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?



$\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ as the fundamental group of $2,3,5,... \infty$?
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 $\infty$.


