There's a body of wisdom (following Beilinson, Bloch, Deligne, ...) relating mixed Tate motives, motivic cohomology, algebraic K-theory, special values of L-functions, and polylogarithms. My understanding is that for smooth projective schemes $X$ over $\mathbf{Q}$, the following hold or are conjectured to:

Certain Ext groups of powers of the Tate motive (base-changed to $X$) are isomorphic to certain motivic cohomology groups of $X$ and to certain pieces of algebraic K-groups of $X$. (I should tensor all these groups with $\mathbf{Q}$. I believe there are further subtleties if we don't do this.)

If instead of extensions of abstract motives as in #1, we look at extensions in categories of realizations (Hodge structures, $p$-adic Galois representations, ...), the map from the algebraic K-groups to the Ext groups in #1 can be described concretely in terms of polylogarithms, or rather the appropriate version of the polylogarithm for the given realization. In particular, for $K_1$, we get extensions of $\mathbf{Q}$ by $\mathbf{Q}(1)$ described by logarithms of nonzero numbers. This is more or less Kummer theory.

The order of vanishing of the L-functions of $X$ at integers are determined by the ranks of these groups.

Now, everything above was about motives over $\mathbf{Q}$ (or over schemes over $\mathbf{Q}$) and with coefficients in $\mathbf{Q}$. My question is then this:

*Is there an analogous picture if we consider motives over a number field $F$ and with coefficients in $F$?

The motive part is easy: instead of Tate motives, we should look at motives associated to algebraic Hecke characters over $F$ with values in $F$. ("Hecke motives"?) These are surely the same as motives of $F$-linear rank $1$. And then we can consider the Ext groups (in the $F$-linear category). Are these isomorphic to $F$-linear analogues of motivic cohomology or algebraic K-groups? Are these maps given by analogues of polylogarithms, and are there relations to analogues of L-functions?

The only new case where I have a clue is where $F$ is an imaginary quadratic field of class number one. Let $E$ be the elliptic curve over $F$ with complex multiplication by $F$. Then there is a "Kummer map" from $E(F)$ to the Galois cohomology group $H^1(G_F,T_l(E))$ (for any prime $l$), which can presumably be viewed as an Ext group of two Hecke motives (though I suppose there is a Weil-Châtelet obstruction to the map being an isomorphism?). So this suggests that in the $F$-linear world, the role of the K-group $K_1(L)=L^*$, for $L$ an extension of $\mathbf{Q}$, would be played by $E(L)$, for $L$ an extension of $F$. I know there is a paper by Beilinson-Levin on elliptic polylogarithms, but I haven't invested the energy to penetrate it. I didn't notice anything in it about complex multiplication, though.

That's all I got. Any ideas?

(All this actually came up rather naturally in some daydreaming about $F$-linear analogues of de Rham-Witt cohomology and topological Hochschild homology, so I'd like to hope there's some connection to reality.)