[proofplan] We work locally at $\mathfrak{p}$ and use the abelian scheme supplied by good reduction. Since $l$ is invertible on the valuation ring, the finite group scheme of $l^n$-torsion is finite étale over that ring. The henselian lifting property for finite étale schemes identifies its geometric generic fibre over the maximal unramified extension with its geometric special fibre, and inertia acts as the identity on the latter. Passing through all levels $l^n$ and then to the inverse limit gives identity inertia action on $T_l(A)$ and hence on $V_l(A)$. [/proofplan] [step:Replace the global assertion by the local inertia action] Let $F := K_{\mathfrak{p}}$ be the completion of $K$ at $\mathfrak{p}$. Let $\mathcal{O}_F$ be its valuation ring, let $k$ be its residue field, and let $q := \operatorname{char} k$ be the residue characteristic. By hypothesis $q \neq l$. Fix a separable closure $\overline{F}$ of $F$. Let \begin{align*} G_F := \operatorname{Gal}(\overline{F}/F) \end{align*} be the absolute [Galois group](/page/Galois%20Group) of $F$. Let $F^{\mathrm{ur}} \subset \overline{F}$ be the maximal unramified extension of $F$, and define the inertia subgroup \begin{align*} I_F := \operatorname{Gal}(\overline{F}/F^{\mathrm{ur}}) \subset G_F. \end{align*} The assertion that $V_l(A)$ is unramified at $\mathfrak{p}$ means precisely that $I_F$ acts as the identity on \begin{align*} V_l(A) = \left(\varprojlim_{n} A[l^n](\overline{F})\right) \otimes_{\mathbb{Z}_l} \mathbb{Q}_l. \end{align*} Thus it is enough to prove that $I_F$ acts as the identity on $A[l^n](\overline{F})$ for every integer $n \geq 1$. [/step] [step:Use good reduction to extend $A$ to an abelian scheme] Since $A/K$ has good reduction at $\mathfrak{p}$, the base change $A_F := A \times_K F$ extends to an abelian scheme \begin{align*} \mathcal{A} \to \operatorname{Spec}\mathcal{O}_F \end{align*} whose generic fibre is $A_F$. More explicitly, $\mathcal{A}$ is a proper smooth group scheme over $\operatorname{Spec}\mathcal{O}_F$ with connected fibres, and there is an isomorphism of abelian varieties over $F$ \begin{align*} \mathcal{A}_F \cong A_F. \end{align*} Let \begin{align*} \widetilde{A} := \mathcal{A}_k \end{align*} denote the special fibre, an abelian variety over $k$. [/step] [step:Show that $l^n$-torsion extends as a finite étale group scheme] Fix an integer $n \geq 1$. Define the multiplication morphism \begin{align*} [l^n]_{\mathcal{A}} : \mathcal{A} \to \mathcal{A} \end{align*} to be multiplication by $l^n$ on the abelian scheme $\mathcal{A}$. Define the finite group scheme \begin{align*} \mathcal{A}[l^n] := \ker([l^n]_{\mathcal{A}}) \end{align*} over $\operatorname{Spec}\mathcal{O}_F$. Because $l$ is invertible in $\mathcal{O}_F$, the multiplication morphism $[l^n]_{\mathcal{A}}$ is finite étale on the abelian scheme $\mathcal{A}$; this is the standard prime-to-residue-characteristic torsion theorem for abelian schemes, cited here as the external result that prime-to-the-base-characteristic torsion of an abelian scheme is finite étale. Hence \begin{align*} \mathcal{A}[l^n] \to \operatorname{Spec}\mathcal{O}_F \end{align*} is finite étale. Its generic fibre is \begin{align*} \mathcal{A}[l^n]_F \cong A_F[l^n], \end{align*} and its special fibre is \begin{align*} \mathcal{A}[l^n]_k \cong \widetilde{A}[l^n]. \end{align*} [/step] [step:Identify prime-to-$\mathfrak{p}$ torsion with the special fibre] Let $\mathcal{O}_F^{\mathrm{sh}}$ be the strict henselization of $\mathcal{O}_F$, let $F^{\mathrm{ur}}$ be its fraction field inside $\overline{F}$, and let $\overline{k}$ be its residue field. Since $\mathcal{A}[l^n]$ is finite étale over the henselian local ring $\mathcal{O}_F$, the finite étale lifting theorem over a henselian local ring gives a canonical bijection \begin{align*} \mathcal{A}[l^n](\mathcal{O}_F^{\mathrm{sh}}) \cong \mathcal{A}[l^n]_k(\overline{k}). \end{align*} Here we cite the standard henselian lifting theorem for finite étale schemes as an external result: for a finite étale scheme over a henselian local ring, sections over the strict henselization are identified with geometric points of the special fibre. Because $\mathcal{A}[l^n]$ is finite over $\mathcal{O}_F$, every section over $F^{\mathrm{ur}}$ extends uniquely over $\mathcal{O}_F^{\mathrm{sh}}$ by properness of finite morphisms. Therefore \begin{align*} A[l^n](F^{\mathrm{ur}}) = \mathcal{A}[l^n](F^{\mathrm{ur}}) \cong \mathcal{A}[l^n](\mathcal{O}_F^{\mathrm{sh}}) \cong \widetilde{A}[l^n](\overline{k}). \end{align*} Moreover, a finite étale scheme over the strictly henselian local ring $\mathcal{O}_F^{\mathrm{sh}}$ is a finite disjoint union of copies of $\operatorname{Spec}\mathcal{O}_F^{\mathrm{sh}}$. Applying this to $\mathcal{A}[l^n]_{\mathcal{O}_F^{\mathrm{sh}}}$, its generic fibre over $F^{\mathrm{ur}}$ is a finite disjoint union of copies of $\operatorname{Spec}F^{\mathrm{ur}}$. Thus every geometric point of $A[l^n]$ is already defined over $F^{\mathrm{ur}}$, and \begin{align*} A[l^n](\overline{F}) = A[l^n](F^{\mathrm{ur}}). \end{align*} [/step] [step:Conclude that inertia acts as the identity at every torsion level] By definition, \begin{align*} I_F = \operatorname{Gal}(\overline{F}/F^{\mathrm{ur}}). \end{align*} Since the previous step gives \begin{align*} A[l^n](\overline{F}) = A[l^n](F^{\mathrm{ur}}), \end{align*} every element $\sigma \in I_F$ fixes every point of $A[l^n](\overline{F})$. Thus $I_F$ acts as the identity on $A[l^n](\overline{F})$ for every integer $n \geq 1$. Taking the inverse limit over $n$ with respect to the multiplication-by-$l$ transition maps gives identity action of $I_F$ on \begin{align*} T_l(A) := \varprojlim_n A[l^n](\overline{F}). \end{align*} Tensoring the $I_F$-representation $T_l(A)$ on which every element acts as the identity with $\mathbb{Q}_l$ over $\mathbb{Z}_l$ gives identity action on \begin{align*} V_l(A) := T_l(A) \otimes_{\mathbb{Z}_l} \mathbb{Q}_l. \end{align*} Therefore $V_l(A)$ is unramified at $\mathfrak{p}$. [/step]