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

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In this article, we consider the group F₁∞(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 F₁∞(N). This enables us to compute the group structure of the rational torsion subgroup C₁∞(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 C₁∞(pⁿ) for a regular prime p.