A function field analogue of Ligozat's theorem for Drinfeld modular units

TL;DR

This paper establishes a function field analogue of Ligozat's theorem for Drinfeld modular units, conjectures a complete characterization, and applies these results to bound the rational cuspidal divisor class group on Drinfeld modular curves.

Contribution

It proves a criterion for Drinfeld modular units analogous to Ligozat's theorem and verifies it for specific levels, advancing understanding of modular units in function fields.

Findings

Established a criterion for Drinfeld modular units on $X_0( frak{n})$

Verified the conjecture for prime power and product of two primes levels

Provided an explicit upper bound for the exponent of the rational cuspidal divisor class group

Abstract

Fix a nonzero level $\mathfrak{n} \in \mathbb{F}_q[T]$. In this paper, we first establish a function field analogue of Ligozat's theorem, which serves as our main result and provides a criterion for Drinfeld modular units on the Drinfeld modular curve $X_0(\mathfrak{n})$. We further conjecture that this criterion characterizes all Drinfeld modular units; we verify the conjecture in the cases of prime power level and of level equal to the product of two primes. Second, as an application of Drinfeld modular units, we investigate the rational cuspidal divisor class group $\mathcal{C}(\mathfrak{n})$ of $X_0(\mathfrak{n})$. We construct an injective map $g$ from the group of degree $0$ rational cuspidal divisors on $X_0(\mathfrak{n})$ to the group of Drinfeld modular units on $X_0(\mathfrak{n})$ tensored with $\mathbb{Q}$ over $\mathbb{Z}$. As a result, we establish an explicit upper bound for the exponent of $\mathcal{C}(\mathfrak{n})$ for general level $\mathfrak{n}$.