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Let me upgrade my comment to an answer: The answer is no, the filtration is not independent of the lifting. In fact the relationship between liftings of $Y$ and filtrations lifting the Hodge filtratio …

For an example of an application of $p$-adic Hodge theory in a geometric setting, I thoroughly recommend reading the beautiful paper P. Berthelot, H. Esnault, K. Rulling, Rational points over finite …

Let me upgrade my comment to an answer: There exist (simple) abelian varieties over $\mathbb{Q}_{p}$ of every dimension $g\geq 2$ with supersingular good reduction and non-semisimple Tate module. Abel …

Fix a prime $p$ and let $X$ be a $\mathbb{Z}_{p}$-scheme. Write $X_{n}:=X\otimes\mathbb{Z}/p^{n}$ and $\phi:X_{1}\rightarrow X_{1}$ for the absolute Frobenius. Let $X\hookrightarrow Z$ be a (suitable) …

Ok, maybe I've figured this out. Hopefully somebody can correct me if this is wrong. Also, I'd still like to know a reference that writes this out in detail, if anybody has one. I'll change the notat …

Deligne cohomology has a geometric interpretation. For example, $H^{2}_{\mathcal{D}}(X,\mathbb{Z}(1))$ is identified with the group $H^{1}(X,\mathcal{O}_{X}^{\ast})$ of isomorphism classes of line bun …

There has been some progress on this question since the question was asked. Apparently it was "known to the experts" that there cannot be an integral $p$-adic cohomology theory which is finitely gener …

As Satan's Minion says, the good reduction case is R. Coleman, A. Iovita, The Frobenius and monodromy operators for curves and abelian varieties, Duke Math. J. 97 (1999), 171--215. For the semistable …