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I'm far from expect in this topic, but I've overheard a lot of number theory talk over the yearshere's my attempt.

As algebraic geometry matured, people learned that one of the deep reasons for importance of Galois group is that it's the fundamental group pi_1(Spec F) (depending on the definition, sometimes this is true only after a profinite completio)completion, I'll omit this fine point further below). This allows us to use the whole apparatus of topology as well as the geometric intuition.

It's very reasonable to study a space in terms of geometric objects that live on it.

Moreover, if we restrict ourselves to the *Abelian part of Galois group, its structure has been completely established by the class field theory. In other words, all one-dimensional representations of Gal Q are known. That begs a natural question about higher-dimensional reps — indeed this goes on by the name of Langlands program.

Another

One more topic in the discussion of the structure of Gal F is about the specific (conjugacy classes) of operators that live there, called Frobenius operators. The amazing thing about them is that they behave, formally, in a way similar to the knot operators in the fundamental group of a threefold without knots. This image is reinforced by the computations of the dimension of Spec Z that give an answer of 3, providing fruitful connections to the theory of real 3-manifolds, knots and their L-functions.

Finally, here's the direction that was explored by many people and was probably made famous through Grothendieck's work. Consider a variety Z := P^1-(0, 1, \infty), the projective plane without three points. It is really interesting as it makes sense over any field. Now suppose we consider it as a scheme over Q and ask for its fundamental group. Then one would have an exact sequence, if I'm not mistaken,

which implies, by standard reasoning, that Gal Q acts on pi_1(Z/\bar Q), which is a very straightforward group — it's simply a normal topological fundamental group of a plane without two points, so it's freely generated by two loops. Grothendieck called it a cartographic group and related it to dessin's d'enfants. The standard references are very well-written in T.'s answers and there are hopefully more questions on this topic coming to MathOverflow ;)

This

As you see, this is a very interesting direction that we should continue to explore. It's One more thing I like about it is that it's also very suitable to MathOverflow format, since there are both many specific questions as well as opportunities to write reviews and to produce new results by collaboration.
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I'm far from expect in this topic, but I've overheard a lot of number theory talk over the years.

First, and that's something quite straightforward, people want to study Gal Q (this is how I will denote it; this common shortcut is defined as Gal F := Gal \bar F/F) because we like Galois groups. For simple fields we know their Galois groups and we know how tremendously important they have been, from solving the equations (Abel et al) to doing group cohomology in class field theory.

As algebraic geometry matured, people learned that one of the deep reasons for importance of Galois group is that it's the fundamental group pi_1(Spec F) (depending on the definition, sometimes this is true only after a profinite completio). This allows us to use the whole apparatus of topology as well as the geometric intuition.

Representations of Gal F thus have a natural geometric meaning as some bundles (local systems) on the coverings of Spec F. This alone would make their study pretty important.

The whole Galois group is very complicated. Fortunately, since the Galois groups of F_q and Q_p (p-adic numbers) are known, we know lots of factorgroups of it. This is the standard topic of algebraic number theory courses.

Moreover, if we restrict ourselves to the Abelian part of Galois group, its structure has been completely established by the class field theory. In other words, all one-dimensional representations of Gal Q are known. That begs a natural question about higher-dimensional reps — indeed this goes on by the name of Langlands program.

The Langlands program is such a huge topic that I don't feel able even starting to talk about it. It might be a good idea to post questions here if you'll feel brave enough to learn it :)

Another direction was explored by many people and was probably made famous through Grothendieck's work. Consider a variety Z := P^1-(0, 1, \infty), the projective plane without three points. It is really interesting as it makes sense over any field. Now suppose we consider it as a scheme over Q and ask for its fundamental group. Then one would have an exact sequence, if I'm not mistaken, 

0  -->  Gal pi_1(Z/\bar Q)  -->  pi_1(Z/Q)  -->  pi_1(Z / \bar Gal Q )
--> 0

which implies, by standard reasoning, that Gal Q acts on pi_1(Z/\bar Q), which is a very straightforward group — it's simply a normal fundamental group of a plane without two points, so it's freely generated by two loops. Grothendieck called it a cartographic group and related it to dessin's d'enfants. The standard references are very well-written in T.'s answers and there are hopefully more questions on this topic coming to MathOverflow ;)

This is a very interesting direction that we should continue to explore. It's also very suitable to MathOverflow, since there are both many specific questions as well as opportunities to write reviews and to produce new results by collaboration.
show/hide this revision's text 2 fixes

I'm far from expect in this topic, but I've overheard a lot of number theory talk over the years.

First, and that's something quite straightforward, people want to study Gal Q (this is how I will denote it; this common shortcut is defined as Gal F := Gal \bar F/F) because we like Galois groups. For simple fields we know their Galois groups and we know how tremendously important they have been, from solving the equations (Abel et al) to doing group cohomology in class field theory.

As algebraic geometry matured, people learned that one of the deep reasons for importance of Galois group is that it's the fundamental group pi_1(Spec F) (depending on the definition, sometimes this is true only after a profinite completion, I'll be skipping details from now on)completio). This allows up us to use the whole apparatus of topology as well as our the geometric intuition.

Representations of Gal F thus have a natural geometric meaning as some bundles (local systems) on the coverings of Spec F. This alone would make their study pretty important.

The whole Galois group is very complicated. Fortunately, since the Galois groups of F_q and Q_p (p-adic numbers) are known, we know lots of factorgroups of it. This is the standard topic of algebraic number theory courses.

Moreover, if we restrict ourselves to the Abelian part of Galois group, its structure has been completely established by the class field theory. In other words, all one-dimensional representations of Gal Q are known. That begs a natural question about higher-dimensional reps — indeed this goes on by the name of Langlands program.

The Langlands program is such a huge topic that I don't feel able even starting to talk about it. It might be a good idea to post questions here if you'll feel brave enough to learn it ;:)

Another direction was explored by many people and was probably made famous through Grothendieck's work. Consider a variety Z := P^1-(0, 1, infty)\infty), the projective plane without three points. It is really interesting as it makes sense over any field. Now suppose we consider it as a scheme over Q and ask for its fundamental group. Then one would have an exact sequence, if I'm not mistaken, 

0 -> Gal Q -> pi_1(Z/Q) -> pi_1(Z / \bar Q)

which implies, by standard reasoning, that Gal Q acts on pi_1(Z/\bar Q), which is a very straightforward group — it's simply a normal fundamental group of a sphere plane without three two points; , so it's freely generated by loop around any two pointsloops. Grothendieck called it a cartographic group and related it to dessin's d'enfantesd'enfants. Searching for these terms The standard references are very well-written in T.'s answers and there are hopefully more questions on this topic coming to MathOverflow will bring you some ;)

This is a very interesting informationdirection that we should continue to explore.

So, let's keep posting on these topicsIt's also very suitable to MathOverflow, shall we?since there are both many specific questions as well as opportunities to write reviews and to produce new results by collaboration.
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