A layer E/F of a formation is a pair of open subgroups E, F such that F is a 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(E/F, AF) whenever E/F is a normal layer. In applications, G is usually 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
- H1(E/F) is trivial, and
- H2(E/F) is cyclic of order |E/F|.
In practice, these cyclic groups come provided with canonical generators uE/F ∈ H2(E/F), called fundamental classes, that are compatible with each other in the sense that the restriction (of cohomology classes) 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(E/F)=1 is sometimes called a field formation. For example, if G is any finite group acting on a field A, then this is a field formation by Hilbert's theorem 90.
Examples of class formations
The most important examples of class formations (arranged roughly in order of difficulty) are as follows:
- Archimedean local class field theory: The module A is the group of non-zero complex numbers, and G is either trivial or is the cyclic group of order 2 generated by complex conjugation.
- Finite fields: The module A is the integers (with trivial G-action), and G is the absolute Galois group of a finite field, which is isomorphic to the profinite completion of the integers.
- Local class field theory of characteristic p>0: The module A is the separable algebraic closure of the field of formal Laurent series over a finite field, and G is the Galois group.
- Non-archimedean local class field theory of characteristic 0: The module A is the algebraic closure of a field of p-adic numbers, and G is the Galois group.
- Global class field theory of characteristic p>0: The module A is the union of the groups of idele classes of separable finite extensions of some function field over a finite field, and G is the Galois group.
- Global class field theory of characteristic 0: The module A is the union of the groups of idele classes of algebraic number fields, and G is the Galois group of the rational numbers (or some algebraic number field) acting on A.
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
- |H0(E/F)| ≥ |E/F|
for cyclic layers E/F. It is usually proved using properties of the Herbrand quotient, in the more precise form
- |H0(E/F)| = |E/F|×|H1(E/F)|.
It is fairly straighforward 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. What is now the 'second' was once the 'first' (see for example p. 49 in this treatment (PDF); this bounds the index of the norms in a class group, in old-fashioned language, and is the part of the main proof that was initially treated by means of L-functions. The historical reason behind this is that the first inequality of genus theory (concerned with 2-torsion in the class groups of quadratic fields) was an upper bound for the number of genera. (discussed at introduction to the Hilbert Zahlbericht (PDF).
The second inequality
The second inequality of class field theory states that
- |H0(E/F)| ≤ |E/F|
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 (When K is the rationals, this is essentially Euler's proof that there are infinitely many primes using the pole at s=1 of the Riemann zeta function.) As each prime in k that is a norm is the product of deg(K/k)= |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(E/F), one shows that the Dirichlet density of primes of k representing the trivial element of this group has density 1/|H0(E/F)|. (This part of the proof is a generalization of Dirichlet's proof that there are infinitely many primes in arithmetic progressions.) But a prime represents a trivial element of the group H0(E/F) 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
- 1/|H0(E/F)| ≥ 1/|E/F|
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(E/F) 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
- H1(E/F)|E/F| = H0(E/F) ≤ |E/F|
- H0(E/F) = |E/F|
- H1(E/F) = 1.
Now a basic theorem about cohomology groups shows that since H1(E/F) = 1 for all cyclic layers, we have
- H1(E/F) = 1
for all normal layers (so in particular the formation is a field formation). This proof that H1(E/F) is always trivial is rather roundabout; no "direct" proof of it (whatever this means) for global fields is known. (For local fields the vanishing of H1(E/F) is just Hilbert's theorem 90.)
For cyclic group, H0 is the same as H2, so H2(E/F) = |E/F| for all cyclic layers. Another theorem of group cohomology shows that since H1(E/F) = 1 for all normal layers and H2(E/F) ≤ |E/F| for all cyclic layers, we have
- H2(E/F)≤ |E/F|
for all normal layers. (In fact, equality holds for all normal layers, but this takes more work; see the next section.)
The Brauer group
The Brauer groups H2(E/*) of a class formation are defined to be the direct limit of the groups H2(E/F) 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(E/F) 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 (though they are related).
The next step is to prove that H2(E/F) 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(E/F).
For cyclic extensions this is already known. The proof for arbitrary extensions uses a homomorphism from the group G onto the profinite completion of the integers, or in other words a compatible sequence of homomorphisms of G onto the cyclic groups of order n for all n. These homomorphisms are constructed using cyclic cyclotomic extensions. This idea was first used by Chebotarev in his proof of Chebotarev's density theorem, and used shortly afterwards by Artin to prove his reciprocity theorem.
The proof of the existence of an element of order |E/F| for an arbitrary layer proceeds by first constructing a suitable auxiliary cyclic extension of degree |E/F| as above; as this is cyclic, there is an element of order |E/F| in its second cohomology, and this element turns out to be essentially an element of H2(E/F).
This shows that the second cohomology group H2(E/F) 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(E/F), called the fundamental class.
It follows from this that the Brauer group H2(E/*) is (canonically) 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
Tate's theorem in group cohomology is as follows. Suppose that A is a module over a finite group G and a is an element of H2(G,A), such that for every subgroup E of G
- H1(E,A) is trivial, and
- H2(E,A) is generated by Res(a) which has order E.
Then cup product with a is an isomorphism
- Hn(G,Z) → Hn+2(G,A).
If we apply the case n=−2 of Tate's theorem to a class formation, we find that there is an isomorphism
- H−2(E/F,Z) → H0(E/F,AF)
for any normal layer E/F. The group H−2(E/F,Z) is just the abelianization of E/F, and the group H0(E/F,AF) 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
- AE → abelianization of E/F,
and taking the limit over all open subgroups F gives a homomophism
- AE → abelianization of E,
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 (for class field theory), 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. These extensions may be non-abelian (though they are extensions of abelian groups by abelian groups); 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 can be shown using what we already know about class fields). This gives enough (abelian) 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 group H0(F, AF) is exactly the idele class group modulo the connected component of the identity, or equivalently the profinite completion of the idele class group. By the Artin isomorphism, this is the abelianization of the Galois group of F.
In the case of characteristic p>0, we need to use Artin-Schreier extensions as well as Kummer extensions.
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 many cases, but a general method for constructing all abelian extensions directly (without first constructing a larger metabelian extension) is not known.
- This is not a Weyl group and has no connection with the Weil-Châtelet group or the Mordell-Weil group
The Weil group of a class formation with fundamental classes uE/F ∈ H2(E/F, AF) is a kind of modified Galois group, introduced by Weil (1951) 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
- 1 → AF → U → E/F → 1
corresponding to the fundamental class uE/F in H2(E/F, AF). 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 (G, A) induces an isomorphism from AG to the abelianization of the Weil group.
- Artin, Emil; Tate, John (2009) , Class field theory, AMS Chelsea Publishing, Providence, RI, ISBN 978-0-8218-4426-7, MR0223335, http://books.google.com/books?isbn=978-0-8218-4426-7
- Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics, 67, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90424-5, MR554237 , esp. chapter XI: Class formations
- Tate, J. (1979), "Number theoretic background", Automorphic forms, representations, and L-functions Part 2,, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: Amer. Math. Soc., pp. 3–26, ISBN 0-8218-1435-4, http://www.ams.org/online_bks/pspum332/
- Weil, André (1951), "Sur la theorie du corps de classes (On class field theory)", Journal of the Mathematical Society of Japan 3: 1–35, ISSN 0025-5645 , reprinted in volume I of his collected papers, ISBN 0-387-90330-5
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