Action of Hecke operators on two distinguished Drinfeld modular forms (Q652223)

In this articles, the action of the Hecke operators \(T_n\), with \(n \in \mathbb{F}_q[T]\) a monic polynomial, on Eisenstein series \(E_{q-1}\) of weight \(q-1\) and the Drinfeld discriminant \(\Delta\) is studied. First uniqueness properties of the expansion analytic functions such as \(E_{q-1}\) or \(\Delta\) with coefficients \(c_m\) indexed by the set of monic polynomials of \(\mathbb{F}_q[T]\), are derived. For eigenforms which admit expansions of this type, the authors note that the eigenvalues of the Hecke operators are related to the coefficients \(c_m\). A consequence is that the Hecke action determines such eigenforms. The paper finishes with a series of remarks, including the statement that the given expansions of \(E_{q-1}\) and \(\Delta\) are determined by their eigenvalues and weights.