Class field theory: an overview of the approach

Class field theory is about the abelian extensions of a number field . Actually, this is strictly speaking global class field theory (there is an analog for abelian extensions of local fields), and there is a similar theory for function fields of transcendence degree 1 over finite fields, but we shall not deal with it.

Let us, however, consider the situation for local fields—which we will later investigate more—as follows. Suppose is a local field and an unramified extension. Then the Galois group is isomorphic to the Galois group of the residue field extension, i.e. is cyclic of order and generated by the Frobenius. But I claim that the group is the same. Indeed, by a basic theorem about local fields that we will prove using abstract nonsense later (but can also be easily proved using successive approximation and facts about finite fields). So is cyclic, generated by a uniformizer of , which has order in this group. Thus we get an isomorphism

sending a uniformizer to the Frobenius.

 

Ideles, idele classes, and applications

As usual, let be a global field. Now we do the same thing that we did last time, but for the ideles.

1. Ideles

First of all, we have to define the ideles. These are only a group, and are defined as the restricted direct product

relative to the unit subgroups of -units (which are defined to be if is archimedean). In other words, an idele is required to satisfy for almost all .

If is a finite set of places containing the archimedean ones, we can define the subset ; this has the product topology and is an open subgroup of . These are called the -ideles. As we will see, they form an extremely useful filtration on the whole idele group.

Dangerous bend: Note incidentally that while the ideles are a subset of the adeles, the induced topology on is not the -topology. For instance, take . Consider the sequence of ideles where is at (where is the -th prime) and 1 everywhere else. Then but not in .

However, we still do have a canonical “diagonal” embedding , since any nonzero element of is a unit almost everywhere. This is analogous to the embedding .

 

The adele ring

Adele – c’est un nom si belle. (Oops, that’s bad French, isn’t it?)

I actually will not be able to finish the proof of the unit theorem here, because I don’t get to the ideles in this post. That will come next time (there are some of the same themes as here).

Let be a global field, i.e. a finite extension of either or . Then we can consider the set absolute values on . In the number field case, these are extensions (up to a power) of the archimedean absolute value on or the -adic absolute values by a theorem of Ostrowski classifying absolute values on . In the function field case, we need another result.

Here’s how we define the adele ring. It is the restricted direct product

where restricted means that any vector is required to satisfy for almost all . This becomes a topological ring if we take a basis of the form

where are open and is the ring of integers, and is a finite set containing the archimedean places. It is clear that addition and multiplication are continuous, and that is locally compact. For finite and containing the archimedean absolute values , there is a subring , and is the union of these subrings.

Since any is contained in for almost all (this is analogous to a rational function on a curve having only finitely many poles), there is an injective homomorphism .

Next, we may define a Haar measure on by taking the product of the Haar measures on , normalized such that for . Thus one gets a (i.e., the) Haar measure on itself.

 

The Dirichlet unit theorem

There is another major result in algebraic number theory that we need to get to!  I have this no longer secret goal of getting to class field theory, and if it happens, this will be a key result.  The hard part of the actual proof (namely, the determination of the rank of a certain lattice) will be deferred until next time; it’s possible to do it with the tools we already have, but it is cleaner (I think) to do it once ideles have been introduced.

Following the philosophy of examples first, let us motivate things with an example. Consider the ring of Gaussian; as is well-known, this is the ring of integers in the quadratic field . To see this, suppose is integral; then so is , and thus are integers. Also the fact must be an integer now means that neither can be of the form for odd.

What are the units in ? If is a unit, so is , so the norm must be a unit in (and hence in ). So if , then and or . So, the units are just the roots of unity.

In general, however, the situation is more complicated. Consider , which is again integrally closed. Then is a unit if and only if its norm to , i.e. is equal to . Indeed, the norm of a unit is still a unit, and since is integrally closed, we find that is a -unit. In particular, the units correspond to the solutions to the Pell equation. There are infinitely many of them.

But the situation is not hopeless. We will show that in any number field, the unit group is a direct sum of copies of and the roots of unity. We will also determine the rank.

 

The Riemann-Roch problem in number fields

This post won’t be as cool as the title sounds. But I will prove something neat, and it will lead to neater things as time goes on (assuming I keep posting on this topic).

We will now return to algebraic number theory, following Lang’s textbook, and study the distribution of points in parallelotopes.

The setup is as follows. will be a number field, and an absolute value (which, by abuse of terminology, we will use interchangeable with “valuation” and “place”) extending one of the absolute values on (which are always normalized in the standard way); we will write for the output at .

Suppose is a valuation of ; we write if extends . Recall the following important formula from the theory of absolute values an extension fields:

Write ; these are the local degrees. From elementary algebraic number theory, we have This is essentially a version of the formula.

The above formalism allows us to deduce an important global relation between the absolute values for .

Theorem 1 (Product formula) If ,

 

The proof of this theorem starts with the case , in which case it is an immediate consequence of unique factorization. For instance, one can argue as follows.