Below is Gauss’ quadratic reciprocity, found in most elementary texts in number theory. In this post, we’ll see how the Hecke operators originate from this theorem.

Theorem (Gauss). Let $\varepsilon(n) = (-1)^{\frac{n-1}{2}}$ and $\omega(n) = (-1)^{\frac{n^2-1}{8}}$. For distinct odd primes $p, q$,

$\displaystyle \left(\frac{p}{q}\right) = \varepsilon(p) .\varepsilon(q). \left(\frac{q}{p}\right),$

$\displaystyle \left(\frac{-1}{p}\right) = \varepsilon(p),$

$\displaystyle \left(\frac{2}{p}\right) = \omega(p).$

Consider the equation

$\textbf{(Q)}: \qquad x^2 = d; \qquad \qquad d\in \mathbb Z \backslash \{0\}.$

Let $a_p(Q)$ be the number of solutions to Q modulo $p$, minus one. Then by definition of the Legendre symbol, $a_p(Q) = \left( \frac{d}{p} \right).$ By property of the Lengendre symbol (or rather, the Jacobi symbol), we have

$a_{mn}(Q) = a_m(Q) . a_n(Q).$                       (*)

Let $N = 4 |d|$. Then it follows from the reciprocity law that $a_p(Q)$ depends only on the value of $p$ modulo $N$. Furthermore, the finite sequence ${a_2(Q), a_3(Q), a_5(Q), \cdots }$ arises as a set of eigenvalues of a linear operator (the Hecke operator) on a finite dimensional complex vector space. We’re going to construct the space.

$V_N := \{ f: (\mathbb Z/n\mathbb Z)^* \to \mathbb C \}$

$T_p : V_N \to V_N, \quad T_p(f)(n) = f(pn) \quad \text{if } p \nmid n \text{ and } 0 \text{ otherwise }.$

Verify that $T_p$ is a linear operator on $V_N$. Now all these operators for varying primes commute with each other. So what is a common eigenvector?

Define $\phi (n) = a_n(Q)$. Then use (*) to show that

$T_p(\phi) = a_p(Q) \phi,$

for all $p$ prime. So, this $\phi$ is indeed, a common eigenvector for all the $T_p$‘s!

I will explain more about Hecke operators on modular forms in a future post. (Edit July 10, 2013: Link to the said post).