# 1-dimensional semi-stable Galois representations with coefficients

For any p-adic local field K, all 1-dim semi-stable 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.

My question is what if we replace the coefficient field to $E \neq Q_p$?

In fact, at the end of the paper by Gerasimos Dousmanis "Rank two filtered $(φ, N)$-modules with Galois descent data and coefficients", 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?

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A one-dimensional $p$-adic Galois representation is crystalline if and only if it's semistable, so this is a duplicate of [mathoverflow.net/questions/61998/] – David Loeffler May 27 '12 at 13:16
@David, actually I also found that post, and am now looking at Conrad's paper, but I suspect if the paper has very "explicit" description of what the characters are...Do you know? – 750am May 27 '12 at 13:30

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.
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.
@Berger, just a quick question, so these $a_i$ will be the Hodge-Tate weights? Thanks! – user24007 May 27 '12 at 14:43