[Paper Review] Hecke operators on rational functions

Juan B. Gil, Sinai Robins|arXiv (Cornell University)|Sep 15, 2003

Advanced Combinatorial Mathematics6 references3 citations

TL;DR

This paper develops the spectral theory of Hecke operators acting on rational functions, establishing that eigenfunctions correspond precisely to rational functions with poles at roots of unity. It proves that the point spectrum of each Hecke operator $U_p$ is $\{\pm p^k \mid k \in \mathbb{N}\} \cup \{0\}$, and derives a structure theorem showing eigenfunctions have coefficients that are quasipolynomials with periodic coefficients tied to roots of unity.

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.

Motivation & Objective

• To develop a complete spectral theory for Hecke operators acting on rational functions, analogous to that in modular forms but in a new algebraic context.
• To characterize the eigenfunctions of Hecke operators $U_p$ acting on rational functions with real coefficients.
• To identify the precise structure of eigenfunctions in terms of poles at roots of unity and quasipolynomial coefficients.
• To decompose the space of rational functions with poles at roots of unity into finite-dimensional eigenspaces graded by weight and level.
• To establish connections between eigenfunctions and arithmetic properties such as multiplicative coefficients and the Riemann zeta function.

Proposed method

• Define the Hecke operator $U_p$ on the space $\mathcal{R}$ of rational functions $f(x) = A(x)/B(x)$ with $\deg A < \deg B$ and $B(0) \neq 0$, by extracting coefficients at indices divisible by $p$.
• Use generating functions and linear recurrence sequences to relate rational functions to quasipolynomials, leveraging the fact that rational functions with poles at roots of unity have quasipolynomial coefficients.
• Prove the involution property: if $f$ is an eigenfunction of $U_p$, then so is $f(1/x)$, with the same eigenvalue.
• Derive the spectrum of $U_p$ as $\{\pm p^k \mid k \in \mathbb{N}\} \cup \{0\}$, using finite-dimensional matrix representations of the action of $U_p$.
• Establish a structure theorem: eigenfunctions have coefficients of the form $a_n = n^{\kappa-1} \sum_{j=1}^{d/\kappa} C_j e^{2\pi i \ell_j n / L}$, where poles are $L$-th roots of unity with multiplicity $\kappa$.
• Use computational tools (e.g., MAPLE 6) to compute explicit eigenfunctions and their eigenvalues for small degrees, organizing them by level $L$ and weight $\kappa$.

Experimental results

Research questions

• RQ1 What is the complete point spectrum of the Hecke operator $U_p$ acting on rational functions?
• RQ2 What structural constraints must a rational function satisfy to be an eigenfunction of $U_p$?
• RQ3 How can eigenfunctions be decomposed into finite-dimensional subspaces indexed by weight $\kappa$ and level $L$?
• RQ4 What is the relationship between eigenfunctions and arithmetic functions such as multiplicative coefficients or the Riemann zeta function?
• RQ5 How do the eigenvalues of $U_p$ relate to the degrees and multiplicities of poles in the denominator of the rational function?

Key findings

• The point spectrum of $U_p$ on $\mathcal{R}$ is $\{\pm p^k \mid k \in \mathbb{N}\} \cup \{0\}$, with no other eigenvalues possible.
• Eigenfunctions of $U_p$ must have denominators $B(x)$ whose roots are roots of unity, satisfying $x^d B(1/x) = (-1)^d B(x)$.
• For $U_2$, eigenfunctions of degree $d=2$ include $f_{2,1}(x) = \frac{x}{(1-x)^2}$ with eigenvalue $\lambda = 2$, and $f_{2,2}(x) = \frac{x}{1+x+x^2}$ with $\lambda = -1$.
• The eigenvalue magnitude for $U_2$ is $|\lambda| = 2^{\kappa-1}$, where $\kappa$ is the common multiplicity of poles (the weight).
• The space $\mathcal{S}_{1,7}(U_2)$ has basis $\left\{ \frac{x+x^2+x^4}{1-x^7}, \frac{x^3+x^5+x^6}{1-x^7} \right\}$, showing explicit construction of eigenfunctions at level 7.
• Eigenfunctions of level $L$ are invariant under the action of $U_p$ only when $\gcd(p,L) = 1$, explaining why $\mathcal{S}_{\kappa,2m}(U_2)$ is empty for all $m$.