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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.2011-08-07
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    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.2012-05-25
  • 0
    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](http://www.springerlink.com/content/n0num5l22436742t/), Invent. Math. 52, 163-185, 1979 and [2, The quartic sum](http://www.springerlink.com/content/t715013442qg5223/), Invent. Math 54., 23-52, 1979.2012-05-09

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