current status of crystalline cohomology?

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?

Edit: U. Jannsen talked recently on "a refinement of crystalline cohomology by using the theory of so-called gauges as introduced earlier by Mazur and Kato and certain syntomic sheaves." Unfortunately I found no preprint on that. Edit: Jannsen on (slides) "a cohomology theory in characteristic p which refines the crystalline cohomology – and works well for torsion" and "a sheaf theory which generalizes the Dieudonné theory – and works well for torsion."

Edit: Go Yamashita talked about "La Theorie de Hodge p-adique pour varietes ouverts" avoiding Falting's almost etale extensions. Unfortunately I found no text where one can read that.

Edit: A short note by Bhargav Bhatt and Aise Johan de Jong on a shortened proof of the comparison theorem between crystalline and de Rham cohomology.

Edit: A new proof of the semistability conjecture by Beilinson and a definition of derived crystals by Gaitsgory and Rozenblyum.

Edit: A p-adic derived de Rham cohomology by Bhargav Bhatt, giving "derived de Rham descriptions of the usual period rings and related maps in p-adic Hodge theory" and "a new proof of Fontaine's crystalline conjecture and Fontaine-Jannsen's semistable conjecture".

Edit: A "a new cohomology theory in characteristic p>0, the so called F-gauge cohomology, a cohomology with values in the category of so-called F-gauges, which refines the cristalline cohomology" by Fontaine, Jannsen.

Edit: An other very interesting and very nice to read history of mathematics talk by Illusie "Grothendieck at Pisa: crystals and Barsotti-Tate groups":

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.