Local Class Field Theory


Today will probably be our last class field theory post. I want to end with a brief description of local class field theory. Let {K} be a global field, and {v\in M_K} a place. We have our standard inclusion {i_v: K_v^*\hookrightarrow \mathcal{J}_K} by putting the element in the {v} component and {1}‘s everywhere else. Suppose {L/K} is abelian. We have the Artin map {\psi_{L/K}: C_K/N_{L/K}(C_L)\stackrel{\sim}{\rightarrow} G=Gal(L/K)}.

What we learned two posts ago is that the image upon composing the maps {K_v\rightarrow \mathcal{J}_K\twoheadrightarrow C_K/N_{L/K}(C_L)\rightarrow G} is exactly the decomposition group {G_w} where {w} lies over {v}. The image of the units {\mathcal{O}_v^*} under the map is the inertia group {I_w}. This gives us two exact sequences that fit together:

\displaystyle \begin{matrix} 1 & \rightarrow & N_{L_w/K_v}(L_w^*) & \rightarrow & K_v^* & \rightarrow & G_w & \rightarrow & 1 \\ & & \cup & & \cup & & & & \\ 1 & \rightarrow & N_{L_w/K_v}(\mathcal{O}_w^*) & \rightarrow & \mathcal{O}_v^* & \rightarrow & I_w & \rightarrow & 1\end{matrix}

This gives us a local Artin map {\psi_{w/v}: K_v^*\rightarrow G_w}. The main theorem of local class field theory is that the map is surjective with kernel {N_{L_w/K_v}(L_w^*)} and moreover the map can be defined independently of localizing the global fields. Just like global class field theory there is an “existence” part of the theorem as well.
This part says that every finite abelian extension of local fields arises as the localization of an extension of global fields and the local Artin maps {\psi_{w/v}} give a bijection between

\displaystyle \left\{ \text{finite index open subgroups in} \ K_v^*\right\} \leftrightarrow \left\{\text{finite abelian ext} \ L_w/K_v \right\}

where the correspondence is {U \leftrightarrow \ker \psi_{w/v}}.
Let’s do the simplest example. Let’s think about the quadratic extensions {\mathbb{Q}_p} by looking at the bijection. The standard construction is that {\mathbb{Q}_p(\sqrt{d})} are the quadratic extensions where {d} is not a square, and {\mathbb{Q}_p(\sqrt{d'})} is the same extension if {d/d'} is a square. Thus there is a nice bijection between the quadratic extensions and the non-trivial elements of {\mathbb{Q}_p^*/(\mathbb{Q}_p^*)^2}.

Local class field theory tells us that the quadradic extensions of {\mathbb{Q}_p} are in bijection with the open index {2} subgroups {U\subset \mathbb{Q}_p^*} via {L/\mathbb{Q}_p\mapsto N_{L/\mathbb{Q}_p}(L^*)}. It turns out that any index {2} subgroup at all must be open because it will contain {(\mathbb{Z}_p^*)^2}.

Now it will be useful to think in different terms, which is actually a more standard modern reformulation of class field theory since it generalizes to “higher dimensions.” There is a bijection between the open index {2} subgroups {U} of {\mathbb{Q}_p^*} and surjective characters {\chi: \mathbb{Q}_p^*\rightarrow \{\pm 1\}} just by taking the kernel of the character. Thus we have reformulated the problem of counting these subgroups to counting characters.

The unit groups have the form (if {p} odd) {\mathbb{Q}_p^*\simeq p^\mathbb{Z}\times \mathbb{Z}_p^*} and {\mathbb{Z}_p^*\simeq \mu_{p-1}\times (1 + p\mathbb{Z}_p)} where {\mu_{p-1}} is thought of via the Teichmuller lift. If {\chi} has order {2}, then it is trivial on {(\mathbb{Q}_p^*)^2\simeq p^{2\mathbb{Z}}\times (\mathbb{Z}_p^*)^2\simeq p^{2\mathbb{Z}}\times (\mu_{p-1})^2\times (1+p\mathbb{Z}_p)}.

Thus we get the result that {\chi} factors through {p^\mathbb{Z}/p^{2\mathbb{Z}}\times \mu_{p-1}/(\mu_{p-1})^2} which is a finite abelian {2}-group of order {4}. Thus the number of non-trivial characters is {3}. This gives us a nice alternate description to the classical Kummer description. It tells us there are exactly {3} quadratic extensions up to isomorphism. These aren’t hard to figure out explicitly either. Just take some {a\in \mathbb{Z}_p^*} which is not a square. The three quadratic extensions are {\mathbb{Q}_p(\sqrt{p})}, {\mathbb{Q}_p(\sqrt{a})}, and {\mathbb{Q}_p(\sqrt{ap})}. A similar description can be computed when {p=2}, but you get {7} in that case.

That is all for class field theory for now. We’ll move on to complex multiplication, and I think we’ve done enough of the basics that we can probably do what we need as we need it now.

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