The story can be generalized to the case of projective variety, however, we consider the most basic case.

Fix an embeding (i.e.'complex place') $\iota:\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}$, this can be viewed as a completion relative to a given archimedian topology. The absolute Galois group $G_\mathbb{Q}$ acts on $\overline{\mathbb{Q}}$, but not continuously relative to the given archimedian topology, hence no continuous action of $G_\mathbb{Q}$ on the completion $\mathbb{C}$ exists. The same thing happens if we consider this p-adically.

Question: Can one make $\overline{\mathbb{Q}}$ into a nontrivial topological space $(X,\tau)$ on which the absolute Galois group $G_\mathbb{Q}$ acts continuously? With such topology given, can one give a proper discription for $(X,\tau)$ and its completion $(\overline{X},\overline\tau)$?

As a possible answer, we could use the height function to define a 'distance' via $d(x,y)=h(x-y)$, and then take the 'open ball' $\{y\in\overline{\mathbb{Q}} | d(x,y)<\delta\}$ (for all $x\in\overline{\mathbb{Q}}, \delta>0$) as a topological sub-basis. Since $d(*,*)$ is not a metric and invarient under $G_\mathbb{Q}$-action , this 'height topoloy' may be 'strange', and I know nothing more.

  • $\begingroup$ Presumably you want some extra conditions on the topology, to rule out e.g. the trivial topology and the discrete topology? $\endgroup$ – Alex Kruckman May 2 '18 at 1:22

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