Hecke operators on rational functions I

Juan B. Gil, Sinai Robins

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

We define Hecke operators Um 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 Um is simply the set {±mk | k ∈ ℕ} ∪ {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.

Original languageEnglish (US)
Pages (from-to)519-554
Number of pages36
JournalForum Mathematicum
Volume17
Issue number4
DOIs
StatePublished - May 2005

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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