Notes on class field theory

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).

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.,

Define the polynomial

and put

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.

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

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

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.

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

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.