Modular units and cuspidal divisor class groups of X-1(N)

Abstract

In this article, we consider the group F-1(infinity)(N) of modular units on X-1(N) that have divisors supported on the cusps lying over infinity of X-0(N), called the infinity-cusps. For each positive integer N, we will give an explicit basis for the group F-1(infinity)(N). This enables us to compute the group structure of the rational torsion subgroup l(1)(infinity)(N) of the Jacobian J(1)(N) of X-1(N) generated by the differences of the infinity-cusps. In addition, based on our numerical computation, we make a conjecture on the structure of the p-primary part of l(1)(infinity)(p(n)) for a regular prime p. (C) 2009 Elsevier Inc. All rights reserved.