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).

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