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js21
js21
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Let $X$ be a smooth projective scheme over $Z_p$$\mathbb{Z}_p$, with special fiber $X_s$ over $\mathbb{F}_p$, generic fiber $X_{\eta}$ over $\mathbb{Q}_p$, and geometric generic fiber $\bar{X_{\eta}}$ over $\bar{\mathbb{Q}_p}$. Is there a way to recover the cardinality of $X_s(\mathbb{F}_p)$ form the collection of $Gal(\bar{\mathbb{Q}_p}/\mathbb{Q}_p)$-modules $H^{i}(\bar{X_{\eta}}, \mathbb{Q}_p)$$H^{i}((\bar{X_{\eta}})_{ét}, \mathbb{Q}_p)$ ?

Let $X$ be a smooth projective scheme over $Z_p$, 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 $H^{i}(\bar{X_{\eta}}, \mathbb{Q}_p)$ ?

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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js21
js21
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  • 38

p-adic etale cohomology

Let $X$ be a smooth projective scheme over $Z_p$, 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 $H^{i}(\bar{X_{\eta}}, \mathbb{Q}_p)$ ?