In a rough way, a category of motives over a field $k$ with coefficients in a field $K$ gives a universal cohomology theory with coefficients in $K$ for algebraic varieties defined over $k$. I had the impression that the choice of coefficients $K=\mathbb{Q}$ was the most standard one and at least the most natural one($\mathbb{Q}$ is the smallest field of characteristic zero).

But in this video conference:


around the time 8.00, Kontsevich says that the "ultimate coefficient field, that you can use in any situation" is $\mathbb{Q}^{CM}$, the union of CM fields, i.e. of imaginary quadratic extensions of totally real number fields.

I don't understand why it is true so my question is:

Why is $\mathbb{Q}^{CM}$ the "ultimate" coefficients fields for motives?

More precisely:

1)I can understand that a bigger coefficient field than $\mathbb{Q}$ can be useful but I would like to know concrete examples of that.

2)I don't know why $\mathbb{Q}^{CM}$ should be enough to cover the eventual answers to 1). I know the usual appearance of CM fields in the theory of complex multiplication of abelian varieties but I don't see a direct relation with the question.

Remark: I am deliberately unprecise on what I consider as "motives". If the answer depends on "details" of the notion (pure/mixed, equivalence relation on cycles...), I would like to know it for the various versions.

  • 2
    $\begingroup$ You should ask Kontsevich. He says many dubious things in this video. $\endgroup$ – anon Jan 3 '15 at 17:56
  • $\begingroup$ This is a guess: it probably has something to do with periods. From section 3 of maths.ed.ac.uk/~aar/papers/kontzagi.pdf, if $f$ is a modular form of positive weight $k$ and $z_0\in\mathbb{H}$ is a CM point, then $\pi^kf(z_0)$ is a period. Among other things, there is a connection between nori motives and periods. $\endgroup$ – Andrew Stout Jan 3 '15 at 18:42
  • $\begingroup$ My impression is (I am not an expert in this particular subject) that sometimes one wants to study certain direct summands of motives that are only defined over certain extensions of $Q$. See Remark 4.17 of this paper arxiv.org/abs/0806.3380; you might also try to read the corresponding paper of Scholl. $\endgroup$ – Mikhail Bondarko Jan 6 '15 at 17:34

I recently learned a fact which seems to answer part of my question: the $CM$ extension $\mathbb{Q}^{CM}$ of $\mathbb{Q}$ is exactly the extension of $\mathbb{Q}$ generated by the Weil numbers (see for example Appendix D of this paper by Drinfeld: http://arxiv.org/abs/1007.4004 ).

Recall that $\alpha \in \overline{\mathbb{Q}}$ is a Weil number if is a $q$-Weil number for some $q$ power of a prime, which means that $|\alpha^{\sigma}|^2 =\alpha^{\sigma} \overline{\alpha^{\sigma}}=q$ for every conjugate $\alpha^{\sigma}$ of $\alpha$ over $\mathbb{Q}$.

The fact that the field generated by a Weil number $\alpha$ is $CM$ is elementary: $\beta = \alpha + \overline{\alpha}$ is totally real and $\alpha$ is a root of $T^2 - \beta T +q$, of discriminant $\beta^2-4q=(\alpha-\overline{\alpha})^2 \leq 0$.

If $X$ is a smooth projective variety over a finite field $\mathbb{F}_q$ then the eigenvalues of the Frobenius acting on the $l$-adic cohomology of $X$ ($l$ prime different from the characteristic $p$ of $\mathbb{F}_q$) are Weil numbers independent of $l$ (Deligne) and equal to the eigenvalues of the Frobenius acting on the crystalline cohomology (Katz-Messing). From the Tate conjecture, all the motivic information about $X$ should be contained in these eigenvalues. This makes clear that $\mathbb{Q}^{CM}$ is a good coefficient for motives over finite fields. The same will be true for motives over global fields as such motive should be determined by its reductions to the various finite places.

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