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

TL;DR

This paper explicitly constructs a basis for modular units on the modular curve $X_1(p)$, investigates their rational subgroups, and analyzes a significant quotient of the cuspidal group for prime levels $p \,\geq\ 5$.

Contribution

It provides an explicit basis for modular units on $X_1(p)$ and uses it to study the structure and quotients of the cuspidal group, advancing understanding of modular curve arithmetic.

Findings

Explicit basis for modular units on $X_1(p)$

Numerical insights into the cuspidal group structure

Identification of a large quotient of the cuspidal group

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.