Modular Units on $X_{1}( p)$ and Quotients of the Cuspidal Group

TL;DR

This work constructs an explicit basis for the group of modular units $\mathcal{F}(p)$ on the modular curve $X_{1}(p)$ with $p\ge5$ by forming products of Siegel functions and proving their invariance under $\Gamma_{1}(p)$. Using this basis, the authors determine the structure of the cuspidal group $\mathcal{C}_{1}(p)$ and its rational subgroup, showing that $\mathcal{C}_{1}^{\infty}(p)=\mathcal{C}_{1}^{\mathbb{Q}}(p)=\mathcal{C}_{1}(p)(\mathbb{Q})$, and providing explicit generators and orders for cuspidal divisors such as $D=P_0-Q_0$ and $D'=P_0-P_{n-1}$. The paper develops sub-bases for divisors supported on particular cusps and for Galois-fixed divisors, enabling efficient computation of the cuspidal groups via linear algebra and Smith normal form. A key technical tool is a detailed circulant-matrix analysis that yields explicit formulas for the orders of cuspidal divisors, including a closed form for the order of $D$. The results yield practical methods to compute and understand the cuspidal structure for many primes and provide insights into the large quotients of the cuspidal group on $X_{1}(p)$.

Abstract

Modular units are functions on modular curves whose divisors are supported on the cusps. They form a free abelian group of rank at most one less than the number of cusps. In this paper we study the group of modular units on $X_{1}( p )$, with prime level $p \ge 5$. We give an explicit basis for this group and study certain rational subgroups of it. We use the basis to numerically investigate the structure of the cuspidal group of $X_{1}( p)$ and its rational subgroup. In the later stages of this paper we use our basis to determine a specific large quotient of the cuspidal group.

Theorems & Definitions (50)

• Definition 1.1
• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 4
• Proposition 5
• Remark 2.1
• Proposition 6
• proof
• Proposition 7