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Notes on class field theory

Kiran S. Kedlaya

Section 7.5 Local-global compatibility

Reference.

[37] VI.5.
Let \(L/K\) be a Galois extension of number fields. So far, we’ve used abstract class field theory to construct reciprocity isomorphisms and to establish the adelic form of the existence theorem (Theorem 7.4.8).
It remains to verify that the “abstract” reciprocity map coincides with the product of the local reciprocity maps (Definition 6.4.4). As noted earlier, this is enough to recover the classical Artin reciprocity law (Proposition 6.4.7); this will finally complete the proof of all of the statements originally asserted in Chapter 2).

Subsection Compatibility for cyclotomic extensions

Definition 7.5.1.

Let \(L/K\) be a Galois extension of number fields. Let
\begin{equation*} r_{L/K}: I_K \to \Gal(L/K)^{\ab} \end{equation*}
be the product of the local reciprocity maps Definition 6.4.4. Meanwhile, let
\begin{equation*} r'_{L/K}: I_K \to \Gal(L/K)^{\ab} \end{equation*}
be the map obtained by inverting the isomorphism \(\Gal(L/K)^{\ab} \to C_K/\Norm_{L/K} C_L\) given by Theorem 7.3.8.
As a base case for our work, we need to know that \(r_{L/K} = r'_{L/K}\) when \(L\) is contained in a small cyclotomic extension. Note that this is very similar to the proof that the map \(v\) is an abstract henselian valuation (Lemma 7.3.6).

Proof.

In the setting of abstract class field theory, \(L/K\) is viewed as an “unramified” extension. Consequently, the reciprocity map \(r'_{L/K}: C_K/\Norm_{L/K} C_L \to \Gal(L/K)\) is described completely by Lemma 5.3.1: it is given by composing the valuation map \(v_K: C_K \to \widehat{\ZZ}\) with the inverse of the map \(d_K: \Gal(K^{\smcy}/K) \cong \widehat{\ZZ}\text{,}\) then projecting from \(\Gal(K^{\smcy}/K)\) to \(\Gal(L/K)\text{.}\) (Note that as per Remark 7.3.11, this does not depend on the artificial choice of the isomorphism \(d_K\text{,}\) because \(v_K\) is defined using the same choice.) Consequently, in this case we end up with the usual Artin map for a cyclotomic extension (Definition 1.1.7), which is compatible with local reciprocity by direct calculation (Example 4.1.4).
For the purposes of illustration, we sketch an alternate approach to that calculation in terms of Lubin-Tate formal groups. This approach has the benefit that it does not depend on global reciprocity, and so can be adapted more easily to extensions which are not cyclotomic.

Proof.

This is straightforward for \(\ell = \infty\text{.}\) For \(\ell \neq \infty, p\text{,}\) we have an unramified extension of local fields, where we know the local reciprocity map takes a uniformizer to a Frobenius. In this case the latter is simply \(\ell\text{.}\)
The hard work is in the case \(\ell=p\text{.}\) For that computation one uses what amounts to a very special case of the Lubin-Tate construction of explicit class field theory for local fields, using formal groups. Put \(K = \QQ_p\text{,}\) \(\zeta = \zeta_{p^m}\) and \(L = \QQ_p(\zeta)\text{.}\)
Suppose without loss of generality that \(u\) is a positive integer, and let \(\sigma \in \Gal(L/K)\) be the automorphism corresponding to \(u^{-1}\text{.}\) Since \(L/K\) is totally ramified at \(p\text{,}\) we have \(\Gal(L/K) \cong \Gal(L^{\unr}/K^{\unr})\text{,}\) and we can view \(\sigma\) as an element of \(\Gal(L^{\unr}/K)\text{.}\) Let \(\phi_L \in \Gal(L^{\unr}/L)\) denote the Frobenius, and put \(\tau = \sigma \phi_L\text{.}\) Then \(\tau\) restricts to the Frobenius in \(\Gal(K^{\unr}/K)\) and to \(\sigma\) in \(\Gal(L/K)\text{.}\) As per Definition 5.2.1, we may compute \(r^{-1}_{L/K}(\sigma)\) as \(\Norm_{M/K} \pi_M\text{,}\) where \(M\) is the fixed field of \(\tau\) and \(\pi_M\) is a uniformizer. We want that norm to be \(u\) times a norm from \(L\) to \(K\text{,}\) i.e.,
\begin{equation*} r^{-1}_{L/K}(\sigma) \in u \Norm_{L/K} L^*\text{.} \end{equation*}
Define the polynomial
\begin{equation*} e(x) = x^p + upx \end{equation*}
and put
\begin{equation*} P(x) = e^{(n-1)}(x)^{p-1} + pu\text{,} \end{equation*}
where \(e^{(k+1)}(x) = e(e^{(k)}(u))\text{.}\) Then \(P(x)\) satisfies Eisenstein’s criterion, so its splitting field over \(\QQ_p\) is totally ramified, any root of \(P\) is a uniformizer, and the norm of said uniformizer is \((-1)^{[L:K]} pu \in \Norm_{L/K} L^*\text{,}\) since \(\Norm_{L/K}(\zeta-1) = (-1)^{[L:K]}(p)\text{.}\)
The punch line is that the splitting field of \(P(x)\) is precisely \(M\text{!}\) Here is where the Lubin-Tate construction comes to the rescue... and where I will stop this sketch. See [37] V.2, V.4 and/or [36] I.3.

Subsection Compatibility for general extensions

Proof.

In the setting of abstract class field theory, \(L/K\) is viewed as a “totally ramified” extension. Consequently, we may set notation as in the proof of Lemma 5.3.2, then apply Proposition 5.2.7 to obtain a commutative diagram
Figure 7.5.5.
in which the horizontal arrows are isomorphisms (Theorem 7.3.8) and the right vertical arrow is an isomorphism, as then is the left vertical arrow. We also have a corresponding diagram on the local side:
Figure 7.5.6.
This means that we can reduce checking the compatibility for \(L/K\) to the corresponding statement for the “unramified” extension \(N/M\text{,}\) to which Lemma 7.5.2 applies.
At last, we obtain the desired compatibility of reciprocity maps, and with it the completion of the proofs from global class field theory. Hooray! (See Remark 7.6.18 for another approach.)

Proof.

By the norm limitation theorem (Theorem 7.3.10), we may assume that \(L/K\) is abelian. By Proposition 5.2.7, we may check the comparison of maps after replacing \(L\) with a larger abelian extension of \(K\text{.}\)
We may split the exact sequence
\begin{equation*} 1 \to \Gal(K^{\ab}/K^{\smcy}) \to \Gal(K^{\ab}/K) \to \Gal(K^{\smcy}/K) \cong \widehat{\ZZ} \to 1 \end{equation*}
by choosing an element of \(\Gal(K^{\ab}/K)\) lifting the generator \(1 \in \widehat{\ZZ}\text{.}\) Using this, we can split \(K^{\ab}\) as the compositum of \(K^{\smcy}\) and an abelian extension which is linearly disjoint from \(K^{\smcy}\text{.}\) Using the previous paragraph, we can split some finite extension of \(L\) as the compositum of linearly disjoint cyclic extensions, one contained in \(K^{\smcy}\) and the others linearly disjoint from \(K^{\smcy}\text{.}\) Applying Lemma 7.5.2 to the first extension and Lemma 7.5.4 to the others, we deduce the desired compatibility for abelian extensions.

Remark 7.5.8.

It’s worth repeating that only now do we know that the product \(r_{L/K}\) of the local reciprocity maps kills principal idèles (Proposition 6.4.5). That fact, which relates local behavior for different primes in a highly global fashion, is the basis of various higher reciprocity laws. See [36], Chapter VIII for details.

Subsection Globalization of local abelian extensions

As a complement to Proposition 7.5.7, we show that every local abelian extension is the completion of a global abelian extension. Over \(\QQ\text{,}\) this holds because the local Kronecker-Weber theorem (Theorem 1.3.4) and the Kronecker-Weber theorem (Theorem 1.1.2) are expressed in terms of the same family of extensions of \(\QQ\text{,}\) namely the cyclotomic extensions; however, in the general case we must take a less explicit approach.

Proof.

We can quickly dispatch the cases where \(v\) is infinite: if \(v\) is complex there is nothing to prove, and if \(v\) is real then we may take \(L = K(\sqrt{-1})\text{.}\) So assume hereafter that \(v\) is finite.
By the existence theorem (Theorem 7.4.8) plus local-to-global compatibility (Proposition 7.5.7), it suffices to produce an open subgroup \(V\) of \(C_K\) of finite index such that the preimage of \(V\) under \(K_v^* \to C_K\) is contained in \(N = \Norm_{M/K_v} M^*\text{.}\) Let \(S\) be the set of infinite places and let \(T = S \cup \{v\}\text{.}\) By Corollary 6.2.11, \(\gotho_{K,T}^*\) is a finitely generated abelian group and \(G = \gotho_{K,T}^* \cap N\) is a subgroup of \(\gotho_{K,T}^*\) of finite index.
Pick a finite place \(u \notin T\text{.}\) The image of \(\gotho_{K,T}^*\) in \(K_u^*\) is a finitely generated subgroup of \(\gotho_{K_u}^*\text{.}\) Hence we can choose a sufficiently small neighborhood \(U\) of the identity in \(\gotho_{K_u}^*\) so as to ensure that \(U \cap \gotho_{K,T}^* \subseteq G\text{.}\)
Now put
\begin{equation*} W = N \times U \times \prod_{w \in S} K_w^* \times \prod_{w \notin S \cup \{u,v\}} \gotho_K^*, \qquad V = K^* W/K^*. \end{equation*}
If \(\alpha_v \in K_v^*\) maps into \(U\text{,}\) then there exists \(\beta \in K^*\) such that \(\alpha_v \beta \in W\text{.}\) On one hand, this implies that \(\alpha_v \beta_v \in N\text{.}\) On the other hand, it implies that \(\beta \in \gotho_{K,T}^*\) and \(\beta_u \in U\text{,}\) so \(\beta \in G\) and so \(\beta_v \in N\text{.}\) Thus \(\alpha_v \in N\text{,}\) as desired.

Exercises Exercises

1.

Prove that Theorem 7.5.9 can be formally promoted to the conclusion that \(L_w= M\text{.}\)
Hint.
Since \(L/K\) is abelian, the kernel of the map \(\Gal(L_w/K_v) \to \Gal(M/K_v)\) is normal in \(\Gal(L/K)\text{;}\) take its fixed field.
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