# Relationship among class field theory, modularity theorem, and the langlands program

Here is some background: I have taken an introductory algebraic number theory course up to valuation-theoretic approach, learned a bit of local class field theory, representation theory, and modular forms.

Recently, I read an introductory exposition on Langlands program by Stephen Gelbart. Then if my understanding is correct, the Langlands program is to find a (conjectured) correspondence $\sigma \mapsto \pi_\sigma$ where $\sigma: G(E/\mathbb{Q}) \rightarrow GL_n(\mathbb{C})$ is a $n$-dimensional representation of the Galois group $G(E/\mathbb{Q})$ with $E/\mathbb{Q}$ Galois, and $\pi_\sigma$ is an "automorphic representation of $GL_n$" satsifying

### $L(s, \sigma) = L(s,\pi_\sigma)$

where the LHS $L$-function is due to Artin and the RHS $L$-function corresponds to the automorhpic form $\pi_\sigma$.

He then gives an application of the above to $\mathbb{Q}(i)$ with $\sigma: G \rightarrow \mathbb{C}^\times$ defined by $\sigma(id) =1, \sigma(cx~conj) = -1$, and an appropriate character $\chi : (\mathbb{Z}/4 \mathbb{Z})^\times \rightarrow \mathbb{C}^\times$. Here the $cx~conj$ is the complex conjugation.

My first question is: does $\chi$ play the role of the corresponding automorphic form $\pi_\sigma$ of $\sigma$ with $n=1$, so the class field theory is just a 1-dimensional class of Langlands program?

The math stackexchange article

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

was beyond my understanding.

Also, I read a commentary of Edward Frankel on Gelbart's article saying that the Taniyama-Shimura conjecture is

“...a special case of the Langlands correspondence for the field of rational numbers: it relates two-dimensional Galois representations on the first etale cohomology of elliptic curves over $\mathbb{Q}$ to the automorphic representations of the group $GL_2$ over the $\mathbb{A}_\mathbb{Q}$ encoded by certain modular forms on the upper half-plane.”

i.e. the $n=2$ case of the Langlands program.

My second question is: does the first $\acute{\text{e}}$tale cohomology of elliptic curves over $\mathbb{Q}$ corresponds to $\sigma$ and the modular forms give rise to the automorphic representation $\pi_\sigma$?

• There is a subtlety, often glossed over in introductory expositions. There are several correspondences, all conjectural. There's a correspondence between $\pi$'s "of Artin type" and $\sigma$ as in your question (complex Galois reps). There's another one between "algebraic" $\pi$'s and motives (e.g. elliptic curves), via $p$-adic Galois representations, and a third between all $\pi$'s and complex representations of the global Langlands group (which doesn't yet exist). Strictly speaking the answer to your second question is no, because $\sigma$ coming from etale cohomology isn't a complex rep. – znt Nov 21 '16 at 7:55
• @libofmath For dimension 1, try to understand the following statement. There is a continuous bijection between finite index Hecke characters $I_K \to \Bbb{C}^\times$ and continuous representations $G_K \to \Bbb{C}^\times$ where $I_K$ is the idele class group. The bijection essentially follows from existence and properties of the Global Artin map. – Ben Lim Nov 21 '16 at 8:55