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.