# On the structure and slopes of Drinfeld cusp forms

## TL;DR

This paper investigates the structure of Drinfeld cusp forms, defining oldforms and newforms, analyzing the Atkin operator's matrix, and providing computational evidence for conjectures on slopes and diagonalizability.

## Contribution

It introduces a new framework for understanding Drinfeld cusp forms, explicitly describes the Atkin operator matrix, and verifies conjectures through computational data.

## Key findings

- Explicit matrix U for Atkin operator on cusp forms
- Verification of conjectures in small weights
- Formulation of bounds for slopes and diagonalizability

## Abstract

We define oldforms and newforms for Drinfeld cusp forms of level $t$ and conjecture that their direct sum is the whole space of cusp forms. Moreover we describe explicitly the matrix $U$ associated to the action of the Atkin operator $\mathbf{U}_t$ on cusp forms of level $t$ and use it to compute tables of slopes of eigenforms. Building on such data, we formulate conjectures on bounds for slopes, on the diagonalizability of $\mathbf{U}_t$ and on various other issues. Via the explicit form of the matrix $U$ we are then able to verify our conjectures in various cases (mainly in small weights).