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

Sheng-Yang Kevin Ho

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 $\mathfrak{n}$.

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

Abstract

Fix a nonzero level . 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 . 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 of . We construct an injective map from the group of degree rational cuspidal divisors on to the group of Drinfeld modular units on tensored with over . As a result, we establish an explicit upper bound for the exponent of for general level .
Paper Structure (5 sections, 12 theorems, 66 equations)

This paper contains 5 sections, 12 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. The transformation law for $D_{\mathfrak{p}\mathfrak{q}}(\gamma' z)$
  3. Proof of Theorem \ref{['Main Theorem']}
  4. The bridge map $g$ between rational cuspidal divisors and modular units on $X_0(\mathfrak{n})$
  5. Applications

Key Result

Theorem 1.1

(Ligozat Ligozat_1975) Let $N$ be a positive integer. Then $g_{\underline{r}}$ is a modular function on $X_0(N)$, i.e., a meromorphic function on $\mathbb{H}\cup \mathbb{P}^1(\mathbb{Q})$ invariant under the action of $\Gamma_0(N)$, if and only if the following conditions are true:

Theorems & Definitions (30)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3: A function field analogue of Ligozat’s theorem
  • Remark 1.4
  • Conjecture 1.5
  • Theorem 1.6
  • Proposition 1.7: cf. ho_rational_2024
  • Remark 1.8
  • Theorem 1.9: cf. papikian_rational_2017 and ho_rational_2024
  • Remark 1.10
  • ...and 20 more