June 5, 2010
So, it turns out there’s another way to prove the second inequality, due to Chevalley in 1940. It’s purely arithmetical, where “arithmetic” is allowed to include cohomology and ideles. But the point is that no analysis is used, which was apparently seen as good for presumably the same reasons that the standard proof of the prime number theorem is occasionally shunned. I’m not going into the proof so much for the sake of number-theory triumphalism but rather because I can do it more completely, and because the ideas will resurface when we prove the existence theorem. Anyhow, the proof is somewhat involved, and I am going to split it into steps. The goal, remember, is to prove that if is a finite abelian extension of degree , then
Here is an outline of the proof:
1. Technical abstract nonsense: Reduce to the case of cyclic of a prime degree and containing the -th roots of unity
2. Explicitly construct a group and prove that
3. Compute the index . The whole proof is too long for one blog post, so I will do step 1 (as well as some preliminary index computations—yes, these are quite fun—today). (more…)
June 3, 2010
(Argh. So, the spacing isn’t working as well as I would like on the post and it reads non-ideally (sorry). So I’ve also included a PDF of the post if it makes things better. -AM)
So, we have defined this thing called the Artin map on the ideals prime to some set of primes. But we really care about the ideles. There has to be some way to relate ideals and ideles. In this post, we give a translation guide between the idealic and ideleic framework. In the good ol’ days, one apparently developed class field theory using only ideal theory, but now the language of the ideles is convenient too (and as we saw, the ideles lend themselves very nicely to computing Herbrand quotients). But they are not as good for the Artin map, unless one already has local class field theory. We don’t—we could if we developed a lot of cohomological machinery and some delightful pieces of abstract nonsense—but that’s not what we’re going to do (at least not until I manage to muster some understanding of said machinery).
1. Some subgroups of the ideles
Fix a number field . Let’s first look at the open subgroups of . For this, we determine a basis of open subgroups in when is a place. When is real, will do. When is complex, (the full thing) is the smallest it gets. When is -adic, we can use the subgroups . Motivated by this, we define the notion of a cycle : by this we mean a formal product of an ideal and real places induced by real embeddings . Say that an idele is congruent to 1 modulo if for all primes and for . We have subgroups consisting of ideles congruent to 1 modulo . Note that in view of the approximation theorem. We define the subgroup consisting of ideles that are congruent to 1 modulo and units everywhere. Fix a finite Galois extension . If is large enough (e.g. contains the ramified primes and to a high enough power), then consists of norms—this is because any unit is a local norm, and any idele in is very close to 1 (or positive) at the ramified primes. These in fact form a basis of open subgroups of . (more…)
May 22, 2010
As usual, let be a global field. Now we do the same thing that we did last time, but for the 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 . (more…)