Finite Galois representations are Geometric? A famous conjecture, the Fontaine-Mazur conjecture, predicts which $p$-adic Galois representations of a number field "come from geometry" are a subquotient of the (Weil) cohomology of a scheme. Two questions on that: what is the current state of the conjecture (I think we know the rank one case by class field theory) and second are there analogues, where we consider representations of the Galois group of a function field or consider complex representations?
To adress the question the titel: do we know that Galois representations with finite image come from geometry?
 A: Ideas from $p$-adic Hodge Theory allow one to be more precise about which cohomology groups one expects to find the corresponding Galois representation. For a finite Galois representation, the representation will necessarily be de Rham with all Hodge--Tate weights zero. So one expects the Galois representation to occur inside $H^0$ of some smooth proper $X$. But $H^0(X,\mathbf{Q}_p)$ is nothing but the free group on the (geometric) components of $X$. Moreover, all of these are defined over a finite extension of $\mathbf{Q}$ and the Galois action on the cohomology group is just comes from the permutation representation on the components.
A very easy example to consider is the scheme $X: f(x) = 0$ for a separable polynomial $f(x) \in \mathbf{Q}[x]$ of degree $d$. The set $X(\mathbf{Q})$ is just the roots of $f(x)$, and the action of the Galois group $\mathrm{Gal}(\overline{\mathbf{Q}})/\mathbf{Q})$ on $X$ factors through the action of $G = \mathrm{Gal}(K/\mathbf{Q})$ where $K$ is the splitting field of $K$, and the representation is just the one arising from the natural permutation representation of $G$ on the roots. For example, if you start with a Galois extension $K/\mathbf{Q}$ of degree $|G|$, and you let $\theta \in K$ be a primitive element and $f(x)$ the minimal polynomial, then the corresponding representation of $G$ on $H^0(X/\overline{\mathbf{Q}},\mathbf{Q}_p) \simeq \mathbf{Q}^{|G|}_p$ is just the regular representation of $G$. Any finite representation $V$  of $G$is a summand of some number of copies of the regular representation, so any finite Galois representation $V$ of $G$ will occur inside the cohomology of $\coprod X$ for some number of copies of this $X$.
A small point: this realizes $V$ as inside some cohomology but not as the entire cohomology. You have to allow this. For example, $V$ could be the non-trivial $1$-dimensional representation of the Galois group of a quadratic extension. This can't be all of $H^0$ because $H^0$ always contains a $G$-invariant vector corresponding to the sum of all the components. But of course the Fontaine-Mazur conjecture only requires that $V$ is a subquotient rather than the entire cohomology.
