Functoriality of the Artin Map

Instead of restating the functoriality of the Artin map, let’s just review the statement through an example. We’ll re-use our example from last time. Let {L} be the splitting field of {x^3-x-1} over {\mathbb{Q}}. We get a non-abelian Galois group {H\simeq S_3} (to keep notation the same, we called this {H} last time). Take the quadratic subextension {K=\mathbb{Q}(\sqrt{-23})}. We have an abelian Galois group {G\simeq \mathbb{Z}/3}. We need the abelianization {H^{ab}\simeq \mathbb{Z}/2}.

By Galois theory we know {H^{ab}} gives us a field extension {L'} sitting between {L} and {\mathbb{Q}}. Class field theory tells us that the conductor {\frak{m}_{L'}=\{(23), w_1\}} because we must pick up all ramification from {\mathbb{Q}(\sqrt{-23})/\mathbb{Q}}. The general argument we gave a few posts ago shows us that {Cl_{\frak{m}_{L'}}(\mathbb{Q})\simeq (\mathbb{Z}/23)^*}. Now {K} is the Hilbert class field, so {Cl(\mathcal{O}_K)\stackrel{\sim}{\rightarrow} G} via the Artin map. We take {\frak{m}=(\sqrt{-23})\mathcal{O}_K} with no embeddings specified.

This gives us the diagram:

\displaystyle \begin{matrix} Cl_\frak{m}(K) & \twoheadrightarrow & G & \stackrel{\Psi_{L/K}}{\rightarrow} & \mathbb{Z}/3 & \rightarrow & 1 \\ id \downarrow & & & & \downarrow & & \\ Cl_\frak{m}(K) & \stackrel{N_{K/F}}{\rightarrow} & (\mathbb{Z}/23)^* & \stackrel{\Psi_{L'/F}}{\rightarrow} & \mathbb{Z}/2 & \rightarrow & 1 \end{matrix}

First, the right vertical arrow is clearly the zero map. The other important part of the diagram is that the norm map is taking a fractional ideal (class) that is relatively prime to {(\sqrt{-23})\mathcal{O}_K} and taking the norm of it which lands you in the units {(\mathbb{Z}/23)^*\simeq \mathbb{Z}/22}. Moreover, the map {\mathbb{Z}/22\rightarrow \mathbb{Z}/2} is the unique surjective one and the image of the norm map must land in the kernel of this by exactness. Interestingly, this tells us that the positive generator of {N_{K/F}(\frak{b})} for any {\frak{b}} prime to {(\sqrt{-23})\mathcal{O}_K} is a square mod 23.

Let’s wrap up today by stating another functoriality result. Given the same setup of {L/K/F} where {G=Gal(L/K)} is abelian and {H=Gal(L/F)} is finite possibly non-abelian. Suppose now that {G\triangleleft H} and {K/F} Galois with {T=Gal(K/F)}. Now {T} acts on {G} as follows. Let {t\in T}. Choose a lift {h_t\in H}. The action is given by {t\cdot g=h_t g h_t^{-1}}. Call this action {\sigma_t}.

We can transfer this Galois action to the ray class group as follows:

\displaystyle \begin{matrix} Cl_\frak{m}(K) & \stackrel{\Psi}{\rightarrow} & G & \rightarrow & 1 \\ \downarrow & & \downarrow \sigma_t & & \\ Cl_\frak{m}(K) & \stackrel{\Psi}{\rightarrow} & G & \rightarrow & 1 \end{matrix}

where the vertical arrow is just the natural map on ideals. Commutativity of the diagram just comes from the standard fact that {hFrob_p(Q)h^{-1}=Frob_p(hQ)}. There is another functoriality we could do, but it doesn’t seem worth it at this point because it is overly complicated and there isn’t a plan to use it anytime soon.


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