Classical version and idelic version of class field theory

Question

Last semester, I took a course about class field theory and I learned about Artin reciprocity, which gives a map from ideal class group to Galois group, $$ \left(\frac{L/K}{\cdot}\right):I_{K}\to Gal(L/K), \,\,\,\,\prod_{i=1}^{m}\mathfrak{p}_{i}^{n_{i}}\mapsto \prod_{i=1}^{m}\left(\frac{L/K}{\mathfrak{p}_{i}}\right)^{n_{i}} $$ where $\left(\frac{L/K}{\mathbb{p}_{i}}\right)$ is a Frobenius map corresponds to prime ideal $\mathfrak{p}$. Today, I learned an adelic version of (global) class field theory, which is $$ \mathbb{A}^{\times}_{F}/\overline{F^{\times}(F_{\infty}^{\times})^{o}} \simeq G_{F}^{ab}$$ where $F$ is number field, $\mathbb{A}_{F}$ is adele over $F$ and $G_{F}^{ab}=Gal(F^{ab}/F)$. I cannot understand how these two are connected. Could anyone can explain explicit relation between these two things?

Answer 1

For more clarity, let us make more precise the definition of the Artin reciprocity map :

1) Over $\mathbf Q$, CFT is the Kronecker-Weber theorem, which says that any finite abelian $L/\mathbf Q$ is contained in a cyclotomic field $\mathbf Q_m = \mathbf Q(\zeta_m)$. Such an $m$ is called a defining modulus for $L/\mathbf Q$ and the conductor $f_L$ of $L/\mathbf Q$ is the smallest (w.r.t. division) defining modulus of $L$. Given a defining modulus $m$ of $L$, set $C_m=(\mathbf Z/m\mathbf Z)^*$, and for $a\in C_m$, define the Artin symbol ($a,L/\mathbf Q$) to be the automorphism of $L$ sending $\zeta_m$ to $\zeta_m^{a}$ , and denote by $I_{L,m}$ its kernel, so as to get an isomorphism $ C_m/I_{L,m} \cong Gal(L/\mathbf Q)$ via the Artin symbol.

2) In classical CFT over a number field $K$, the previous notions can be generalized, but in a very non obvious way. Define a $K$-modulus $\mathfrak M$ to be the formal product of an ideal of the ring of integers $A_K$ and some infinite primes of $K$ (implicitly raised to the first power). In the sequel, for simplification, we'll "speak as if" $\mathfrak M$ was an ideal. Denote by $A_{\mathfrak M}$ the group of fractional prime to $\mathfrak M$ and by $R_{\mathfrak M}$ the subgroup of principal fractional ideals $(x)$ s.t. $x$ is "congruent to" $1$ mod $\mathfrak M$ , and put $C_{\mathfrak M}=A_{\mathfrak M}/R_{\mathfrak M}$. For a finite abelian extension $L/K$, define $I_{L/K,\mathfrak M}=N(C_{L,\mathfrak M})$ , where $N_{L/K}$ is the norm of $L/K$ . A defining $K$-modulus of $L/K$ is s.t. $(C_{\mathfrak M}:I_{L/K,\mathfrak M})=[L:K]$, and the conductor $f_{L/K}$ is the "smallest" defining $K$-modulus of $L/K$. For a finite $K$-prime $\mathfrak P$, coprime with $\mathfrak M$, it can be shown that there exists an unique Artin symbol $(\mathfrak P , L/K) \in G(L/K)$ characterized by $(\mathfrak P, L/K)(x)\equiv x^{N\mathfrak P}$ mod $\mathfrak PA_L$ for any $x\in A_L$, with $N=N_{K/\mathbf Q}$. This definition can be extended multiplicatively to $C_{\mathfrak M}$, and the Artin reciprocity law is the isomorphism $C_{\mathfrak M}/I_{L/K,\mathfrak M} \cong G(L/K)$ via the Artin symbol.

3) In idelic CFT over a number field $K$, the previous $C_{\mathfrak M}$ 's are replaced by idèle class groups. The idèle group $J_K$ is the group of invertible elements of the adèle ring of $K$ (equipped with the "restricted product topology") and the idèle class group $C_K$ is the quotient $J_K/K^*$ . Write $C'_K=C_K/D_K$ , where $D_K$ = the connected component of identity = the subgroup of infinitely divisible elements of $C_K$. For a $K$-modulus ${\mathfrak M}$, let $I_{\mathfrak M} = J_{\mathfrak M}.K^*/K^*$, where $J_{\mathfrak M}$ is the subgroup of idèles which are "congruent" to 1 mod $\mathfrak M$. Given an abelian $L/K$, a defining $K$-modulus $\mathfrak M$ is such that $I_{\mathfrak M}$ is contained in $N_{L/K}C_L$. The Artin global reciprocity map $(.,L/K)$ is defined as follows : by the Chinese Remainder theorem, for any $j \in J_K$, there exists $x \in K^*$ s.t. $j$ is "congruent to" $x$ mod ${\mathfrak M}$; then define $(j, L/K)$ to be the product of the elements $(L/K, \mathfrak P)^{n_\mathfrak P}$ , where $n_\mathfrak P = ord (jx^{-1})_\mathfrak P$, for all $\mathfrak P$ coprime to $\mathfrak M$. It is easy to see that this can be "passed to the quotient" to define a map $(., L/K) : C'_K \to G(L/K)$ s.t. $C'_K/N_{L/K}C'_L \cong G(L/K)$ . This is the Artin reciprocity law in idelic terms. Now that we are rid of the cumbersome modulii $\mathfrak M$, we can take projective limits along the finite abelian extensions of $K$ to get a canonical isomorphism $C'_K \cong G(K^{ab}/K)$, which you can check to coincide with the (rather unexploitable) expression that you gave.

Needless to say, almost all the properties explained above are very elaborate and difficult theorems.