Bernoulli determinants and cuspidal subgroups
arXiv:2607.13536
summary
The paper derives an explicit formula for the size of the rational cuspidal class group of the modular curve X₁(N) using a determinant involving the second Bernoulli polynomial, and proposes a higher-weight analogue of this group.
Abstract
We give an explicit formula for the order of the rational cuspidal class group of the modular curve for an arbitrary integer . The proof relies on results of Streng on the group of modular units on , 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.
Topics & keywords
#modular curves#cuspidal class groups#bernoulli polynomials#modular units#determinant formulasX1(N)rational cuspidal class groupsecond Bernoulli polynomialmodular unitshigher weight analogue