Hecke Operators on Drinfeld Cusp Forms

Abstract

In this paper, we study the Drinfeld cusp forms for ${\Gamma }_1(T)$ and $\Gamma (T)$ using Teitelbaum's interpretation as harmonic cocycles. We obtain explicit eigenvalues of Hecke operators associated to degree one prime ideals acting on the cusp forms for ${\Gamma }_1(T)$ of small weights and conclude that these Hecke operators are simultaneously diagonalizable. We also show that the Hecke operators are not diagonalizable in general for ${\Gamma }_1(T)$ of large weights, and not for $\Gamma (T)$ even of small weights. The Hecke eigenvalues on cusp forms for $\Gamma (T)$ with small weights are determined and the eigenspaces characterized.

28 pages To appear in JNT