Class formation


In mathematics, a class formation is a topological group acting on a module satisfying certain conditions. Class formations were introduced by Emil Artin and John Tate to organize the various Galois groups and modules that appear in class field theory.

Definitions

A formation is a topological group G together with a topological G-module A on which G acts continuously.
A layer E/F of a formation is a pair of open subgroups E, F of G such that F is a finite index subgroup of E. It is called a normal layer if
F is a normal subgroup of E, and a cyclic layer if in addition the quotient group is cyclic.
If E is a subgroup of G, then AE is defined to be the elements of A fixed by E.
We write
for the Tate cohomology group
Hn whenever E/F is a normal layer.
In applications, G is often the absolute Galois group of a field, and in particular is profinite, and the open subgroups therefore correspond to the finite extensions of the field contained in some fixed separable closure.
A class formation is a formation
such that for every normal layer E/F
In practice, these cyclic groups come provided with canonical generators uE/F ∈ H2,
called fundamental classes, that are compatible with each other in the sense that
the restriction of a fundamental class is another fundamental class.
Often the fundamental classes are considered to be part of the structure of a class formation.
A formation that satisfies just the condition H1=1 is sometimes called a field formation.
For example, if G is any finite group acting on a field L and A=L×, then this is a field formation by Hilbert's theorem 90.

Examples

The most important examples of class formations are as follows:
It is easy to verify the class formation property for the finite field case and the archimedean local field case, but the remaining cases are more difficult. Most of the hard work of class field theory consists of proving that these are indeed class formations. This is done in several steps, as described in the sections below.

The first inequality

The first inequality of class field theory states that
for cyclic layers E/F.
It is usually proved using properties of the Herbrand quotient, in the more precise form
It is fairly straightforward to prove, because the Herbrand quotient is easy to work out, as it is multiplicative on short exact sequences, and is 1 for finite modules.
Before about 1950, the first inequality was known as the second inequality, and vice versa.

The second inequality

The second inequality of class field theory states that
for all normal layers E/F.
For local fields, this inequality follows easily from Hilbert's theorem 90 together with the first inequality and some basic properties of group cohomology.
The second inequality was first proved for global fields by Weber using properties of the L series of number fields, as follows. Suppose that the layer E/F corresponds to an extension k⊂K of global fields. By studying the Dedekind zeta function of K one shows that the degree 1 primes of K have Dirichlet density given by the order of the pole at s=1, which is 1 As each prime in k that is a norm is the product of deg= |E/F| distinct degree 1 primes of K, this shows that the set of primes of k that are norms has density 1/|E/F|. On the other hand, by studying Dirichlet L-series of characters of the group H0, one shows that the Dirichlet density of primes of k representing the trivial element of this group has density
1/|H0|.
But a prime represents a trivial element of the group H0 if it is equal to a norm modulo principal ideals, so this set is at least as dense as the set of primes that are norms. So
which is the second inequality.
In 1940 Chevalley found a purely algebraic proof of the second inequality, but it is longer and harder than Weber's original proof. Before about 1950, the second inequality was known as the first inequality; the name was changed because Chevalley's algebraic proof of it uses the first inequality.
Takagi defined a class field to be one where equality holds in the second inequality. By the Artin isomorphism below, H0 is isomorphic to the abelianization of E/F, so equality in the second inequality holds exactly for
abelian extensions, and class fields are the same as abelian extensions.
The first and second inequalities can be combined as follows. For cyclic layers, the two inequalities together prove that
so
and
Now a basic theorem about cohomology groups shows that since H1 = 1 for all cyclic layers, we have
for all normal layers.
This proof that H1 is always trivial is rather roundabout; no "direct" proof of it for global fields is known.
For cyclic group, H0 is the same as H2, so H2 = |E/F| for all cyclic layers.
Another theorem of group cohomology shows that since H1 = 1
for all normal layers and H2 ≤ |E/F| for all cyclic layers, we have
for all normal layers.

The Brauer group

The Brauer groups H2 of a class formation are defined to be the direct limit of the groups H2 as F runs over all open subgroups of E. An easy consequence of the vanishing of H1 for all layers is that the groups H2 are all subgroups of the Brauer group. In local class field theory the Brauer groups are the same as Brauer groups of fields, but in global class field theory the Brauer group of the formation is not the Brauer group of the corresponding global field.
The next step is to prove that H2 is cyclic of order exactly |E/F|; the previous section shows that it has at most this order, so it is sufficient to find some element of order |E/F| in H2.
The proof for arbitrary extensions uses a homomorphism from the group G onto the profinite completion of the integers with kernel G∞, or in other words a compatible sequence of homomorphisms of G onto the cyclic groups of order n for all n, with kernels Gn. These homomorphisms are constructed using cyclic cyclotomic extensions of fields; for finite fields they are given by the algebraic closure, for non-archimedean local fields they are given by the maximal unramified extensions, and for global fields they are slightly more complicated. As these extensions are given explicitly one can check that they have the property that H2 is cyclic of order n, with a canonical generator. It follows from this that for any layer E, the group H2 is canonically isomorphic to Q/Z. This idea of using roots of unity was introduced by Chebotarev in his proof of Chebotarev's density theorem, and used shortly afterwards by Artin to prove his reciprocity theorem.
For general layers E,F there is an exact sequence
The last two groups in this sequence can both be identified with Q/Z and the map between them is then multiplication by |E/F|. So the first group is canonically isomorphic to Z/nZ. As H2 has order at most Z/nZ is must be equal to Z/nZ ).
This shows that the second cohomology group H2 of any layer is cyclic of order |E/F|, which completes the verification of the axioms of a class formation.
With a little more care in the proofs, we get a canonical generator of H2, called the fundamental class.
It follows from this that the Brauer group H2 is isomorphic to the group Q/Z, except in the case of the archimedean local fields R and C when it has order 2 or 1.

Tate's theorem and the Artin map

in group cohomology is as follows. Suppose that A is a module over a finite group G and a is an element of H2, such that for every subgroup E of G
Then cup product with a is an isomorphism
If we apply the case n=−2 of Tate's theorem to a class formation, we find that there is an isomorphism
for any normal layer E/F. The group H−2 is just the abelianization of E/F, and the group H0 is AE modulo the group of norms of AF. In other words, we have an explicit description of the abelianization of the Galois group E/F in terms of AE.
Taking the inverse of this isomorphism gives a homomorphism
and taking the limit over all open subgroups F gives a homomorphism
called the Artin map. The Artin map is not necessarily surjective, but has dense image. By the existence theorem below its kernel is the connected component of AE, which is trivial for class field theory of non-archimedean local fields and for function fields, but is non-trivial for archimedean local fields and number fields.

The Takagi existence theorem

The main remaining theorem of class field theory is the Takagi existence theorem, which states that every
finite index closed subgroup of the idele class group is the group of norms corresponding to some abelian extension.
The classical way to prove this is to construct some extensions with small groups of norms, by first adding in lots of roots of unity, and then taking Kummer extensions and Artin–Schreier extensions. These extensions may be non-abelian ; however, this does not really matter, as the norm group of a non-abelian Galois extension is the same as that of its maximal abelian extension. This gives enough extensions to show that there is an abelian extension corresponding to any finite index subgroup of the idele class group.
A consequence is that the kernel of the Artin map is the connected component of the identity of the idele class group, so that the abelianization of the Galois group of F is the profinite completion of the idele class group.
For local class field theory, it is also possible to construct abelian extensions more explicitly using Lubin–Tate formal group laws. For global fields, the abelian extensions can be constructed explicitly in some cases: for example, the abelian extensions of the rationals can be constructed using roots of unity, and the abelian extensions of quadratic imaginary fields can be constructed using elliptic functions, but finding an analog of this for arbitrary global fields is an unsolved problem.

Weil group

The Weil group of a class formation with fundamental classes uE/F ∈ H2 is a kind of modified Galois group, introduced by and used in various formulations of class field theory, and in particular in the Langlands program.
If E/F is a normal layer, then the Weil group U of E/F is the extension
corresponding to the fundamental class uE/F in H2. The Weil group of the whole formation is defined to be the inverse limit of the Weil groups of all the layers
G/F, for F an open subgroup of G.
The reciprocity map of the class formation induces an isomorphism from AG to the abelianization of the Weil group.