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Let $X$ be a smooth and proper variety over $\mathbb{Q}$. Then for each prime $p$ we have the representation $R_p=H^i_{et}(X\times \overline{\mathbb{Q}_p}, \mathbb{Q}_p)$ of $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$. Is there an elementary way (in particular avoiding the comparison theorems) to show that the Hodge-Tate representations among $R_p$'s all have the same Hodge-Tate weights (one does not necessarily have to show that all $R_p$'s are Hodge-Tate though they are by Faltings's comparison theorem)?

Let $X$ be a smooth and proper variety over $\mathbb{Q}$. Then for each prime $p$ we have the representation $R_p=H^i_{et}(X\times \overline{\mathbb{Q}_p}, \mathbb{Q}_p)$ of $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$. Is there an elementary way (in particular avoiding the comparison theorems) to show that the Hodge-Tate representations among $R_p$'s all have the same Hodge-Tate weights (one does necessarily have to show that all $R_p$'s are Hodge-Tate though they are by Faltings's comparison theorem)?

Let $X$ be a smooth and proper variety over $\mathbb{Q}$. Then for each prime $p$ we have the representation $R_p=H^i_{et}(X\times \overline{\mathbb{Q}_p}, \mathbb{Q}_p)$ of $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$. Is there an elementary way (in particular avoiding the comparison theorems) to show that the Hodge-Tate representations among $R_p$'s all have the same Hodge-Tate weights (one does not necessarily have to show that all $R_p$'s are Hodge-Tate though they are by Faltings's comparison theorem)?

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Let $X$ be a smooth and proper variety over $\mathbb{Q}$. Then for each prime $p$ we have the representation $H^i_{et}(X\times \overline{\mathbb{Q}_p}, \mathbb{Q}_p)$$R_p=H^i_{et}(X\times \overline{\mathbb{Q}_p}, \mathbb{Q}_p)$ of $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$. Is there an elementary way (in particular avoiding the comparison theorems) to show that thesethe Hodge-Tate representations among $R_p$'s all have the same Hodge-Tate weights (one does necessarily have to show that all $R_p$'s are Hodge-Tate though they are by Faltings's comparison theorem)?

Let $X$ be a smooth and proper variety over $\mathbb{Q}$. Then for each prime $p$ we have the representation $H^i_{et}(X\times \overline{\mathbb{Q}_p}, \mathbb{Q}_p)$ of $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$. Is there an elementary way to show that these representations all have the same Hodge-Tate weights?

Let $X$ be a smooth and proper variety over $\mathbb{Q}$. Then for each prime $p$ we have the representation $R_p=H^i_{et}(X\times \overline{\mathbb{Q}_p}, \mathbb{Q}_p)$ of $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$. Is there an elementary way (in particular avoiding the comparison theorems) to show that the Hodge-Tate representations among $R_p$'s all have the same Hodge-Tate weights (one does necessarily have to show that all $R_p$'s are Hodge-Tate though they are by Faltings's comparison theorem)?

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Independence of $p$ of Hodge-Tate weights

Let $X$ be a smooth and proper variety over $\mathbb{Q}$. Then for each prime $p$ we have the representation $H^i_{et}(X\times \overline{\mathbb{Q}_p}, \mathbb{Q}_p)$ of $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$. Is there an elementary way to show that these representations all have the same Hodge-Tate weights?