10
$\begingroup$

Suppose $k$ is a number field, i.e. an extension of $\mathbb{Q}$ of finite degree, so we have a natural inclusion $\mathbb{Q} \rightarrow k$, which induces a morphism, \begin{equation} \text{Spec}\,k \rightarrow \text{Spec}\,\mathbb{Q} \end{equation}

I have a naive (probably wrong) question to bother the mathoverflow community: is there a (mixed)-motive (over $\mathbb{Q}$ with coefficient ring $\mathbb{Q}$ ) associated to the $\mathbb{Q}$-scheme $\text{Spec}\,k$? If so, what is its Betti ($\ell$-adic) realisation? Please forgive my naiveness.

$\endgroup$
13
$\begingroup$

Write $X=\mathrm{Spec}\, k$, which is a $0$-dimensional variety. Motives of $0$-dimensional varieties are called Artin motives, and they are pure. The Betti realization is the Betti cohomology of $X$, which is $$ H^0(X(\mathbb{C}),\mathbb{Q})=\mathbb{Q}^{X(\mathbb{C})}=\mathbb{Q}^{\mathrm{Hom}(k,\mathbb{C})}. $$ There is an involution induced by complex conjugation on $\mathbb{C}$.

To define the $\ell$-adic realization, we need to choose an algebraic closure $\overline{\mathbb{Q}}$ of $\mathbb{Q}$ (of course the one inside $\mathbb{C}$ is a common choice). The basechange $X_{\overline{\mathbb{Q}}}:=X\times_{\mathrm{Spec}\,\mathbb{Q}}\mathrm{Spec}\,\overline{\mathbb{Q}}$ can be identified with the set $\mathrm{Hom}(k,\overline{\mathbb{Q}})$ (viewed as a constant scheme over $\overline{\mathbb{Q}}$), and the $\ell$-adic realization is then $$ H^0_{et}(X_{\overline{\mathbb{Q}}},\mathbb{Q}_\ell)=\mathbb{Q}_{\ell}^{\mathrm{Hom}(k,\overline{\mathbb{Q}})}. $$ This space comes with an action of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.

Artin motives are discussed in Section 9.4 of [1] and Section 6 of [2].

[1] Huber, Annette; Müller-Stach, Stefan, Periods and Nori motives, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 65. Cham: Springer (ISBN 978-3-319-50925-9/hbk; 978-3-319-50926-6/ebook). xxiii, 372 p. (2017). ZBL1369.14001.

[2] Deligne, Pierre; Milne, J.S., Tannakian categories, Hodge cycles, motives, and Shimura varieties, Lect. Notes Math. 900, 101-228 (1982). ZBL0477.14004.

$\endgroup$
  • 2
    $\begingroup$ The Galois representation in question is the induced representation from the subgroup of the Galois group associated to $k$ by the Galois correspondence, as one can tell from unwinding the definitions. $\endgroup$ – Will Sawin Oct 8 '17 at 0:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.