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In an introductory lecture note by T. Saito,

http://www.ms.u-tokyo.ac.jp/~t-saito/pp/GR2.pdf

he said that the absolute Galois group $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ could be seen as the fundamental group of the set set of primes $2,3,5,... \infty$. The local systems over $2,3,5,... \infty$ could be seen as determined by the representations of $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. Could anyone elaborate on this analogy? In what sense does the analogy catch the essence of Galois representations?

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    $\begingroup$ In my understanding this is only a vague analogy with what happen for the field of rational functions $\mathbb{C}(X)$ on a projective curve $X$ over $\mathbb{C}$ and the fact that $\mathbb{Q}$ is the field of rational functions over $Spec(\mathbb{Z})$ which is $\{2,3,5, ... \}$ which we would like to think about as an algebraic curves over a field. In this perspective, the place at infinity seem to behave as a 'compactification' in an impressive number of situations (product formula for local norms, Arakalov geometry...) so that $\{2,3,5,\dots,\infty\}$ should be some sort of projective curve. $\endgroup$ – Simon Henry Oct 26 '17 at 11:08
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Be aware that this is a somewhat vague comment by Saito which can be interpreted in numerous ways. I would not take what he has written too seriously.

I would guess that he means that $$\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) = \lim_U \,\pi_1(U)$$ where the limit is over all open subsets $U \subset \mathrm{Spec} \mathbb{Z} = \{2,3,5, \ldots \}$. (Note that you have misquoted him, as he does indeed talk about open subsets of the set of all primes). Saito is being a bit naughty as $\mathrm{Spec} \mathbb{Z}$ is more than just a set, as it also comes equipped algebro-geometric structure given by its structure sheaf, which is obviously crucial.

The fundamental group $\pi_1(U)$ is the Galois group of the maximal algebraic field extension of $\mathbb{Q}$ which is unramified at all the primes in $U$. You can also throw in the infinite place if you like to account for ramification at $\infty$.

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    $\begingroup$ Like with most analogies, some features of the Galois group are clarified by this analogy, while others are obscured. This is a good way to explain why Galois representations are interesting and natural objects (for the same reason ODE's are interesting and natural, basically) and to describe the relation of the Galois groups to its inertia groups (they generate it, but other than that the structure is quite complicated) but it's not a very good way to understand technical issues like $p$-adic Hodge theory and the Fontaine-Mazur conjecture. $\endgroup$ – Will Sawin Oct 26 '17 at 12:05

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