Modular unit and cuspidal divisor class groups of X_1(N)
Abstract
Description
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 $p$.
43 pages
43 pages