User a m - MathOverflow

Period rings for Galois representations

2011-02-07T23:06:48Z

I have some questions concerning period rings for Galois representations.

First, consider the case of $p$-adic representations of the absolute Galois group $G_K$, where $K$ denote a $p$-adic field. Among all these representations, we can distinguish some of them, namely those which are Hodge-Tate, de Rham, semistable or crystalline. This is due to Fontaine who constructed some period rings : $B_{HT}$, $B_{dR}$, $B_{st}$ and $B_{crys}$.

Constructing the ring $B_{HT}$ is not very difficult and it is quite natural.

Does someone have any idea where $B_{dR}$ comes from ?

For $B_{crys}$, I guess it was constructed to detect the good reduction of (proper, smooth ?) varieties. I don't know anything of crystalline cohomology but does someone have a simple explanation of the need to use the power divided enveloppe of the Witt vectors of the perfectisation (?) of $\mathcal{O}_{\mathbb{C}_p}$ ?

As for the ring $B_{st}$, once you have $B_{crys}$, I think the idea of Fontaine was to add a period from Tate's elliptic curve, which have bad semistable reduction. Does someone knows if Fontaine was aware that adding just this period will be sufficient or was it a good surprise ?

Finally, why there is no period rings for global $p$-adic Galois representations ?

Answer by A M for When is an extension of characters de Rham?

2011-02-16T23:24:02Z

Take a look at section 16 of this document of Laurent Berger : http://www.umpa.ens-lyon.fr/~lberger/barcelone/BergerBarcelone.pdf

Induced representations and $(\varphi, \Gamma)-modules

2010-12-18T12:02:38Z

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $L$ be a finite Galois extension of $K$. Let $V$ be a $p$-adic representation of $G_L$ Let $D$ be the $(\varphi, \Gamma)$-module associated to $V$ by Fontaine's functor. Is there an easy way to describe the $(\varphi, \Gamma)$-module associated to $Ind_{G_L}^{G_K} V$ in terms of $D$ ?

Cohomology and tensor product

2010-11-20T18:01:45Z

Let $G$ be a profinite group, $A$ a free $\mathbb{Z}_p$-module of finite rank with a continuous action of $G$ and $B$ any $\mathbb{Z}_p$-module (I am not supposing it to be free), with the trivial action of $G$.

I am looking at continuous cohomology groups. I tought that $H^i(G,A \otimes B)$ would be isomorphic to $H^i(G,A) \otimes B$, which is false in general.

Is there a condition on $B$ making this statement true ?

local to global Galois representation

2010-10-12T20:02:46Z

Let $\rho_p : \mbox{Gal}(\overline{\mathbb{Q}}_p / {\mathbb{Q}_p}) \to \mbox{GL}_n(\mathbb{Q}_p)$ be a de Rham $p$-adic representation. Can one find a representation $\rho : \mbox{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mbox{GL}_n(\mathbb{Q}_p)$ such that $\rho$ is geometric (in the sense of Fontaine-Mazur) and such that the restriction of $\rho$ to $\mbox{Gal}(\overline{\mathbb{Q}}_p / {\mathbb{Q}_p})$ is $\rho_p$ ?

Comment by A M

2011-05-19T10:16:40Z

What do you mean by "$\rho_E$ is invariant under $Gal(E/F)$" ?

Comment by A M

2011-01-24T18:26:03Z

For Laurent : will this book be an extended version of your Galois trimester notes or something else ?

Comment by A M

2011-01-17T19:48:41Z

Neil, I am interested in a proof too, is that too much work to post the proof here or secret or is it possible for you to share it ? Thanks by advance.

Comment by A M

2010-12-20T18:12:11Z

Thanks, it is exactly what I wanted to know. So basically, on the $(\varphi, \Gamma)$-module side, the construction is the "obvious" one.

Comment by A M

2010-12-02T19:45:50Z

I don't know if jjj knows where to find Tate's original article so I may add that it is in : Proceedings of a Conference on Local Fields, Springer, 1967.

Comment by A M

2010-11-20T23:19:10Z

I am pretty aware that the reduction mod p of the cohomology is not the cohomology of the reduction mod p, which is quite strange to me...