The rational cuspidal subgroup of with M squarefree

For a positive integer N, let (Figure presented.) be the modular curve over (Figure presented.) and (Figure presented.) its Jacobian variety. We prove that the rational cuspidal subgroup of (Figure presented.) is equal to the rational cuspidal divisor class group of (Figure presented.) when (Figure presented.) for any prime p and any squarefree integer M. To achieve this, we show that all modular units on (Figure presented.) can be written as products of certain functions (Figure presented.), which are constructed from generalized Dedekind eta functions. Also, we determine the necessary and sufficient conditions for such products to be modular units on (Figure presented.) under a mild assumption.