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 \(\phi_K\colon K^* \to \Gal(K^{\ab}/K)\) 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.
For uniqueness, see Remark 4.1.9. For existence, see the discussion in Section 4.3, particularly Remark 4.3.7 and Remark 4.3.9.
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.
Let us see more explicitly what Theorem 4.1.2 says when \(K=\QQ_p\text{.}\) By local Kronecker-Weber (Theorem 1.1.5), 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^*\text{.}
\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}\text{.}
\end{equation*}
However, it will be more convenient to think of the image as sitting inside
\begin{equation*}
\Gal\left(\bigcup_n \QQ(\zeta_{p^n-1})/\QQ\right) \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\colon \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*}
which acts as \(p \mapsto p\) on the second factor. The correct choice is to take the map \(a \mapsto a^{-1}\) on the first factor; this can be verified using the Lubin-Tate construction (Theorem 4.6.16).
From this example, we also obtain an explicit description of the local reciprocity map for any cyclotomic extension of any local field. See Proposition 4.3.11.
Subsection The local existence theorem
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.
Let \(K\) be a local field. 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 4.3.18.
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 \(L/K\) be a (not necessarily Galois) extension of local fields. 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, particularly Remark 4.3.8.
Remark 4.1.8.
In Theorem 4.1.7, we have \(\Norm_{L/K} L^* \subseteq \Norm_{M/K} M^*\) because
\begin{equation*}
\Norm_{L/K} = \Norm_{M/K} \circ \Norm_{L/M}\text{.}
\end{equation*}
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.
Let \(K\) be a local field. 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; see Exercise 2 for the case \(K = \QQ_p\) and Theorem 4.6.16 for the general case.
In any case, we see that the requirements of Theorem 4.1.2 uniquely determine the image of the map \(\phi_K\) on every uniformizer \(\pi\) of \(K\text{.}\) Since these uniformizers generate \(K^*\text{,}\) we deduce the uniqueness aspect of Theorem 4.1.2.
Remark 4.1.10.
At this point, for \(L/K\) a finite extension of local fields, we can combine the local reciprocity law with the norm limitation theorem to obtain an isomorphism
\begin{equation*}
K^*/\Norm_{L/K} L^* \to \Gal(L/K)^{\ab} = G^{\ab}\text{;}
\end{equation*}
this can be reinterpreted as an isomorphism
\begin{equation*}
H^0_T(G, L^*) \to H^{-2}_T(G, \ZZ)\text{.}
\end{equation*}
In fact we will run this backward: we will use cup products (Proposition 4.3.1) to construct an isomorphism of this form (Theorem 4.1.11) and then deduce both the local reciprocity law and the norm limitation theorem.
Theorem 4.1.11.
For any 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, particularly Remark 4.3.7.
Subsection The local invariant map
We will first prove the following.
Theorem 4.1.12.
For any local field \(K\text{,}\) there is a canonical isomorphism
\begin{equation*}
\inv_K\colon H^2(\Gal(\overline{K}/K), \overline{K}^*) \to \QQ/\ZZ\text{.}
\end{equation*}
Moreover, the inflation map
\begin{equation*}
H^2(\Gal(K^{\unr}/K), (K^{\unr})^*) \to H^2(\Gal(\overline{K}/K), \overline{K}^*)
\end{equation*}
is an isomorphism.
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}\colon H^2(\Gal(L/K), L^*) \to \frac{1}{n}\ZZ/\ZZ\text{,}
\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 Definition 4.1.15.)
To use Theorem 4.1.12 to prove Theorem 4.1.11 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 4.3.4.
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
Remark 4.1.16.
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\text{.}
\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 \(d\colon G \to \widehat{\ZZ}\) (defining “unramified” extensions of \(K\)), a continuous \(G\)-module \(A\) (playing the role of \(\overline{K}^*\)), and a homomorphism \(v\colon A^G \to \widehat{\ZZ}\) (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.
For \(K = \QQ_p\text{,}\) take \(\pi = p\) in Remark 4.1.9. Show that in this case \(K_\pi = \QQ(\zeta_{p^\infty})\text{.}\)
Hint.
The key point is that for any positive integer \(e\text{,}\) \(\Norm_{\QQ_p(\zeta_{p^e})/\QQ_p} (1 - \zeta_{p^e}) = p\text{.}\)
2.
Show that for each \(a \in \ZZ_p\text{,}\) the power series
\begin{equation*}
[a] := (1+T)^{a} - 1 = \sum_{n=1}^\infty \frac{a(a-1) \cdots (a-n+1)}{n!} T^n
\end{equation*}
has coefficients in \(\ZZ_p\text{.}\) Then show that for \(\pi \in \ZZ_p\) a uniformizer and \(e\) a positive integer, \([\pi^e] = u_e P_e\) where \(P_e\) is a monic polynomial of degree \(p^e\) and \(u_e\) is a power series in \(\ZZ_p\) with constant term \(1\text{.}\) Finally, deduce that \(P_e/P_{e-1}\) is an irreducible polynomial whose roots belong to \(K_\pi\text{.}\)
Hint.
The second assertion is a special case of the Weierstrass preparation theorem.
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 \(K^*\) of finite index.
Hint.
In fact already \(\Norm_{L/K} K^*\) is open; compare Exercise 2. 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 \(K^*\) of finite index.
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.
6.
Let \(K\) be a finite extension of \(\QQ_p\text{.}\) Show that \(K^*\) can be viewed as a closed subspace of \(K \times K\) via the inclusion \(x \mapsto (x,x^{-1})\text{,}\) and deduce from this that \(K^*\) is a locally compact abelian group for the subspace topology. It can also be viewed as a subspace of \(K\text{,}\) but not as a closed subspace; this distinction will show up more seriously when we talk about adèles and idèles (Remark 6.2.3).
