Deligne, Goncharov and Levine have constructed a Tannakian category of mixed Tate motives, MTM($\mathcal{O}\_{K,S}$), over the ring of integers of a number field $K$ unramified outside a finite set of places $S$.


In particular there is a category MTM($\mathbb{Z}$) of mixed Tate motives unramified over $\mathbb{Z}$. Its objects are extensions of pure Tate motives $\mathbb{Q}(n)$ where $n$ is an integer. They arise from motivic sheaves on the moduli spaces of marked genus 0 curves.


The $\ell$-adic realization of the pure Tate motive $\mathbb{Q}(n)$ is the $Gal (\overline{\mathbb{Q}}/\mathbb{Q})$-module $\mathbb{Q}\_\ell$, the Galois action being given by the $n$-th power of the $\ell$-adic cyclotomic character $\chi\_\ell$.


In Langlands philosophy every motive should correspond to an automorphic form. For pure Tate motives $\mathbb{Q}(n)$ class field theory shows how: they correspond to powers of idèle class characters.



> 
> Which automorphic form/representation does a *truly* mixed Tate motive correspond to?
> 
> 
> 


For example we know that $M^1\_n := Ext^1(\mathbb{Q}(0), \mathbb{Q}(n)) = K\_{2n-1}(\mathbb{Z})\otimes \mathbb{Q}$ for $n > 1$ where $K\_.$ are the Milnor K-groups. What makes $M^1\_n$ automorphic?



Automorphy of mixed Tate motives over $\mathbb{Z}$
This answer is a slight addition to Joel's and David's.


In the theory of Galois representations, there is a general philosophy that $p$-adic phenomena (say in Hida families or eigenvarieties) reflect corresponding mod $p$ phenomena. So before asking if all extensions of $\mathbb Q\_p(n)$ by $\mathbb Q\_p$ can be constructed automorphically, one could ask the corresponding question mod $p$. Here is a slightly more general version, for the number field $\mathbb Q$: if $\psi$ and $\chi$ are two mod $p$ Dirichlet characters (of some conductor), thought of as Galois characters, whose product is odd, can any extension of $\chi$ by $\psi$ be realised in the reduction mod $p$ of a lattice in the (irreducible) $p$-adic Galois representation attached to some cusp form.


I think that the answer to this (and more general version of this) question is believed to be yes (folklore), and perhaps is even known to be yes in the particular case I just discussed. Questions of this type are certainly discussed in Skinner--Wiles: I think they will show that if you have a $p$-adic lattice in an irreducible representation whose reduction gives some extension class, then there is a cusp form giving rise to this extension. (They will need to assume ordinary at $p$, but actually recent work of L. Pan will let you replace ordinary by a more general Fontaine--Mazur-type condition.) Part of their technique is to move from one possible extension class to another. (Kisin used to call this ``Skinner--Wiles tunnelling''.)


I don't know if it's known that every extension of $\chi$ by $\psi$ actually can be lifted to a lattice that is ordinary (or, more generally, geometric) at $p$, though. If it were, one would get a positive answer to the mod $p$ version of the problem in some cases.


On a different note, I believe that Harder (and probably many others too) studied extension classes of automorphic motives arising from Eisenstein cohomology classes, and in this way constructed (or proposed construtions for) actual mixed motives, rather than just Galois representations. The first example of this would be looking at cohomology of open modular curves, or of complete modular curves relative to the cusps. One obtains mixed motives this way, but by Manin--Drinfeld, if you work over $\mathbb Q$ you get split extensions --- non-splitness only occurs with torsion coefficients. I think you can get more interesting things if you both delete some cusps, and then work relative to others (so that the weight filtration on cohomology has length three rather than just two), although it's been a long time since I thought about it (Mazur mentioned these ideas to me, in conjunction with Harder's name, when I was a grad student). I don't know what happens on higher dimensional Shimura varieties, but work of Harris and Zucker on mixed Hodge theory of Shimura varieties is probably relevant.


