For any number field $K$, the Fontaine-Mazur conjecture predicts that any potentially semistable $p$-adic representation of the absolute Galois group $G_K$ of $K$ that is almost everywhere unramified comes from algebraic geometry (i.e., is a subquotient of the etale cohomology of some variety over $K$, up to Tate twist). As far as I can see, the only cases where any progress has been made concerns the case that $K$ is totally real or CM.
This made me wonder: Is the Fontaine-Mazur conjecture known to be true for $1$-dimensional representations for any number field $K$? For CM fields, the theory of CM abelian varieties gives varieties whose cohomology realizes nontrivial characters (and I guess that easy variations should produce all characters). What are the geometric objects appearing for other fields?
[edit: The word 'geometric' is avoided now, see the comments.]
Best Answer
Let $\chi$ be a one-dimensional geometric (in the sense of FM) $p$-adic Galois representation of $G_K$ and let $\psi$ be the Hecke character of $K$ associated to $\chi$ by class field theory. The fact that $\chi$ is de Rham (=pst) at all primes above $p$ imples that $\psi$ is an algebraic Hecke character. Generally, the only algebraic Hecke characters of $K$ are of the form $(\text{finite order})\cdot\mathcal{N}^n$ where $\mathcal{N}$ is the norm character. Under class field theory, $\mathcal{N}$ corresponds to the cyclotomic character, so it comes from geometry; additionally, any finite order character comes from geometry (it arises as the subquotient of the $H^0$ of a zero-dimensional variety). The only time there are more algebraic Hecke characters is when $K$ contains a CM field. Denoting $L$ the maximal CM field in $K$, every algebraic Hecke character of $K$ is of the form $(\text{finite order})\cdot(\psi_L\circ\mathcal{N}_{K/L})$ where $\psi_L$ is an algebraic Hecke character of $L$ and $\mathcal{N}_{K/L}$ is the norm from $K$ to $L$. Again, finite order characters come from geometry, so this case is reduced to the CM case. As you've mentioned the CM case has been dealt with, so Fontaine–Mazur is true for $\mathrm{GL}(1)$.