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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?

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Once again, not an expert, but as far as I know (for the $2$-dimensional case): 1) yes, we take the character, 2) we only expect this for odd representations, 3) yes, $N$ is the level. –  Qiaochu Yuan Aug 7 '11 at 13:39
    
I am not an expert, but I have been told of two references: C.R. Matthews, Gauss sums and elliptic functions 1. The Kummer sum, Invent. Math. 52, 163-185, 1979 and 2, The quartic sum, Invent. Math 54., 23-52, 1979. –  user31019 May 9 '12 at 23:14
    
This is suggested in the introduction to Diamond-Shurman, but I haven't read far enough into that particular book to see whether they go back to it. –  Dylan Moreland May 25 '12 at 17:05

1 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.


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.

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Matt: Maybe you want to include something about primitive Dirichlet characters vs. newforms. –  KCd May 10 '12 at 3:44
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I was wondering when you would answer this. Thank you! –  Dylan Moreland May 10 '12 at 3:50
    
@KCd: Dear Keith, I'm not sure I have the energy for that; your comment can stand (for now, at least) as a cryptic explanation of this point! Best wishes, –  Matt E May 10 '12 at 4:04

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