The rational eigenfunctions of the hecke operator up

Abstract

We study the Hecke operator U_p on rational functions. As results, we obtain an explicit description of the rational eigenfunctions of the Hecke operator U_p. In this thesis, we extend the vector space of rational functions from real to complex, that is, the Taylor coefficients of rational functions are complex numbers. We have proved the generalized Spectral Theorem. Based on the Structure Theorem, we have proved a theorem which provides an explicit form of rational eigenfunctions of Up. Moreover, for the vector space of rational eigenfunctions for any fixed eigenvalue p^k, we find the basis for the Taylor coefficients whose generating functions are eigenfunctions. In short, we have determined the spectrum of eigenvalues and the explicit form of corresponding eigenfunctions of Hecke operator Up.