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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$\begingroup$ By "elementary" I guess you mean without using the Hodge-Tate or de Rham comparison theorems? Then the answer is no. $\endgroup$– David HansenCommented Feb 3, 2020 at 11:18
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$\begingroup$ Why is this conclusion true? $\endgroup$– user141691Commented Feb 4, 2020 at 6:25
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2$\begingroup$ Because the only known way to prove that these representations are Hodge-Tate is to use the Hodge-Tate comparison theorem... $\endgroup$– David HansenCommented Feb 4, 2020 at 9:42
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