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

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

2007-12-05

arxiv.org/abs/0712.0629v1

Mathematics

Number Theory

43 pages

Scientific paper

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 $p$.

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