Why is Class Field Theory the same as Langlands for GL_1?

Question

I've heard many people say that class field theory is the same as the Langlands conjectures for GL_1 (and more specifically, that local Langlands for GL_1 is the same as local class field theory). Could someone please explain why this is true?

My background is as follows: I understand the statements of class field theory (in other words, that abelian extensions correspond to open subgroups of the idele class group, and the quotient is the Galois group of that abelian extension). I know what modular forms are and what a group representation is, but not much more than that. So I'm looking to see why the statement of class field theory that I know is essentially the same as a certain statement about L-functions, representations, or automorphic forms, in such a way that a more advanced mathematician could easily recognize the latter statement as Langlands in dimension 1.

Answer 1

What you are looking for is the correspondence between algebraic Hecke characters over a number field $F$ and compatible families of $l$-adic characters of the absolute Galois group of $F$. This is laid out beautifully in the first section of Laurent Fargues's notes here.

EDIT: In more detail, as Kevin notes in the comments above, an automorphic representation of $GL(1)$ over $F$ is nothing but a Hecke character; that is, a continuous character $$\chi:F^\times\setminus\mathbb{A}_F^\times\to\mathbb{C}^\times$$ of the idele class group of $F$. You can associate $L$-functions to these things: they admit analytic continuation and satisfy a functional equation. This is the automorphic side of global Langlands for $GL(1)$.

How to go from here to the Galois side? Well, let's start with the local story. Fix some prime $v$ of $F$; then the automorphic side is concerned with characters $$\chi_v:F_v^\times\to\mathbb{C}^\times$$ Local class field theory gives you the reciprocity isomorphism $$rec_v:W_{F_v}\to F_v^\times,$$ where $W_{F_v}$ is the Weil group of $F_v$. Then $\chi_v\circ rec_v$ gives you a character of $W_{F_v}$. This is local Langlands for $GL(1)$. The matching up local $L$-functions and $\epsilon$-factors is basically tautological.

We return to our global Hecke character $\chi$. Recall that global class field theory can be interpreted as giving a map (the Artin reciprocity map) $$Art_F:F^\times\setminus\mathbb{A}_F^\times\to Gal(F^{ab}/F),$$ where $F^{ab}$ is the maximal abelian extension of $F$. Local-global compatibility here means that, for each prime $v$ of $F$, the restriction $Art_F\vert_{F_v^\times}$ agrees with the inverse of the local reciprocity map $rec_v$.

Since $Art_F$ is not an isomorphism, we do not expect every Hecke character to be associated with a Galois representation. What is true is that $Art_F$ induces an isomorphism from the group of connected components of the idele class group to $Gal(F^{ab}/F)$. In particular, any Hecke character with finite image will factor through the reciprocity map, and so will give rise to a character of $Gal(F^{ab}/F)$. This is global Langlands for Dirichlet characters (or abelian Artin motives).

But we can say more, supposing that we have a certain algebraicity (or arithmeticity) condition on our Hecke character $\chi$ at infinity. The notes of Fargues referenced above have a precise definition of this condition; I believe the original idea is due to Weil. The basic idea is that the obstruction to $\chi$ factoring through the group of connected components of the idele class group (and hence through the abelianized Galois group) lies entirely at infinity. The algebraicity condition lets us "move" this persnickety infinite part over to the $l$-primary ideles (for some prime $l$), at the cost of replacing our field of coefficients $\mathbb{C}$ by some finite extension $E_\lambda$ of $\mathbb{Q}_l$. This produces a character

$$\chi_l:F^\times\setminus\mathbb{A}_F^\times\to E_\lambda^\times$$

that shares its local factors away from $l$ and $\infty$ with $\chi$, but now factors through $Art_F$. Varying over $l$ gives us a compatible family of $l$-adic characters associated with our automorphic representation $\chi$ of $GL(1)$. The $L$-functions match up since their local factors do.

Answer 2

This is a deep question, but here's a stab at it. Your take on class field theory is the traditional one, that certain abelian groups are isomorphic. A lot of effort was put into looking for a generalization to nonabelian groups. But the key fact in the abelian case turns out to be not that certain groups are isomorphic, but that their dual groups (homomorphisms into $\mathbb C^\times$) are isomorphic. In a sense this is trivial, because a finite abelian group is isomorphic to its dual. But it lets you see how to generalize to the nonabelian case: forget about the groups themselves, and think instead about their representations. One of the pieces of the Langlands program is that $n$ dimensional representations of Galois groups correspond to automorphic representations of GL(n).

This is barely skimming the surface. I recommend 'An Elementary Introduction to the Langlands Program' by Stephen Gelbart in the AMS Bulletin, v. 10 1984 pp 177-220 (as well as his other excellent expository papers).