Number Theory Seminar

In 1977 Mazur proved that the rational torsion subgroup of the Jacobian of the modular curve $X_0(N)$, $N > 5$ prime, is generated by the linear equivalence class of the difference of the two cusps. More generally, it is conjectured that for a general $N$, the rational torsion subgroup of the Jacobian of $X_0(N)$ is generated by cusps.  In this talk, we'll discuss a generalisation of this to other modular curves, namely certain covers of $X_0(N)$, indexed by subgroups of $(\mathbf{Z}/N\mathbf{Z})^\times$.