  In this article, we consider the group $F_1^\infty(N)$ of modular
  units on $X_1(N)$ that have divisors supported on the cusps lying
  over $\infty$ of $X_0(N)$, called the $\infty$-cusps. For each
  positive integer $N$, we will give an explicit basis for the group
  $F_1^\infty(N)$. This enables us to compute the group structure of
  the rational torsion subgroup $C_1^\infty(N)$ of the Jacobian
  $J_1(N)$ of $X_1(N)$ generated by the differences of the
  $\infty$-cusps. In addition, based on our numerical
  computation, we make a conjecture on the structure of the
  $p$-primary part of $C_1^\infty(p^n)$ for a regular prime $p$.
