The rational cuspidal divisor class group of X0(N)

Abstract

For any positive integer N, we completely determine the structure of the rational cuspidal divisor class group of X0(N), which is conjecturally equal to the rational torsion subgroup of J0(N). More specifically, for a given prime ℓ, we construct a rational cuspidal divisor Zℓ(d) for any non-trivial divisor d of N. Also, we compute the order of the linear equivalence class of Zℓ(d) and show that the ℓ-primary subgroup of the rational cuspidal divisor class group of X0(N) is isomorphic to the direct sum of the cyclic subgroups generated by the linear equivalence classes of Zℓ(d).