Bernoulli determinants and cuspidal subgroups

We give an explicit formula for the order of the rational cuspidal class group of the modular curve $X_1(N)$ for an arbitrary integer $N$.

The proof relies on results of Streng on the group of modular units on $X_1(N)$, and requires computing a certain determinant involving the second Bernoulli polynomial.

We also define a higher weight analogue of the cuspidal class group and speculate that its order is related to a similar determinant defined using a higher degree Bernoulli polynomial.