Modular Units on $X_1(p)$ and Quotients 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.