My question is really about how to think of Hecke operators as helping to generalize quadratic reciprocity.
Quadratic reciprocity can be stated like this: Let $\rho: Gal(\mathbb{Q})\rightarrow GL_1(\mathbb{C})$ be a $1$-dimensional representation that factors through $Gal(\mathbb{Q}(\sqrt{W})/\mathbb{Q})$. Then for any $\sigma \in Gal(\mathbb{Q})$, $\sigma(\sqrt{W})=\rho(\sigma)\sqrt{W}$. Define for each prime number $p$ an operator on the space of functions from $(\mathbb{Z}/4|W|\mathbb{Z})^{\times}$ to $\mathbb{C}^{\times}$ by $T(p)$ takes the function $\alpha$ to the function that takes $x$ to $\alpha(\frac{x}{p})$. Then there is a simultaneous eigenfunction $\alpha$, with eigenvalue $a_p$ for $T(p)$, such that for all $p\not|4|W|$ $\rho(Frob_p)=a_p$. (and to relate it to the undergraduate-textbook-version of quadratic reciprocity, one need only note that $\rho(Frob_p)$ is just the Legendre symbol $\left( \frac{W}{p}\right)$.)
Now I'm trying to understand how people think of generalizations of this. First, still in the one dimensional case, let's say we are not working over a quadratic field. What would the generalization be? What would take the place of $4|W|$? Would the space of functions that the $T(p)$'s work on still thes space of functions from $(\mathbb{Z}/N\mathbb{Z})^{\times}$ to $\mathbb{C}^{\times}$? What is this $N$?
Now let's jump to the $2$-dimensional case. Here we have the actual theory of Hecke operators. However, as I understand it, there is a basis of simultaneous eigenvalues only for the cusp forms. Now I'm finding it hard to match everything up: are we dealing just with irreducible $2$-dimensional representations? Instead of $\rho$ do we take the character? Would we say that for each representation there's a cusp form such that it's a simultaneous eigenfunction and such that $\xi(Frob_p)=a_p$ (the eigenvalues) where $\xi$ is the character of $\rho$? This should probably be for all $p$ that don't divide some $N$. What is this $N$? Does it relate to the cusp forms somehow? Is it their weight? Their level?
In other words:
Questions
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What is the precise statement of the generalization (in the terminology above) of quadratic reciprocity for the $1$-dimensional case?
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What is the precise statement of the generalization (in the terminology above) of quadratic reciprocity for the $2$-dimensional case?
Best Answer
The one-dimensional generalization of quadratic reciprocity is class field theory (over $\mathbb Q$, if you want to restrict to that case, where it is known as the Kronecker--Weber theorem).
Here is a formulation which is useful for comparing with the two-dimensional version; it is helpful to split it into two parts:
Given a Dirichlet character $\chi: (\mathbb Z/N\mathbb Z)^{\times} \to \mathbb C^{\times},$ there is a (uniquely determiend) Galois character $\psi: G_{\mathbb Q} \to \mathbb C^{\times}$, unramified outside $N$, such that $\psi(Frob_p) = \chi(p)$ for each prime $p$ not dividing $N$.
Every continuous character $\psi: G_{\mathbb Q} \to \mathbb C^{\times}$ is associated to some Dirichlet character as in the previous bullet point.
As you observe, one can think of multiplication by $p$ on $(\mathbb Z/N\mathbb Z)^{\times}$ as a "Hecke operator at $p$" (for $p$ not dividing $N$), and then the Dirichlet characters are precisely the normalized Hecke eigenforms (i.e. the functions $(\mathbb Z/N\mathbb Z)^{\times} \to \mathbb C$ that are eigenforms for all the Hecke operators, normalized so that $\psi(1) = 1$).
Now for the two-dimensional version:
For every weight one cupsidal Hecke eigenform $f$ of level $N$ (i.e. an eigenform for all the $T_p$ with $p$ not dividing $N$) there is a (uniquely determined) continuous irreducible representation $\rho:G_{\mathbb Q} \to GL_2(\mathbb C)$, unramified outside $N$, such that, for each $p$ not dividing $N$, the char. poly of $\rho(Frob_p)$ is equal to the $p$th Hecke polynomial of $f$, i.e. equal to $X^2 - a_p X + \varepsilon(p)$, where $a_p$ is the $T_p$-eigenvalue of $f$, and $\epsilon$ is the nebentypus character of $f$. As a slight aside, note that the determinant of $\rho$ is a one-dimensional character, and the preceding condition shows that $\det \rho$ is associated to the Dirichlet character $\varepsilon$ via the abelian correspondence already considered. Note also that, necessarily for the nebentypus of a weight one eigenform, one has $\varepsilon(-1) = -1$, and hence $\det\rho(c) = -1$ (where $c \in G_{\mathbb Q}$) is complex conjugation).
If $\rho:G_{\mathbb Q} \to GL_2(\mathbb C)$ is a continuous irreducible representation such that $\det\rho(c) = -1$, then $\rho$ arises from a weight one cuspidal eigenform as in the preceding bullet point.
Remarks:
The first bullet point in the one-dimensional case follows from the isomorphism $Gal(\mathbb Q(\zeta_N)/\mathbb Q) = (\mathbb Z/N\mathbb Z)^{\times}$, which is essentially equivalent to the irreducibility of the $N$th cyclotomic polynomial, due to Gauss.
The second bullet point in the one-dimensional case follows from the Kronecker--Weber theorem, which states that every abelian extension of $\mathbb Q$ is contained in a cyclotomic extension.
The first bullet point in the two-dimensional case is a theorem of Deligne and Serre.
The second bullet point in the two-dimensional case was conjectured by Langlands (in fact it is a very particular case of a much more general conjecture of Langlands, which strenghtens Artin's conjecture on the holomorphicity of Artin $L$-functions), and was proved in full generality by Khare, Wintenberger, and Kisin (after much partial progress by others).
There is a closely related story for modular forms of weight $> 1$, but then one must introduce $\ell$-adic representations, rather than just complex ones. For more on this you may want to look at some of the posts on the Langlands program linked to here.