Synopsis of Class Field Theory Roughly following Neukirch

Let K be a number field, and denote places of K by 𝔭.

For finite places 𝔭, let 𝒪_{𝔭} denote the valuation ring of K_{𝔭}, and let

U_{𝔭} = \{ \begin{matrix}𝒪_{𝔭}^{∗} & if 𝔭 ∤ ∞, \\ ℝ_{>0}^{∗} ⊆ ℝ^{∗} = K_{𝔭} & if 𝔭 is real, and \\ K_{𝔭}^{∗} & if 𝔭 is complex. \end{matrix}

Then the group of idèles I_{K} is the restricted direct product of K_{𝔭}^{∗} with respect to the subgroups U_{𝔭}. (This is not quite the book’s definition, due to differences at the real places, but there are only finitely many of those, so in the end it makes no difference.)

We have K^{∗} ⊆ I_{K} as a discrete subgroup (embedded diagonally).

For all finite sets S of places of K there are subgroups

I_{K}^{S} = ∏_{𝔭∈S} K_{𝔭}^{∗} × ∏_{𝔭∉S} U_{𝔭}

of I_{K} . If S ⊆ S′ then I_{K}^{S} ⊆ I_{K}^{S′} . Also I_{K} = ⋃ I_{K}^{S} .

For n ∈ ℕ we define subgroups of K_{𝔭}^{∗}:

V_{𝔭}^{(n)} = \{ \begin{matrix}K_{𝔭}^{∗} & if 𝔭 is finite and n = 0, \\ 1 + 𝔭^{n}𝒪_{𝔭} & if 𝔭 is finite and n > 0, \\ ℝ_{>0}^{∗} & if 𝔭 is real, and \\ K_{𝔭}^{∗} & if 𝔭 is complex \end{matrix}

and

U_{𝔭}^{(n)} = V_{𝔭}^{(n)} ∩ U_{𝔭} = \{ \begin{matrix}𝒪_{𝔭}^{∗} & if 𝔭 is finite and n = 0, \\ 1 + 𝔭^{n}𝒪_{𝔭} & if 𝔭 is finite and n > 0, \\ ℝ_{>0}^{∗} & if 𝔭 is real, and \\ K_{𝔭}^{∗} & if 𝔭 is complex. \end{matrix}

These groups are open subgroups of U_{𝔭} of finite index, and every open subgroup of U_{𝔭} contains U_{𝔭}^{(n)} for sufficiently large n.

A module 𝔪 of K is a nonzero ideal in 𝒪_{K} . We often write 𝔪 = ∏ 𝔭^{n_{𝔭}} , and let n_{𝔭} = 0 for infinite 𝔭. If α = (α_{𝔭})_{𝔭} and β are idèles, then we say that α ≡ β (mod^{+} 𝔪) if α_{𝔭}/β_{𝔭} ∈ V_{𝔭}^{(n_{𝔭})} for all 𝔭 (this is only a restriction for finite 𝔭 | 𝔪 and for real places).

Let J_{K}^{𝔪} be the group of fractional ideals of K prime to 𝔪, let P_{K}^{𝔪} be the group of principal fractional ideals of K generated by some x ∈ K with x ≡ 1 (mod^{+} 𝔪) , and let Cl_{K}^{𝔪} = J_{K}^{𝔪}/P_{K}^{𝔪}.

The group of idèle classes is C_{K} := I_{K}/K^{∗}.

There is a natural surjection I_{K} ↠ J_{K} with kernel I_{K}^{S_{∞}} , so J_{K} ≅ I_{K}/I_{K}^{S_{∞}} and Cl_{K} ≅ I_{K}/K^{∗}I_{K}^{S_{∞}} ≅ C_{K} / (K^{∗}I_{K}^{S_{∞}}/K^{∗}) .

In the Arakelov context, we have a natural surjection I_{K} ↠ \overline{J}(𝒪_{K}) with kernel

I_{K}^{0} = ∏{α_{𝔭} ∈ K_{𝔭}^{∗} : ∥α_{𝔭}∥_{𝔭} = 1} .

For an idèle α, we define its absolute norm N(α) = ∏ N(𝔭)^{v_{𝔭}(α_{𝔭})} = ∏ ∥α_{p}∥_{𝔭}^{−1} . Since N(x) = 1 for all x ∈ K^{∗}, this descends to a well-defined continuous function on C_{K} , and we define C_{K}^{0} = {α ∈ C_{K} : N(α) = 1}. It is compact. Note that this is not a similar definition to I_{K}^{0} .

We let I_{K}^{𝔪} = ∏ U_{𝔭}^{(n_{𝔭})} . Then C_{K}^{𝔪} := K^{∗}I_{K}^{𝔪}/K^{∗} is the congruence subgroup of C_{K} , and C_{K}/C_{K}^{𝔪} is the ray class group. We have C_{K}/C_{K}^{𝔪} ≅ Cl_{K}^{𝔪} .

Let L/K be a finite extension of number fields. We define a norm N_{K}^{L}: I_{L} → I_{K} by

(N_{K}^{L}α)_{𝔭} = ∏_{𝔮|𝔭} N_{K_{𝔭}}^{L_{𝔮}}α_{𝔮} .

This map is continuous and has open image. It extends to a map C_{L} → C_{K} , also denoted N_{K}^{L} .

Let L/K be an abelian extension and let G = Gal(L/K) . For finite primes 𝔭 of K not ramifying in L, there is a well-defined Frobenius element σ ∈ G depending only on 𝔭. We define the Artin symbol (𝔭, L/K) to be this Frobenius element, and extend this multiplicatively to all fractional ideals of K not involving ramified primes. A module 𝔪 is admissible for L/K if N_{K_{𝔭}}^{L_{𝔮}}L_{𝔮}^{∗} ⊇ U_{𝔭}^{(n_{𝔭})} for all places 𝔭 of K and all (equivalently, at least one) 𝔮 | 𝔭. It is a deep theorem that there exists an admissible module involving only ramified primes.

If 𝔪 is admissible for L/K, then it is known that C_{K}/N_{K}^{L}C_{L} ≅ J_{K}^{𝔪}/P_{K}^{𝔪}N_{K}^{L}J_{L}^{𝔪𝒪_{L}} , and that the Artin symbol is trivial on P_{K}^{𝔪}. Therefore, the Artin symbol defines an Artin map from C_{K} to Gal(L/K) .

Theorem (Global Class Field Theory). Let K be a number field. Then:

(a). The map L ↦ N_{K}^{L}C_{L} induces an inclusion-reversing one-to-one correspondence between finite abelian extensions L of K and open subgroups H of C_{K} (equivalently, closed subgroups of finite index).

(b). If L/K is finite abelian and H = N_{K}^{L}C_{L}, then the Artin map induces an

isomorphism C_{K}/H →^{∼} Gal(L/K) . (c). If L and L′ are finite abelian over K and H and H′ are the corresponding subgroups of C_{K} , then LL′ corresponds to H ∩ H′ and L ∩ L′ corresponds to HH′.

(d). A finite place 𝔭 of K is unramified in L if and only if H ⊇ K^{∗}U_{𝔭}/K^{∗} (where U_{𝔭} is embedded in I_{K} by extending by 1’s at places other than 𝔭). (e). A (finite or infinite) place 𝔭 of K splits completely in L if and only if

H ⊇ K^{∗}K_{𝔭}^{∗}/K^{∗}.