I will discuss the Brill-Noether theory of a general elliptic 𝐾3 surface using wall-crossing with respect to Bridgeland stability conditions. As an application, I will provide an example of a general 𝑘-gonal curve from the perspective of Hurwitz-Brill-Noether theory. This is joint work with Gavril Farkas and Andrés Rojas.

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 $(\mathbf{Z}/N\mathbf{Z})^\times$.