How does one see Hecke Operators as helping to generalize Quadratic Reciprocity?

Question

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

1. What is the precise statement of the generalization (in the terminology above) of quadratic reciprocity for the $1$-dimensional case?

2. What is the precise statement of the generalization (in the terminology above) of quadratic reciprocity for the $2$-dimensional case?

Answer 1

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.

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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.