Structure of cuspidal groups beyond the X₀(N) family

Characterize the abstract structure of cuspidal groups, or of suitable subgroups of cuspidal groups, for families of modular curves other than the modular curves X₀(N).

Background

The paper notes that the structure of cuspidal groups is well studied for modular curves X₀(N), where modular units constructed from Dedekind eta functions can often be used to determine the group structure. By contrast, the appearance of Bernoulli numbers in the orders of vanishing of modular units on intermediate modular curves makes analogous structural calculations more difficult.

The paper derives a direct-sum decomposition for the cuspidal group of the intermediate modular curves X_H(p) and gives explicit computations for bounded prime levels. However, the quoted passage identifies the broader structural determination of cuspidal groups, or relevant subgroups, for families beyond X₀(N) as unresolved. The discussion singles out cases with small index k as potentially more tractable.