Hecke operators on rational functions I
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Juan B. Gil
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.
© de Gruyter
Articles in the same Issue
- Hecke operators on rational functions I
- n-Cotilting and n-tilting modules over ring extensions
- Reduction for the projective arclength functional
- Extensions, dilations and functional models of Dirac operators in limit-circle case
- Numerical constraints for embedded projective manifolds
- Some results on Qp spaces, 0 < p < 1, continued
- On the probabilistic ζ-function of pro(finite-soluble) groups
- Generating automorphism groups of chains
Articles in the same Issue
- Hecke operators on rational functions I
- n-Cotilting and n-tilting modules over ring extensions
- Reduction for the projective arclength functional
- Extensions, dilations and functional models of Dirac operators in limit-circle case
- Numerical constraints for embedded projective manifolds
- Some results on Qp spaces, 0 < p < 1, continued
- On the probabilistic ζ-function of pro(finite-soluble) groups
- Generating automorphism groups of chains
