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 and . For distinct odd primes ,

Consider the equation

Let be the number of solutions to Q modulo , minus one. Then by definition of the Legendre symbol, By property of the Lengendre symbol (or rather, the Jacobi symbol), we have

(*)

Let . Then it follows from the reciprocity law that depends only on the value of modulo . Furthermore, the finite sequence **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.

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

Define . Then use (*) to show that

for all prime. So, this is indeed, a common eigenvector for all the ‘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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