1-dimensional semi-stable Galois representations with coefficients - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T05:35:13Z http://mathoverflow.net/feeds/question/98103 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/98103/1-dimensional-semi-stable-galois-representations-with-coefficients 1-dimensional semi-stable Galois representations with coefficients 750am 2012-05-27T11:51:20Z 2012-05-27T14:43:23Z <p>For any p-adic local field K, all 1-dim <strong>semi-stable</strong> Galois repn: $G_K \to Q_p^{*}$ are just $Q_p(n)\otimes \mu$, where $Q_p(n)$ is the Tate twist of cyclotomic character, and $\mu$ an unramified charater.</p> <p><strong>My question</strong> is what if we replace the coefficient field to $E \neq Q_p$?</p> <p>In fact, at the end of the paper by Gerasimos Dousmanis <a href="http://arxiv.org/abs/0711.2137" rel="nofollow">"Rank two filtered $(φ, N)$-modules with Galois descent data and coefficients"</a>, the filtered $(\varphi, N)$ modules of all such 1-dim repns are all classified. My question really is, how do we write out the representations explicitly? </p> http://mathoverflow.net/questions/98103/1-dimensional-semi-stable-galois-representations-with-coefficients/98115#98115 Answer by Laurent Berger for 1-dimensional semi-stable Galois representations with coefficients Laurent Berger 2012-05-27T14:14:23Z 2012-05-27T14:30:11Z <p>If $E \neq Q_p$ then there may be more $1$-dimensional crystalline representations that the ones you mention. By Lubin-Tate theory, every character of $G_K$ can be written as an unramified character times a character of $O_K^\times$ (after making proper choices and identifications). The algebraic characters of $O_K^\times$ are then crystalline and if $E$ contains $K$, then these provide examples of crystalline characters.</p> <p>EDIT : oops sorry, I did not read the question carefully enough, I did not see that you were asking for an explicit description of <em>all</em> such representations. As David Loeffler pointed out, the answer is in Brian Conrad's paper (now in appendix B, the paper having grown since last time I mentioned it). </p> <p>Here is what it basically says, using the same identifications as above: assume that $E$ contains $K^{gal}$ (it's usually harmless to assume that the coefficient field is large enough). If $s$ runs through the set of embeddings $s : K \to E$ and the $a_s$ are integers, then $x \mapsto \prod_s s(x)^{a_s}$ gives rise to a crystalline character of $G_K$ and they're all of this type times an unramified character.</p> http://mathoverflow.net/questions/98103/1-dimensional-semi-stable-galois-representations-with-coefficients/98119#98119 Answer by 750am for 1-dimensional semi-stable Galois representations with coefficients 750am 2012-05-27T14:43:23Z 2012-05-27T14:43:23Z <p>@Berger, just a quick question, so these $a_i$ will be the Hodge-Tate weights? Thanks!</p>