5 added 9 characters in body

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 $\mathbb{Q}_p$ an 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.

4 added 3 characters in body

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 $\mathbb{Q}_p$ an $l$-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.

3 backticks

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 $\mathbb{Q}_p$ an $l$-th root of $1$, where $p$ and $l$ are primes, to get the extension $K|\mathbb{Q}_p$. 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 $\operatorname{Gal}(\bar{\mathbb{Q}}_p|\mathbb{Q}_p)\to\mathbb{Q}_p^\times$, \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 $\operatorname{Gal}(\bar{\mathbb{Q}}_p|\mathbb{Q}_p)$, \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 $\mathbb{Q}p$\mathbb{Q}_p$; the actual definition is in terms of a certain ring $\mathbf{B}{\text{cris}}$, \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 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.