# What is the classification of characters in $p$-adic Hodge theory?

Let $K$ be a $p$-adic field and $\chi : Gal_K \rightarrow \mathbb{Q}_p^\times$ be a character. I know that $\chi$ is Hodge-Tate of weight $0$ iff $\chi(I_K)$ is finite (by Sen's theory), and that it is Hodge-Tate of weight $k$ iff $\chi.\chi_p^{-k}$ is HT of weight $0$.

Is there a similar description for De Rham, Semi-stable and Cristalline ?

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Crystalline characters, in your case, are exactly the twists of unramified characters, see for example this MO question mathoverflow.net/questions/61998/crystalline-characters. – ChrisLazda Aug 27 '13 at 21:36

The de Rham characters are the same as the Hodge-Tate ones. The semistable ones are the same as the crystalline ones, and in your notation they are the de Rham ones for which $(\chi \cdot \chi_p^{-k})(I_K)$ is trivial (and not merely finite). This can be found for example in Fontaine and Mazur's paper.