Modular units and cuspidal divisor class groups of X1(N)

In this article, we consider the group of modular units on X1(N) that have divisors supported on the cusps lying over ∞ of X0(N), called the ∞-cusps. For each positive integer N, we will give an explicit basis for the group . This enables us to compute the group structure of the rational torsion subgroup of the Jacobian J1(N) of X1(N) generated by the differences of the ∞-cusps. In addition, based on our numerical computation, we make a conjecture on the structure of the p-primary part of for a regular prime p.