On the rational cuspidal subgroup and the rational torsion points of $J_0(pq)$

Abstract

For two distinct prime numbers p, q, we compute the rational cuspidal subgroup C(pq) of Jo(pq) and determine the e-primary part of the rational torsion subgroup of the old subvariety of Jo(pq) for most primes e. Some results of Berkovic on the nontriviality of the Mordell-Weil group of some Eisenstein factors of Jo(pq) are also refined.