Modular units and cuspidal divisor classes on X0(n2M) with n|24 and M squarefree

Abstract

For a positive integer N, let C(N) be the subgroup of J0(N) generated by the equivalence classes of cuspidal divisors of degree 0 and C(N)(Q):=C(N)∩J0(N)(Q) be its Q-rational subgroup. Let also CQ(N) be the subgroup of C(N)(Q) generated by Q-rational cuspidal divisors. We prove that when N=n2M for some integer n dividing 24 and some squarefree integer M, the two groups C(N)(Q) and CQ(N) are equal. To achieve this, we show that all modular units on X0(N) on such N are products of functions of the form η(mτ+k/h), mh2|N and k∈Z and determine the necessary and sufficient conditions for products of such functions to be modular units on X0(N).