Hecke operators on rational functions

TL;DR

This paper introduces Hecke operators acting on rational functions, explores their eigenfunctions, and reveals connections to Dirichlet characters, linear recurrences, and modular form analogies, uncovering structural properties of these functions.

Contribution

It defines new Hecke operators on rational functions, characterizes their eigenfunctions, and establishes links to Dirichlet characters and modular form concepts.

Findings

Eigenfunctions have spectra {+/- m^k, 0}

Eigenfunctions involve Dirichlet characters mod L

Rational functions with quasi-polynomial coefficients are characterized

Abstract

We define Hecke operators U_m that sift out every m-th Taylor series coefficient of a rational function in one variable, defined over the reals. We prove several structure theorems concerning the eigenfunctions of these Hecke operators, including the pleasing fact that the point spectrum of the operator U_m is simply the set {+/- m^k, k in N} U {0}. It turns out that the simultaneous eigenfunctions of all of the Hecke operators involve Dirichlet characters mod L, giving rise to the result that any arithmetic function of m that is completely multiplicative and also satisfies a linear recurrence must be a Dirichlet character times a power of m. We also define the notions of level and weight for rational eigenfunctions, by analogy with modular forms, and we show the existence of some interesting finite-dimensional subspaces of rational eigenfunctions (of fixed weight and level), whose union gives all of the rational functions whose coefficients are quasi-polynomials.