One last question (hopefully) is : do we really need class field theory for this ? The main point of the whole thing is the description of de Rham characters of the Galois group but I don't know if this relies on class field theory ?

I was being a bit silly. Of course our finite order character $\eta : \text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to \mathbf{Q}_p^\times$ can be seen as a character to a $\mathbf{C}^\times$ and thus to a number field since it takes values in the roots of unity of $\mathbf{Q}_p$. Viewing this number field as a finite dimensional vector space over $\mathbf{Q}$ gives us the desired "Artin motive".

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 ?

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 $\eta$.

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 $\overline{\mathbf{Q}_p})$.

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).

@LSpice Ok I'll keep that in mind :) Thanks for you help !

Thank you very much YCor ! Thank you also for the comment on the non-intrinsicness of nilpotency it corrected a big misconception of mine :) @LSpice I think I am happy with just having $X$ be in the Lie algebra of a unipotent subgroup. Would I gain something by knowing that it is in the Lie algebra of the unipotent radical of a Parabolic subgroup ?

Dear LSpice, thank you very much for your answer. Since I'm not very at ease with the dynamic method can you confirm the following argument ? The whole point is that $X \in \mathfrak{g}$ lies the unipotent radical of some parabolic of $G$ associated to a cocharacter $\lambda$ of $G$ if and only if $\lim\limits_{t\to 0} X^{\lambda(t)} = 0$. And then we use this fact in both directions by saying : 1) the result is true over $\overline{k}$ thus there exists a cocharater defined over $\overline{k}$ 2) by Kempf's result we can choose it to be defined over $k$ 3) the result is true over $k$.

It means that for any representation $V$ of $G$ with induced representation $\rho : \mathfrak{g} \to \mathfrak{gl}(V)$, $\rho(X)$ is nilpotent is the usual sense. But I imagine it is equivalent to the fact there exists a faithful representation of $\rho$ of $\mathfrak{g}$ such that $\rho(X)$ is nilpotent (and in particular it does not depend on $G$) but I'm not 100% sure.