Hecke Operators and Drinfeld Cusp Forms of Level

Abstract

We use a linear algebra interpretation of the action of Hecke operators on Drinfeld cusp forms to prove that when the dimension of the $\mathbb {C}_\infty $ -vector space $S_{k,m}(\mathrm {{GL}}_2(\mathbb {F}_q[t]))$ is one, the Hecke operator $\mathbf {T}_t$ is injective on $S_{k,m}(\mathrm {{GL}}_2(\mathbb {F}_q[t]))$ and $S_{k,m}(\Gamma _0(t))$ is a direct sum of oldforms and newforms.

Notes on Atkin–lehner Theory for Drinfeld Modular Forms

“…Our methods are completely different from those of [4,6]. We are very optimistic that our methods can be used when dim S k,l (GL 2 (A)) ≥ 3.…”

“…For even characteristic, Conjecture 1.1(iii) is not true [2]. Assuming that the characteristic p is odd, Conjecture 1.1 is proved in [4,6] in some special cases, using harmonic cocycles, the trace maps Tr and Tr , and the linear algebra interpretation of the Hecke operators T p and U p . By studying the action of the T T -operators on the Fourier coefficients of Drinfeld modular forms, we prove the following result.…”