Modular unit and cuspidal divisor class groups of X_1(N)

TL;DR

This paper explicitly constructs modular units on $X_1(N)$ supported on $ty$-cusps, computes the structure of the associated torsion subgroup of the Jacobian, and conjectures about its $p$-primary parts.

Contribution

It provides an explicit basis for the group of modular units supported on $ty$-cusps and determines the structure of the related torsion subgroup of the Jacobian.

Findings

Explicit basis for $F_1^ty(N)$ constructed.

Structure of $C_1^ty(N)$ computed.

Conjecture on the $p$-primary part of $C_1^ty(p^n)$ for regular primes.

Abstract

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 $p$.