Let $(G, X)$ be a Shimura datum of Hodge type. Suppose that $K \le G(\mathbb A_f)$ is such a compact open subgroup that its $p$th component $K_p = \mathcal G(\mathbb Z_p)$ is a hyperspecial subgroup of $G(\mathbb Q_p)$, where $\mathcal G$ is a reductive extension of $G$ to $\mathbb Z_p$.

Then one can extend the embedding $G \hookrightarrow GSp(V)$ to $\mathcal G \hookrightarrow GL(V_{\mathbb Z_p})$ for some $\mathbb Z_p$-lattice $V_{\mathbb Z_p}$ (cf. Kisin, Integral models for Shimura varieties of abelian type, Lemma 2.3.1).

Let point $x \in Sh_K(G, X)$ be a closed point in characteristic zero. Then the $p$-adic Tate module $T_p \mathcal A_x$ (where $\mathcal A \to Sh_K(G, X)$ is the universal abelian variety) should be "canonically isomorphic to $V_{\mathbb Z_p}$ (up to action of $K_p$)" (cf. e.g. beginning of Section 3.2 here). Why?

I guess it is something trivial, but I can't see it. What follows at once from the "moduli description" of Shimura varieties of Hodge type is that $V_p \mathcal A_x := T_p \mathcal A_x \otimes_{\mathbb Z_p} \mathbb Q_p$ is canonically (up to $K_p$-action) isomorphic to $V_{\mathbb Z_p} \otimes_{\mathbb Z_p} \mathbb Q_p$.

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    $\begingroup$ Without the choice of the lattice, all you have is a family of abelian varieties up to isogeny. The choice of a $\mathbb{Z}$-lattice fixes a canonical family of abelian varieties within the isogeny class, characterized precisely by the property that their homology can be identified with this lattice at each geometric point. The $\mathbb{Z}_p$-lattice does the same, but only determines a family up to prime-to-$p$ isogeny. In other words, it only pins down the $p$-adic Tate module in the way that Kisin says. $\endgroup$ – Keerthi Madapusi Pera Apr 8 at 22:38
  • $\begingroup$ I think I understand... Thank you! $\endgroup$ – Jędrzej Garnek Apr 9 at 13:31

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