Can a p-adic representation and its twist by a non-crystalline character both have nontrivial $D_{cris}$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T08:10:02Z http://mathoverflow.net/feeds/question/83953 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83953/can-a-p-adic-representation-and-its-twist-by-a-non-crystalline-character-both-hav Can a p-adic representation and its twist by a non-crystalline character both have nontrivial $D_{cris}$? Kevin Ventullo 2011-12-20T15:30:32Z 2011-12-20T16:36:05Z <p>For a continuous irreducible representation</p> <p>$\rho: G_{\mathbb{Q}_p}\rightarrow GL_n(\overline{\mathbb{Q}_p})$,</p> <p>is it possible for both $D_{cris}(\rho)$ and $D_{cris}(\chi\otimes\rho)$ to be nonzero, where $\chi$ is some non-crystalline character? </p> http://mathoverflow.net/questions/83953/can-a-p-adic-representation-and-its-twist-by-a-non-crystalline-character-both-hav/83959#83959 Answer by Laurent Berger for Can a p-adic representation and its twist by a non-crystalline character both have nontrivial $D_{cris}$? Laurent Berger 2011-12-20T16:36:05Z 2011-12-20T16:36:05Z <p>Let me use Colmez' article "Representations triangulines" as a reference. Let $V$ be a repn which satisfies your condition.</p> <p>By proposition 4.3, $V$ is trianguline. By proposition 4.10, the HT weight of $\chi$ has to be an integer. You can then assume that $\chi$ has finite order, and this implies that $V$ is potentially crystalline on an abelian extension of $Q_p$ (aka crystabelline).</p> <p>Conversely, it seems likely that one can give examples of irreducible crystabelline representations $V$ such that $D_{cris}(V)$ and $D_{cris}(V \otimes \chi)$ are both nonzero, with $\chi$ of finite order (see the examples in 2.4 of Berger-Breuil's "Sur quelques representations potentiellement cristallines de $GL_2(Q_p)$").</p>