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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:

  1. 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.)

  2. 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.

  3. 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.)

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The quick answer is: yes there is, conjecturally everything carries over. See papers by K.Kato, D.Burns and M.Flach, T.Fukaya and K.Kato... –  Olivier Apr 22 '10 at 12:10
    
Thanks! But unfortunately I wasn't able to find which papers you're referring to. I found lots on noncommutative Iwasawa theory and equivariant zeta-functions, but as far as I can tell, these are different issues. If you could point me to a specific paper, or explain why these things are the same as what I was asking about, I'd be grateful. –  JBorger Apr 22 '10 at 20:04
    
I've come across the articles by Nekovář and Fontaine–Perrin-Riou in motives I that describe the Beilinson conjecture (and Bloch–Kato) for motives over a number field $F$ with coefficients in $\mathbf{Q}$. I would guess that one could do this with coefficients in some number field $E$: as far as the Deligne conjecture on special values of $L$-functions, the period doesn't depend on the choice of a place of $E$, but I don't know... –  Rob Harron Apr 29 '10 at 0:02
    
I wonder how would that relate to topological Hochschild homology - by turning to brave new rings? –  Thomas Riepe Apr 29 '10 at 9:56
    
@ Rob H.: Thanks. I think that forgetting the coefficients would change things quite a bit. The second article you mention explicitly says they will ignore coefficients, but then they add that Fontaine's Seminaire Bourbaki talk from 1992 doesn't ignore them! Fontaine does consider the $F$-linear Ext groups I mentioned, but he seems to stop short of fully $F$-linearizing everything along the lines I wondered about above. I'll have to think about whether this is because the full picture I sketched above (i.e. $F$-linear $K$-groups, $L$-functions, etc) doesn't exist. –  JBorger Apr 30 '10 at 4:15

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