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Let $A$ be a polarized abelian variety over a local field $K$ with residue characteristic $p$. In the course of proving that a polarized abelian variety $A/K$ has semi-stable reduction iff for all $\ell\ne p$ the action of the inertia group $I$ on the $\ell$-Tate module of $A$ is unipotent, Grothendieck nicely describes the relation between the "fixed part" $V=T_l(A)^I$ of the Tate module and the "toric part" $W=T_l(T)$: namely, we have $W=V\cap V^\perp$, where orthogonality is with respect to the Weil-pairing. See SGA7-IX, diagram 2.5.4 (which is mislabeled).

My question is, do we have the same picture mod $m$ for all $(m,p)=1$? In other words can we define such a "toric part" $W$ sitting inside $V=A[m]^I$ satisfying $W=V\cap V^\perp$? It is not immediately clear from Grothendieck's proof, where he views $T_\ell(T)$ as a locally-constant $\ell$-adic sheaf over $\mathcal{O}_K$, and not as the Tate module of a torus lifting the toric part of the special fiber of the NĂ©ron model of $A$.

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