Cuspidal class number of the tower of modular curves X 1(Np n )

Abstract

We consider a cuspidal class number, which is the order of a subgroup of the full cuspidal divisor class group of X 1(Np n ) with $${p\nmid N}$$ and n ≥ 1. By studying the second generalized Bernoulli numbers, we obtain results similar to ones (Ferrero and Washington in Ann Math (2) 109(2):377–395, 1979; Washington in Invent Math 49:87–97, 1978) about the relative class numbers of cyclotomic $${\mathbb{Z}_p}$$ -extension of an abelian number field.