Modular Units and Cuspidal Divisor Class Groups of X1(n)

Abstract. In this article, we consider the group F∞ 1 (N) 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 F∞ 1 (N). This enables us to compute the group structure of the rational torsion subgroup C∞ 1 (N) 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 C∞ 1 (pn) for a regular prime p.