2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/143580In 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 $p$.43 pagesNumber TheoryAlgebraic Geometry11G16; 11G18; 11F03; 14G05text