Usage of algebraic geometry in understanding the total Galois group of the rational

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?

• Well noone fully understands the absolute Galois group (in fact it is often remarked that one can only write down two elements explicitly). Maybe what your proffessor means is that we can understand certain properties of this group via its representations. Quite a bit is known and conjectured about these representations (falling under results in the Langlands program) but basic known examples come from Global Class Field Theory (which is essentially about 1-dim reps) and Elliptic curves (which produce certain 2-dim reps). – fretty Apr 5 '13 at 18:29
• There are various good google searches you can do on these things to get to the results however neither path is easy and requires a lot of study about specifics. – fretty Apr 5 '13 at 18:30

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.

• Thank you! I shall look into it. – edo arad Apr 6 '13 at 11:37

One of the ways one studies the Galois group $G_\mathbf{Q}=\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$, as with any group, is by studying continuous representations $\rho:G_\mathbf{Q}\rightarrow\mathrm{GL}_n(F)$ where $F$ is some topological field (or even a topological ring). This is a Galois representation.

One especially interesting $F$ is $\overline{\mathbf{Q}}_p$, the algebraic closure of the field of $p$-adic numbers (the image of a continuous representation of $G_\mathbf{Q}$ with values in $\mathrm{GL}_n(\overline{\mathbf{Q}}_p)$ actually lands in $\mathrm{GL}_n(E)$ for a finite extension $E$ of $\mathbf{Q}_p$ inside the algebraic closure, so it is equivalent to study representations over such finite extensions). It is especially interesting because the topology of $\mathrm{GL}_n(\overline{\mathbf{Q}}_p)$ is fairly compatible with the profinite group $G_\mathbf{Q}$, whereas, when $F=\mathbf{C}$, a Galois representation into $\mathrm{GL}_n(\mathbf{C})$ must have finite image for topological reasons. These are called $p$-adic Galois representations. Examples of such representations arise from algebraic geometry in the following way: if $X$ is a proper smooth scheme over $\mathbf{Q}$, then the $p$-adic etale cohomology $H^d_{\mathrm{et}}(X_{\overline{\mathbf{Q}}},\overline{\mathbf{Q}}_p)$ is a finite-dimensional $\overline{\mathbf{Q}}_p$-vector space with a continuous action of $G_\mathbf{Q}$. An irreducible Galois representation is said to come from algebraic geometry if it is isomorphic to an irreducible subquotient of (a Tate twist of) some such etale cohomology group. Examples of Galois representations which come from algebraic geometry are the $p$-adic representations attached by Eichler-Shimura and Deligne to classical cuspidal newforms of weight $k\geq 2$.

There is a famous conjecture (the Fontaine-Mazur conjecture) which asserts that an irreducible $p$-adic Galois representation which is "geometric" comes from algebraic geometry in the sense above. The notion of a geometric $p$-adic Galois representation is technical, but roughly speaking it means a $p$-adic Galois representation that looks like it comes from algebraic geometry (I guess, although to be honest I don't have much intuition for the de Rham property which is one of the conditions in the definition of the term "geometric," and there are others on this site who could explain this a lot better than I can). It is known that Galois representations which come from algebraic geometry are geometric, and many cases of the converse have been proved for $\mathrm{GL}_2$ over $\mathbf{Q}$ and over some totally real fields (I think). This is still a huge area of active research in number theory.

• Upon rereading the question I realized that I misinterpreted it. I'll leave this answer for now but if people think it should be deleted then I will do so. – Keenan Kidwell Apr 5 '13 at 18:44
• Nooo, don't delete it, it is quite interesting and definitely answers (one interpretation of) the question. – Martin Brandenburg Apr 5 '13 at 18:50
• Thank you! it is very interesting! Actually, I dont realy understand why this doesnt answer my question. – edo arad Apr 6 '13 at 11:38
• A beautiful answer. – Piotr Pstrągowski Oct 5 '13 at 1:00