Rational eigenfunctions of the Hecke operators

Abstract

We study the action of the Hecke operators $U_n$ on the space $\mathcal R$ of rational functions in one variable, over $\mathbb C$. The main goal is to give a complete classification of the eigenfunctions of $U_n$. We accomplish this by introducing certain number-theoretic directed graphs, called Zolotarev Graphs, which extend the well-known permutations due to Zolotarev. We develop the theory of these Zolotarev graphs, using them to decompose the eigenfunctions of $U_n$ into certain natural finite-dimensional vector spaces of rational functions, which we call the eigenspaces. In this context, we prove that the dimension of each eigenspace is equal to the number of nodes of a cycle that belongs to its corresponding Zolotarev graph. We prove that the number of leaves of this Zolotarev graph equals the dimension of the kernel of $U_n$. We then give a novel number-theoretic formula for the number of cycles of fixed length, in each Zolotarev graph. We also study the simultaneous eigenfunctions for all of the $U_n$, and give explicit bases for all of them. In the process, we answer many questions that were set out in the work of Gil and Robins (2005). We also discover certain strong relations between these graphs and the kernel of $U_n$ acting on a subspace of $\mathcal R$; in particular, we give several equivalent conditions for the diagonalizibility of $U_n$. Finally, we prove that the classical Artin Conjecture on primitive roots is equivalent to a new conjecture here, that infinitely many of these eigenspaces have dimension $1$.