The Fontaine-Mazur cojectures says that if $\chi : \text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \mathbf{Q}_p^\times$ is a character which is unramified almost everywhere and de Rham at $p$ then it appears as sub-quotient of the étale cohomology of some algebraic variety over $\mathbf{Q}$.

My question is : Can I explicitely find a variety $X$ and integer $r$ such that $\chi$ is a subquotient of $H^i_{\text{ét}}(X_{\overline{\mathbf{Q}}},\mathbf{Q}_p(r))$ ?

From what I understand $X$ will be of dimension $0$ and thus $i=0$ too. I have found these notes of Fargues : https://webusers.imj-prg.fr/~laurent.fargues/Motifs_abeliens.pdf

They are focused on the much more complicated case where $\mathbf{Q}$ is replaced by any number field and I can't seem to extract the answer to my question. In these notes it says that the Hecke character associated to $\chi$ by class field theory will be of the form $\eta*N^k$ where $\eta$ has finite order and $N$ is the norm (this I am fine with) and this implies that $\chi$ is the $p$-adic étale realization of $M(k)$ where $M$ is the simple Artin motive associated to $\eta$. Unfortunately I don't really know what this last part means and I would really like and explicit variety and understand how the $p$-adic étale cohomology gives rise to our original character.

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    $\begingroup$ Does the section on Artin Motives in Chapter 6 of Deligne and Milne, Tannakian Categories, answer your question? $\endgroup$ – anon May 19 at 22:36
  • $\begingroup$ Thanks Anon ! It does help but here is one thing I don't understand : given a character of finite order of $\text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ over $\mathbf{C}$ how do I get the finite dimensional representation of $\text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ over $\mathbf{Q}$ which then gives me the desired Artin motive. I think this is linked to viewing $\chi$ as a character over some number field and the notes of Fargues emphasize that there is some care to be taken with the coefficients (going between coefficients over $\mathbf{Q}_p$, $\mathbf{C}$ and number fields). $\endgroup$ – Jdoe May 20 at 6:33
  • $\begingroup$ In my question I mentioned the fact that one can associate an Hecke character to $\chi$ but this requires going from $\mathbf{Q}_p$ to $\mathbf{C}$ and already this seems highly non canonical (choose an isomorphism between $\mathbf{C}$ and $\overline{\mathbf{Q}_p})$. $\endgroup$ – Jdoe May 20 at 8:13
  • $\begingroup$ I found the following notes which are very helpful : math.stanford.edu/~conrad/modseminar/pdf/L11.pdf. What I understand now is the following : any continuous character $\chi : \text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \mathbf{Q}_p^{\times}$ which is de Rham at $p$ is of the form $\eta \epsilon^n$ where $\eta$ is a finite order character and $\epsilon$ is the cyclotomic character. Thus what I am left to understand is what variety over $\mathbf{Q}$ has $p$-adic étale cohomology isomorphic to $\eta$. $\endgroup$ – Jdoe May 20 at 17:10
  • $\begingroup$ This should clearly have an easy answer using "Artin motives" but what I am stumling on is the fact that $\eta$ has values in $\mathbf{Q}_p^\times$ and thus is not a finite dimensional representation over $\mathbf{Q}$. I think this can be solved by finding a model over $\mathbf{Q}$ but I haven't yet managed to do so. Any ideas ? $\endgroup$ – Jdoe May 20 at 17:12

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