Period rings for Galois representations - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T23:45:52Z http://mathoverflow.net/feeds/question/54708 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/54708/period-rings-for-galois-representations/54790#54790 Answer by Keerthi Madapusi Pera for Period rings for Galois representations Keerthi Madapusi Pera 2011-02-08T16:50:38Z 2011-02-08T16:50:38Z <p>Beilinson has recently discovered a new proof of the de Rham comparison isomorphism. You can find a write-up here: <a href="http://arxiv.org/abs/1102.1294" rel="nofollow">http://arxiv.org/abs/1102.1294</a>. Here, he shows that $B_{dR}$ naturally shows up when you consider the p-adic completion (in a suitable sense) of the derived de Rham cohomology of $\mathcal{O}_{\bar{K}}$ over $\mathcal{O}_K$. </p> <p>Also, $A_{cris}$ naturally shows up as (more or less) the global sections of the structure sheaf over the crystalline site for $\mathcal{O}_{\bar{K}}$ over $W(k)$ ($k$ is the residue field of $K$). There is a very nice explanation of this in R. S. Lodh's thesis: <a href="http://www.math.utah.edu/~remi/research/thesispt1formatted.pdf" rel="nofollow">http://www.math.utah.edu/~remi/research/thesispt1formatted.pdf</a>.</p> http://mathoverflow.net/questions/54708/period-rings-for-galois-representations/87481#87481 Answer by SGP for Period rings for Galois representations SGP 2012-02-03T21:39:48Z 2012-02-04T22:30:12Z <p>Beilinson's results (two papers, one mentioned by Keerthi and <a href="http://arxiv.org/abs/1111.3316" rel="nofollow">the other here</a>) have been generalised by Bhargav Bhatt; <a href="http://math.uchicago.edu/~drinfeld/p-adic_periods/Bhatt-p-adic_derived_de_Rham.pdf" rel="nofollow">his paper</a> also introduces a global period ring $A_{ddR}$ for global Galois representations!! The ring $A_{ddR}$ is a filtered $\hat{Z}$-algebra equipped with a Gal$(\bar{Q}/{Q})$-action. </p> <p>A (one of the many) beautiful result in this paper is the following theorem:</p> <p>Let $X$ be a semistable variety over $Q$. Then the log de Rham cohomology of a semistable model for $X$ is isomorphic to the $\hat{Z}$-etale cohomology of $X_{\bar{Q}}$, once both sides are base changed to a localization of $A_{ddR}$ (whle preserving all natural structures on either side).</p>