User a m - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T05:22:28Z http://mathoverflow.net/feeds/user/10001 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54708/period-rings-for-galois-representations Period rings for Galois representations A M 2011-02-07T23:06:48Z 2012-02-04T22:30:12Z <p>I have some questions concerning period rings for Galois representations.</p> <p>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 $B_{crys}$.</p> <p>Constructing the ring $B_{HT}$ is not very difficult and it is quite natural. </p> <p>Does someone have any idea where $B_{dR}$ comes from ?</p> <p>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 $\mathcal{O}_{\mathbb{C}_p}$ ?</p> <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 ?</p> <p>Finally, why there is no period rings for global $p$-adic Galois representations ?</p> http://mathoverflow.net/questions/55663/when-is-an-extension-of-characters-de-rham/55673#55673 Answer by A M for When is an extension of characters de Rham? A M 2011-02-16T23:24:02Z 2011-02-16T23:24:02Z <p>Take a look at section 16 of this document of Laurent Berger : <a href="http://www.umpa.ens-lyon.fr/~lberger/barcelone/BergerBarcelone.pdf" rel="nofollow">http://www.umpa.ens-lyon.fr/~lberger/barcelone/BergerBarcelone.pdf</a></p> http://mathoverflow.net/questions/49794/induced-representations-and-varphi-gamma-modules Induced representations and $(\varphi, \Gamma)-modules A M 2010-12-18T12:02:38Z 2010-12-19T08:01:00Z <p>Let$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$?</p> http://mathoverflow.net/questions/46756/cohomology-and-tensor-product Cohomology and tensor product A M 2010-11-20T18:01:45Z 2010-11-20T18:01:45Z <p>Let$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$.</p> <p>I am looking at continuous cohomology groups. I tought that$H^i(G,A \otimes B)$would be isomorphic to$H^i(G,A) \otimes B$, which is false in general.</p> <p>Is there a condition on$B$making this statement true ?</p> http://mathoverflow.net/questions/41940/local-to-global-galois-representation local to global Galois representation A M 2010-10-12T20:02:46Z 2010-10-12T20:18:20Z <p>Let$\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$?</p> http://mathoverflow.net/questions/65411/how-can-we-extend-galois-representations Comment by A M A M 2011-05-19T10:16:40Z 2011-05-19T10:16:40Z What do you mean by &quot;$\rho_E$is invariant under$Gal(E/F)$&quot; ? http://mathoverflow.net/questions/53036/books-you-would-like-to-read-if-somebody-would-just-write-them/53044#53044 Comment by A M A M 2011-01-24T18:26:03Z 2011-01-24T18:26:03Z For Laurent : will this book be an extended version of your Galois trimester notes or something else ? http://mathoverflow.net/questions/52322/de-rham-cohomology-of-formal-groups Comment by A M A M 2011-01-17T19:48:41Z 2011-01-17T19:48:41Z Neil, I am interested in a proof too, is that too much work to post the proof here or secret or is it possible for you to share it ? Thanks by advance. http://mathoverflow.net/questions/49794/induced-representations-and-varphi-gamma-modules/49858#49858 Comment by A M A M 2010-12-20T18:12:11Z 2010-12-20T18:12:11Z Thanks, it is exactly what I wanted to know. So basically, on the$(\varphi, \Gamma)\$-module side, the construction is the &quot;obvious&quot; one. http://mathoverflow.net/questions/48079/hodge-tate-decomposition-for-formal-groups/48083#48083 Comment by A M A M 2010-12-02T19:45:50Z 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. http://mathoverflow.net/questions/46756/cohomology-and-tensor-product Comment by A M A M 2010-11-20T23:19:10Z 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...