Let $K$, $L$ be finite extensions of the $p$-adic numbers. Suppose $\chi:G\_K\rightarrow L^{\times}$ is crystalline. It is my understanding that if either $K$ or $L=\mathbb{Q}\_p$, then $\chi$ must be a Tate twist of an unramified character. Is there a classification of crystalline characters in general?



Crystalline Characters
Hi Kevin,


1. Go to <http://math.stanford.edu/~conrad/>
2. Download "Grunwald--Wang for global character groups"
3. Read appendix A, especially prop A.3


The answer (note that $K$ and $L$ are switched in Brian's paper) is that once you've identified your character as a character of $K^\times$ via local class field theory, it should be "algebraic" on $O\_K^\times$.


