in the local theorem must be a non-archimedean local field. Here $K_{\mathfrak{p}}$ is the completion of the number field $K$ at the finite prime $\mathfrak{p}$, so it is a non-archimedean local field. Its residue characteristic is the rational prime $p$ specified in the statement. The variety to which the theorem is applied must be an abelian variety over that local field; this is $A_{\mathfrak{p}} := A \times_{\operatorname{Spec} K} \operatorname{Spec} K_{\mathfrak{p}}$, which remains an abelian variety after [field extension](/page/Field%20Extension). The [local Néron-Ogg-Shafarevich criterion](/theorems/???) therefore applies and states that, for every rational prime $l \ne p$, \begin{align*} A_{\mathfrak{p}} \text{ has good reduction over } K_{\mathfrak{p}} \iff I_{\mathfrak{p}} \text{ acts as the identity on } V_lA. \end{align*} The first condition is the local translation of good reduction at $\mathfrak{p}$: an abelian variety over $K$ has good reduction at $\mathfrak{p}$ exactly when its base change to $K_{\mathfrak{p}}$ has good reduction over the valuation ring of $K_{\mathfrak{p}}$. The second condition is the local translation of unramifiedness: the representation $V_lA$ is unramified at $\mathfrak{p}$ exactly when the inertia subgroup $I_{\mathfrak{p}}$ acts as the identity after restricting from $G_K$ to the decomposition subgroup at $\mathfrak{p}$.[/guided]

[step:Deduce both implications and the independence of $l$] Suppose first that $A$ has good reduction at $\mathfrak{p}$. Then $A_{\mathfrak{p}}$ has good reduction over $K_{\mathfrak{p}}$, so the local criterion gives that $I_{\mathfrak{p}}$ acts as the identity on $V_lA$ for every rational prime $l \ne p$. Hence $V_lA$ is unramified at $\mathfrak{p}$ for every such $l$. Conversely, suppose that for one rational prime $l_0 \ne p$, the representation $V_{l_0}A$ is unramified at $\mathfrak{p}$. Then $I_{\mathfrak{p}}$ acts as the identity on $V_{l_0}A$. Applying the local criterion with $l = l_0$ gives that $A_{\mathfrak{p}}$ has good reduction over $K_{\mathfrak{p}}$, and therefore $A$ has good reduction at $\mathfrak{p}$. The first paragraph then shows that $V_lA$ is unramified at $\mathfrak{p}$ for every rational prime $l \ne p$. This proves the stated equivalence and the final assertion. [/step]