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?


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.


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