Timeline for Is the $\mathbb R$-split component of a torus defined over $\mathbb Q$ also defined over $\mathbb Q$?

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May 8, 2017 at 3:25	comment	added	nfdc23		@WillSawin:Ah, ok. By saturation considerations with lattices inside $\mathbf{Q}$-vector spaces, it suffices to work in the isogeny category of tori and with rationalized character groups. So the question for $\mathbf{T}_s$ is if for a continuous representation of $\Gamma={\rm{Gal}}(\overline{\mathbf{Q}}/\mathbf{Q})$ on a finite-dimensional $\mathbf{Q}$-vector space $V$ and a chosen complex conjugation $\tau\in\Gamma$, the subspace $V^{\tau}$ is $\Gamma$-stable. That is easily contradicted using $S_3$-representations, such as in your answer.
May 7, 2017 at 15:08	comment	added	Will Sawin		@nfdc23 I don't think he means $\mathbf T_s$ should be defined and split over $\mathbb Q$. An analogous statement that's actually true is that if $X$ is a scheme over $\mathbb Q$ that over $\mathbb C$ is isomorphic to a union of copies of points and $\mathbb A^1$, while none of the points and the $\mathbb A^1$s need be defined over $\mathbb Q$, the union of all the points and the union of all the $\mathbb A^1$s both are (because they are Galois-invariant). The subtle point is that this kind of argument works over $\mathbb C$ but not over $\mathbb R$.
May 7, 2017 at 4:44	answer	added	Will Sawin		timeline score: 6
May 7, 2017 at 4:30	history	asked	Jerry	CC BY-SA 3.0