Definition 4.1.1.
For \(K\) a local field, let \(K^{\ab}\) be the maximal abelian extension of \(K\text{.}\)
Section 4.1 Overview of local class field theory
Reference.
Subsection The local reciprocity law
The main theorem of local class field theory is the following.
Theorem 4.1.2. Local Reciprocity Law.
Let \(K\) be a local field. Then there is a unique map \(\phi_K: K^* \to \Gal(K^{\ab}/K)\) satisfying the following conditions:
- for any uniformizer \(\pi\) of \(K\) and any finite unramified extension \(L\) of \(K\text{,}\) \(\phi_K(\pi)\) acts on \(L\) as the Frobenius automorphism;
- for any finite abelian extension \(L\) of \(K\text{,}\) the group of norms \(\Norm_{L/K} L^*\) is in the kernel of \(\phi_K\text{,}\) and the induced map \(K^*/\Norm_{L/K} L^* \to \Gal(L/K)\) is an isomorphism.
Proof.
See the discussion in Section 4.3.
Definition 4.1.3.
The map \(\phi_K\) in Theorem 4.1.2 is variously called the local reciprocity map or the norm residue symbol.
Example 4.1.4.
Using the local Kronecker-Weber theorem (Theorem 1.1.5), the statement of Theorem 4.1.2 can be explicitly verified for \(K=\QQ_p\text{.}\) To wit, we have \(K^{\ab} = K_1 K_2\) where \(K_1 = \bigcup_n \QQ_p(\zeta_{p^n})\) and \(K_2 = \bigcup_n \QQ_p(\zeta_{p^n-1})\text{,}\) and \(\Gal(K^{\ab}/K) \cong \Gal(K_1/K) \times \Gal(K_2/K)\text{.}\) Since \(p\) is totally ramified in \(K_1\text{,}\) we have
\begin{equation*}
\Gal(K_1/K) \cong \Gal(\QQ(\zeta_{p^\infty})/\QQ) \cong \ZZ_p^*.
\end{equation*}
Since \(p\) is unramified in \(K_2\text{,}\) we have
\begin{equation*}
\Gal(K_2/K) \cong \Gal(\overline{\FF}_p/\FF_p) \cong \widehat{\ZZ}.
\end{equation*}
However, it will be more convenient to think of the image as sitting inside
\begin{equation*}
\Gal(\bigcup_n \QQ(\zeta_{p^n-1})/\QQ) \cong \widehat{\ZZ}^*/\ZZ_p^* \cong \prod_{q \neq p} \ZZ_q^*
\end{equation*}
(here using global Kronecker-Weber and Artin reciprocity). That is, we are looking for a map
\begin{equation*}
\phi_K: \ZZ_p^* \times p^{\ZZ} \cong \QQ_p^* \to \Gal(K_1/K) \times \Gal(K_2/K) \subset \ZZ_p^* \times \prod_{q \neq p} \ZZ_q^*;
\end{equation*}
the map we want is the identity on the first factor and the map \(p \mapsto p\) on the second factor. See Exercise 1.
The local reciprocity law is an analogue of the Artin reprocity law for number fields. We also get an analogue of the existence of ray class fields.
Theorem 4.1.5. Local existence theorem.
For every finite (not necessarily abelian) extension \(L\) of \(K\text{,}\) \(\Norm_{L/K} L^*\) is an open subgroup of \(K^*\) of finite index. Conversely, for every (open) subgroup \(U\) of \(K^*\) of finite index, there exists a finite abelian extension \(L\) of \(K\) such that \(U = \Norm_{L/K} L^*\text{.}\)
Proof.
For the first assertion, see Exercise 3 (or Exercise 4 for the case of characteristic \(p\)). For the second assertion, see Theorem 4.3.11.
Remark 4.1.6.
In Theorem 4.1.5, the topology on \(K^*\) is the one given by taking the disjoint union of the sets \(\pi^n \gotho_K^*\) for \(n \in \ZZ\text{,}\) where \(\pi \in K^\times\) is a uniformizer. In fact, it is only necessary to keep track of this topology in the function field case; for \(K\) a finite extension of \(\QQ_p\text{,}\) one can show that every subgroup of \(K^*\) of finite index is open.
Another way to identify the correct topology on \(K^*\) is to equip \(K\) with its usual topology (the norm topology defined by an extension of the \(p\)-adic absolute value) and then take the subspace topology for the inclusion of \(K^*\) into \(K \times K\) given by \(x \mapsto (x, x^{-1})\text{.}\) While this does coincide with the subspace topology for the inclusion of \(K^*\) into \(K\text{,}\) there are good reasons not to view it this way; see Exercise 6.
The local existence theorem says that if we start with a nonabelian extension \(L\text{,}\) then \(\Norm_{L/K} L^*\) is also the group of norms of an abelian extension. But which one? The following theorem gives the answer.
Theorem 4.1.7. Norm limitation theorem.
Let \(M\) be the maximal abelian subextension of \(L/K\text{.}\) Then \(\Norm_{L/K} L^* = \Norm_{M/K} M^*\text{.}\)
Proof.
See the discussion in Section 4.3.
Remark 4.1.8.
In Theorem 4.1.7, it is evident that \(\Norm_{L/K} L^* \subseteq \Norm_{M/K} M^*\) because \(\Norm_{L/K} = \Norm_{M/K} \circ \Norm_{L/M}\text{.}\) Since the group \(\Norm_{L/K} L^*\) can be shown directly to be an open subgroup of finite index (see Exercise 3), Theorem 4.1.5 implies that it has the form \(\Norm_{N/K} N^*\) for some finite abelian extension \(N\) of \(K\text{.}\) Theorem 4.1.2 then implies that \(M \subseteq N\text{.}\) The subtle point that remains to be proven is that the inclusion \(M \subseteq N\) is actually an equality.
Remark 4.1.9.
For each uniformizer \(\pi\) of \(K\text{,}\) let \(K_\pi\) be the composite of all finite abelian extensions \(L\) such that \(\pi \in \Norm_{L/K} L^*\text{.}\) Then the local reciprocity map implies that \(K^{\ab} = K_\pi \cdot K^{\unr}\text{.}\)
It turns out that \(K_\pi\) can be explicitly constructed as the extension of \(K\) by certain elements, thus giving a generalization of local Kronecker-Weber to arbitrary local fields! These elements come from Lubin-Tate formal groups, which we will not discuss further.
Note that for \(L/K\) a finite extension of local fields, the map
\begin{equation*}
K^*/\Norm_{L/K} L^* \to \Gal(L/K) = G
\end{equation*}
obtained by combining the local reciprocity law with the norm limitation theorem is in fact an isomorphism of \(G = G^{\ab} = H^{-2}_T(G, \ZZ)\) with \(K^*/\Norm_{L/K} L^* = H^0_T(G, L^*)\text{.}\) We will in fact show something stronger, from which we will deduce both the local reciprocity law and the norm limitation theorem.
Theorem 4.1.10.
For any finite Galois extension \(L/K\) of local fields with Galois group \(G\text{,}\) there is a canonical isomorphism \(H^i_T(G, \ZZ) \to H^{i+2}_T(G, L^*)\text{.}\)
Proof.
See the discussion in Section 4.3.
Remark 4.1.11.
The map in Theorem 4.1.10 can be written in terms of the cup product in group cohomology (see [36], Proposition II.1.38). We will not develop this point of view here.
Subsection The local invariant map
We will first prove the following.
Theorem 4.1.12.
For any local field \(K\text{,}\) there exist canonical isomorphisms
\begin{gather*}
H^2(\Gal(K^{\unr}/K), (K^{\unr})^*) \to H^2(\Gal(\overline{K}/K), \overline{K}^*)\\
\inv_K: H^2(\Gal(\overline{K}/K), \overline{K}^*) \to \QQ/\ZZ.
\end{gather*}
Proof.
This will follow from Proposition 4.2.1.
Definition 4.1.13.
In Theorem 4.1.12, the first map is an inflation homomorphism; the second map is called the local invariant map. More precisely, for \(L/K\) finite of degree \(n\text{,}\) we have an isomorphism
\begin{equation*}
\inv_{L/K}: H^2(\Gal(L/K), L^*) \to \frac{1}{n}\ZZ/\ZZ,
\end{equation*}
and these isomorphisms are compatible with inflation. (In particular, we don’t need to prove the first isomorphism separately. But that can be done, by considerations involving the Brauer group; see below.)
To use Theorem 4.1.12 to prove Theorem 4.1.10 and hence the local reciprocity law (Theorem 4.1.2) and the norm limitation theorem (Theorem 4.1.7), we employ the following theorem of Tate.
Theorem 4.1.14.
Let \(G\) be a finite group and \(M\) a \(G\)-module. Suppose that for each subgroup \(H\) of \(G\) (including \(H=G\)), \(H^1(H,M) = 0\) and \(H^2(H,M)\) is cyclic of order \(\#H\text{.}\) Then there exist isomorphisms \(H^i_T(G, \ZZ) \to H^{i+2}_T(G, M)\) for all \(i\text{;}\) these are canonical once you fix a choice of a generator of \(H^2(G,M)\text{.}\)
Proof.
See Theorem 4.3.1.
Definition 4.1.15.
For any field \(K\text{,}\) the group \(H^2(\Gal(\overline{K}/K), \overline{K}^*)\) is called the Brauer group of \(K\text{.}\) See Section 7.6 for further discussion.
Subsection Abstract class field theory
Having derived local class field theory once, we will do it again a slightly different way in Chapter 5. In the course of proving the above results, we will show (among other things) that if \(L/K\) is a cyclic extension of local fields,
\begin{equation*}
\#H^0_T(\Gal(L/K), L^*) = [L:K], \qquad \#H^{-1}_T(\Gal(L/K), L^*) =1.
\end{equation*}
It turns out that this alone is enough number-theoretic input to prove local class field theory! More precisely, we will identify “minimal” properties of a field \(K\) with \(G = \Gal(\overline{K}/K)\text{,}\) a surjective continuous homomorphism \(d: G \to \widehat{\ZZ}\) (defining “unramified” extensions of \(K\)), a continuous \(G\)-module \(A\) (playing the role of \(\overline{K}^*\)), and a homomorphism \(v: A^G \to \widehat{\ZZ}\) (playing the role of the valuation map) that will suffice to yield the reciprocity map. See Section 5.4 for the continuation of this discussion.
Exercises Exercises
1.
Building on Example 4.1.4, verify Theorem 4.1.2 in the case \(K = \QQ_p\text{.}\)Hint.
The first assertion of Theorem 4.1.2 follows from global Artin reciprocity (Definition 1.1.7). To check the second assertion for \(L = \QQ(\zeta_n)\text{,}\) use the fact that \(\Norm_{\QQ_p(\zeta_{p^m})/\QQ_p} (1 - \zeta_{p^m}) = p\) for any positive integer \(m\text{.}\) Alternatively, see Lemma 7.5.3.
2.
For \(K = \QQ_p\text{,}\) take \(\pi = p\) in Remark 4.1.9. Determine \(K_\pi\text{,}\) again using local Kronecker-Weber.Hint.
You should get \(K_\pi = \QQ(\zeta_{p^\infty}).\)
3.
Prove that for any finite extension \(L/K\) of finite extensions of \(\QQ_p\text{,}\) \(\Norm_{L/K} L^*\) is an open subgroup of \(K^*\text{.}\)Hint.
Show that already \(\Norm_{L/K} K^*\) is open! The corresponding statement in positive characteristic is more subtle; see Exercise 4.
4.
Prove that for any finite extension \(L/K\) of finite separable extensions of \(\FF_p((t))\text{,}\) \(\Norm_{L/K} L^*\) is an open subgroup of \(K^*\text{.}\)Hint.
Reduce to the case of a cyclic extension of prime degree. If the degree is prime to \(p\text{,}\) you may imitate Exercise 3; otherwise, that approach fails because \(\Norm_{L/K} K^*\) lands inside the subfield \(K^p\text{,}\) but you can use this to your advantage to make an explicit calculation.
5.
A quaternion algebra over a field \(K\) is a central simple algebra over \(K\) of dimension 4. If \(K\) is not of characteristic 2, any such algebra has the form
\begin{equation*}
K \oplus Ki \oplus Kj \oplus Kk, \qquad i^2 = a, j^2 = b, ij = -ji = k
\end{equation*}
for some \(a,b \in K^*\text{.}\) (For example, the case \(K = \RR\text{,}\) \(a=b=-1\) gives the standard Hamilton quaternions.) A quaternion algebra is split if it is isomorphic to the ring of \(2 \times 2\) matrices over \(K\text{.}\) Show that if \(K\) is a local field, then any two quaternion algebras which are not split are isomorphic to each other.Hint.
While this can be done using elementary methods, it will also follow from Theorem 4.1.12 via the cohomological description of Brauer groups; see Lemma 7.6.2.
