# Tate module is canonically isomorphic to a $\mathbb Z_p$-lattice on Shimura Variety of Hodge Type

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 $$V_{\mathbb Z_p} \otimes_{\mathbb Z_p} \mathbb Q_p$$.

• 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. – Keerthi Madapusi Pera Apr 8 '19 at 22:38
• I think I understand... Thank you! – Jędrzej Garnek Apr 9 '19 at 13:31