Quadratic reciprocity in Langlands program

Question

I know quadratic reciprocity is the easiest example of langlands correspondence. Langlands correspondence gives some relation between automorphic forms and artin representations. My question is: what are the automorphic forms and the artin representation that are linked via quadratic reciprocity?

Answer 1

For simplicity let $\chi$ be a Dirichlet character modulo $q^k$ a prime power, for an idele $x\in \Bbb{A_Q}^\times$, $x = x_\infty \prod_p x_p$ ($x_\infty \in \Bbb{R}^\times,x_p=p^{v_p(x)} a_p \in \Bbb{Q}_p^\times, a_p \in \Bbb{Z}_p^\times$, $v_p(x)=0$ for all but finitely many $p$) let $$\chi(x) = \chi(a_q \bmod q^k) \text{sign}(x_\infty)^{\frac{1-\chi(-1)}{2}} \prod_{p \ne q} \chi(p)^{-v_p(x)}$$ Note it is well-defined and it is a (continuous) homomorphism $\Bbb{A_Q}^\times \to \Bbb{C}^\times$.

The point is that if $y \in \Bbb{Q}^\times$ and $x_\infty = y,\forall p, x_p=y$ (the diagonal embedding of $\Bbb{Q}^\times$ into the ideles) then $\chi(x) = 1$.

Thus $\chi$ is a continuous homomorphism $\Bbb{A_Q^\times/Q^\times} \to \Bbb{C}^\times$, ie. an automorphic form for $GL_1(\Bbb{A_Q})$.

The Langlands reciprocity is saying $L(s,\chi) = \prod_p \frac{1}{1-\chi(p) p^{-s}}$ (which is the L-function of that automorphic form) is the L-function of some Galois representation $Gal(\overline{\Bbb{Q}}/\Bbb{Q}) \to GL_1(\Bbb{C})$, which here is $\rho(\sigma) = \chi(n)$ if $\sigma(\zeta_{q^k})= \zeta_{q^k}^n$, the Galois representation appearing in the Dedekind zeta function $\zeta_{\Bbb{Q}(\zeta_{q^k})}(s) = \prod_{\chi \bmod q^k} L(s,\widetilde{\chi})$ of the cyclotomic field $\Bbb{Q}(\zeta_{q^k}) = \overline{\Bbb{Q}}^{\ker(\rho)}$.

Integrating $\chi(x)(|x|_\infty \prod_p |x_p|_p)^s$ with respect to the Haar measure on $\Bbb{A_Q^\times/Q^\times}$ and using that $\chi(x^{-1}) = \overline{\chi(x)}$ provides the functional equation. For automorphic forms over $GL_n$, homomorphism is replaced by left $GL_n(\Bbb{Q})$-invariant and generating an irreducible representation (which guarantees the Euler product).