Let $X$ be a smooth projective scheme over $\mathbb{Z}_p$, with special fiber $X_s$ over $\mathbb{F}_p$, generic fiber $X_{\eta}$ over $\mathbb{Q}_p$, and geometric generic fiber $\bar{X_{\eta}}$ over $\bar{\mathbb{Q}_p}$. Is there a way to recover the cardinality of $X_s(\mathbb{F}_p)$ form the collection of $Gal(\bar{\mathbb{Q}_p}/\mathbb{Q}_p)$-modules $H^{i}((\bar{X_{\eta}})_{ét}, \mathbb{Q}_p)$ ?
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4$\begingroup$ I think the answer is yes thanks to $p$-adic Hodge theory: after tensoring with $B_{cris}$ and taking Galois invariants, we get $H^i_{cris}(X/\mathbb{Z}_p)$ with the Frobenius, which then we can use to count points in the special fiber. $\endgroup$– Piotr AchingerCommented Jul 19, 2014 at 12:50
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