Question

The great references given on Ilya's question make me wonder about the current status of the many conjectures and open questions in Illusie's survey from 1994 on crystalline cohomology. Obviously (just compare Illusie's survey from 1975 with that above or with Chambert-Loir's survey from 1998), there is very intense work on that and the connections between the various cohomology theories attacking the case "l=p". Some more recent surveys only on Fontaine's p-adic Hodge theory are already linked to in the answers to Ilya's question, Le Stum's book (Errata) covers rigid chohomology. Among the open issues mentioned in Illusie's survey are finiteness theorems, crystalline coefficients, geometric semistability, the identity of characteristic polynomials of the Frobenius of different theories,... What is the current status of these? Which new theories have been created the past decade, how fit they together and which new questions emerged?

Answer 1

This is a "big-picture" question, but allow me to illustrate some recent progress by taking a small example close to my heart.

Let us adjoin to the field $\mathbb{Q}_p$ a primitive $l$-th root of $1$, where $p$ and $l$ are primes, to get the extension $K|\mathbb{Q}_p$. We notice that this extension is unramified if $l\neq p$ but ramified if $l=p$. When we adjoin all the $l$-power roots of $1$, we get the $l$-adic cyclotomic character $\chi_l:\operatorname{Gal}(\bar{\mathbb{Q}}_p|\mathbb{Q}_p)\to\mathbb{Q}_l^\times$ which is unramified if $l\neq p$ but ramified if $l=p$. But we cannot just say that $\chi_p$ is ramified and be done with it. We have to somehow express the fact that $\chi_p$ is a natural and a "nice" character, not an arbitrary character $\operatorname{Gal}(\bar{\mathbb{Q}}_p|\mathbb{Q}_p)\to\mathbb{Q}_p^\times$, of which there are very many because the topologies on the groups $\operatorname{Gal}(\bar{\mathbb{Q}}_p|\mathbb{Q}_p)$, $\mathbb{Q}_p^\times$ are somehow "compatible".

The fact that $\chi_p$ is a "nice" character is expressed by saying that it is crystalline. In general, we can talk of crystalline representions of $\operatorname{Gal}(\bar{\mathbb{Q}}_p|\mathbb{Q}_p)$ on finite-dimensional spaces over $\mathbb{Q}_p$; the actual definition is in terms of a certain ring $\mathbf{B}_{\text{cris}}$, constructed by Fontaine, which can be understood in terms of crystalline cohomology.

My illustrative example is about the $l$-adic criterion for an abelian variety $A$ over $\mathbb{Q}_p$ to have good reduction. For $l\neq p$, this can be found in a paper by Serre and Tate in the Annals, and it is called the Néron-Ogg-Shafarevich criterion. It says that $A$ has good reduction if and only if the representation of $\operatorname{Gal}(\bar{\mathbb{Q}}_p|\mathbb{Q}_p)$ on the $l$-adic Tate module $V_l(A)$ is unramified.

What happens when $l=p$ ? It is too much to expect that $V_p(A)$ be an unramified representation when $A$ has good reduction; we have seen that even $\chi_p$ is not unramified. What Fontaine proved is that the $p$-adic representation $V_p(A)$ is crystalline (if $A$ has good reduction). To complete the analogy with the case $l\neq p$, Coleman and Iovita proved in a paper in Duke that, conversely, if the representation $V_p(A)$ is crystalline, then the abelian variety $A$ has good reduction.

I hope you find this enticing.

Answer 2

Kedlaya gave a talk in August in which he mentioned some work of Daniel Caro on finiteness for rigid cohomology with coefficients (some of which is on the ArXiv). On the same page, you can find notes from his talks on semistable reduction.