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Posting my comment as an answer: See the top of page 216 in S. Zucker, Generalized intermediate Jacobians and the theorem on normal functions, Invent. Math. 33 (1976), no.3, 185–222. Note that Zucker …

Let me add a different answer. Let $p$ be a prime with $p>\dim X$. Let $\mathcal{K}_{i}$ denote the higher $K$-sheaf on $X$, and let $S\mathcal{K}_{i}:=\mathrm{im}((\mathcal{O}_{X}^{\times})^{\oplus i …

This may not be precisely what you want, but a function field analogue of Beilinson's conjectures is formulated in R. Sreekantan, Non-Archimedean regulator maps and special values of $L$-functions, Cy …

This is an old question but since it hasn't received much attention, let me just point out "the next" example beyond that given in the question: Let $k$ be a perfect field of characteristic $p>0$, and …

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 …

I like chapter 6 of Lewis, James D. A survey of the Hodge conjecture, Second edition, Appendix B by B. Brent Gordon, CRM Monogr. Ser., 10, American Mathematical Society, Providence, RI, 1999. It has l …

I shall try to stick to the notation in Katz's paper. Let $k$ be a perfect field of characteristic $p>0$. Let $S_{\infty}$ be a $p$-adically complete and separated smooth formal $W(k)$-scheme and $f:X …

This answer just amounts to adding a reference to Marc Hoyois' comment: there is a discussion of precisely this on pages 393-394 in Scholl's An introduction to Kato's Euler systems, London Math. Soc. …

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 …

Apologies for being a bit late, but let me try to expand on my comment. An excellent source for this material is James Lewis' "user-friendly" survey article [Lew14]. First recall that for $X$ smooth o …

$\require{AMScd}$I'll expand a little on my comment to give an answer to David's follow up question: Firstly, the general relationship is described in Chiarellotto's Duke 1999 paper "Rigid cohomology …

Let $k$ be an algebraically closed field of characteristic $p>0$ and write $W:=W(k)$ for its ring of Witt vectors. Consider a smooth cubic fourfold $X_{0}\subset\mathbb{P}^{5}_{k}$ and let $F(X_{0})$ …

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 …