On the Rational Cuspidal Divisor Class Groups of Drinfeld Modular Curves $X_0(\mathfrak{p}^r)$

TL;DR

This paper explicitly determines the structure of the rational cuspidal divisor class groups of Drinfeld modular curves $X_0(rak{p}^r)$ for prime power levels, relating divisors to $ riangle$-quotients and using the Drinfeld discriminant function.

Contribution

It provides an explicit description of the rational cuspidal divisor class groups for all prime levels and exponents, connecting them with $ riangle$-quotients and the Drinfeld discriminant.

Findings

Explicit structure of $rak{p}^r$-level class groups determined

Connection established between divisors and $ riangle$-quotients

Results hold for arbitrary prime $rak{p}$ and $r geq 2$

Abstract

Let $\mathcal{C}(\mathfrak{p}^r)$ be the rational cuspidal divisor class group of the Drinfeld modular curve $X_0(\mathfrak{p}^r)$ for a prime power level $\mathfrak{p}^r\in \mathbb{F}_q[T]$. We relate the rational cuspidal divisors of degree $0$ on $X_0(\mathfrak{p}^r)$ with $\Delta$-quotients, where $\Delta$ is the Drinfeld discriminant function. As a result, we are able to determine explicitly the structure of $\mathcal{C}(\mathfrak{p}^r)$ for arbitrary prime $\mathfrak{p}\in \mathbb{F}_q[T]$ and $r\geq 2$.