TITLE: Usage of algebraic geometry in understanding the total Galois group of the rational
QUESTION [6 upvotes]: A professor of mine remarked in class that: "The tools via which we understand the total Galois group of the rationals comes from algebraic geometry". 
Could anyone shed some light on this remark, or just give me reference?

REPLY [11 votes]: Actually this refers to a big program in arithmetic algebraic geometry. Original references are Grothendieck's SGA, Esquisse d'un programme and La Longue Marche à travers la théorie de Galois (just a few thousands pages...). A recent reference is Szamuely's book Galois groups and fundamental groups. Specifically sections 4.7 and 5.6 might be of interest. If $k$ is a field and $X$ is a proper variety over $k$, then there is a homomorphism from the absolute Galois group of $k$ into the group of outer automorphisms of the étale fundamental group of $X$ (induced by the fundamental exact sequence). Belyi's Theorem implies that this homomorphism is injective for $k=\mathbb{Q}$. Thus we get lots of geometric representations of the absolute Galois group of $\mathbb{Q}$. This is also related to Dessins d'enfants and the Grothendieck-Teichmüller group. I won't even attempt to summarize these fascinating and sophisticated theories. Instead I refer to MO/1909 and MO/64065.