I will explain some new connections between the $abc$ conjecture and modular forms. In particular, I will outline a proof of a new unconditional estimate for the $abc$ conjecture, which lies beyond the existing techniques in this context. The proof involves a number of tools such as Shimura curves, CM points, analytic number theory, and Arakelov geometry. It also requires some intermediate results of independent interest, such as bounds for the Manin constant beyond the semi-stable case. If time permits, I will also explain some results towards Szpiro's conjecture over totally real number fields which are compatible with the discriminant term appearing in Vojta's conjecture for algebraic points of bounded degree.[-]

In this talk we will see that there are only finitely many modular curves that admit a smooth plane model. Moreover, if the degree of the model is greater than or equal to 19, no such curve exists. For modular curves of Shimura type we will show that none can admit a smooth plane model of degree 5, 6 or 7. Further, if a modular curve of Shimura type admits a smooth plane model of degree 8 we will see that it must be a twist of one of four curves.

This is joint work with Samuele Anni and Eran Assaf.[-]

We investigate the geometry of Galois deformation rings in the defect zero setting but when the Taylor-Wiles hypothesis does not hold. In particular, we consider the question of whether or not the map from the local deformation ring to the global deformation ring is a local complete intersection map and the role the Taylor-Wiles hypothesis plays in this question. We exhibit an example in the context of classical weight two modular forms where this does not hold and shows that a resulting Tor algebra acts on the cohomology of a modular orbifold. This is joint work in progress with Preston Wake.[-]