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 $\pi_\sigma$?

Any comments or inspirations are appreciated. Thank you in advance.

P.S. I have posted this questions in math stackexchange, but I wasn't getting any answer. I read a tip to migrate the question to math overflow. Please let me know if there is any problem.