On Drinfeld cusp forms of prime level

TL;DR

This paper investigates the structure of Drinfeld cusp forms of prime level over function fields, defining oldforms and newforms, and proving decomposition and Hecke operator properties in specific cases.

Contribution

It introduces a definition for oldforms and newforms of prime level Drinfeld cusp forms and proves a decomposition and injectivity of Hecke operators in certain cases.

Findings

Space of cuspforms of level t decomposes into oldforms and newforms.

Hecke operator T_t is injective on cusp forms of level 1.

Provides evidence supporting existing conjectures in the field.

Abstract

Let $(P_d)$ be any prime of $\mathbb{F}_q[t]$ of degree $d$ and consider the space of Drinfeld cusp forms of level $P_d$, i.e. for the modular group $\Gamma_0(P_d)$. We provide a definition for oldforms and newforms of level $P_d$. Moreover, when the dimension of the vector space of oldforms is one and $P_1=t$ we prove that the space of cuspforms of level $t$ is the direct sum of oldforms and newforms and that the Hecke operator $\mathbf{T}_t$ acting on Drinfeld cusp forms of level 1 is injective, thus providing more evidence for the conjectures presented and stated in [2] and [3].