Rational torsion points on Jacobians of modular curves (Q2804243)

The cuspidal subgroup \({\mathcal C}(N)\) of the Jacobian \(J_0(N)\) of the modular curve \(X_0(N)\) is the subgroup generated by the degree zero divisors on \(X_0(N)\) of the form \(\alpha-\beta\), where \(\alpha\) and \(\beta\) are cusps. It was shown by \textit{B. Mazur} in [Publ. Math., Inst. Hautes Étud. Sci. 47, 33--186 (1977; Zbl 0394.14008)] that \({\mathcal C}(N)={\mathcal T}(N)\), the group of rational torsion points in \(J_0(N)\). More recently, \textit{M. Ohta} [Tokyo J. Math. 37, No. 2, 273--318 (2014; Zbl 1332.11061)] proved that the corresponding \(\ell\)-primary subgroups \({\mathcal T}(N)[\ell^\infty]\) and \({\mathcal C}(N)[\ell^\infty]\) also coincide, for \(\ell\geq 5\) and also for \(\ell=3\) if \(3\) does not divide~\(N\).NEWLINENEWLINEOhta's proof involves analysing a quotient of the ring \({\mathbb T}(N)\otimes{\mathbb Z}_\ell\), where \({\mathbb T}(N)\) is generated by the Hecke operators \(T_r\) for primes \(r\) not dividing \(N\) and the Atkin-Lehner operators \(\omega_p\) for \(p|N\). The ideal concerned is the Eisenstein ideal \({\mathcal I}_0\) generated by the \(T_r-r-1\). In this paper, the author modifies Ohta's proof by replacing \(\omega_p\) with \(T_p\). The corresponding ring \({\mathbb T}(N)'\) and the quotient \({\mathbb T}(N)'\otimes{\mathbb Z}_\ell/{\mathcal I}_0\) do not have such a clear structure in general, but the case \(N=pq\) for primes \(p\neq q\) can be analysed.NEWLINENEWLINEDoing this, the author is able to prove a slight extension of Ohta's result: namely, it remains valid for \(\ell=p=3\) and \(N=3q\), unless perhaps \(q\equiv 1\pmod 9\) and \(3^{(q-1)/3}\equiv 1\pmod q\).