Improved bounds for Szpiro's conjecture

In the direction of Szpiro's conjecture for all elliptic curves $E$ over $\mathbb{Q}$, the current best bound is $\log Δ\ll N \log N$ where $Δ$ is the absolute value of the minimal discriminant of $E$ and $N$ is the conductor of $E$ (the implicit constant is absolute). This bound dates back to 2013. We obtain the stronger bound $\log Δ\ll N\log \log N$ which was previously known only under GRH. In addition, for semistable elliptic curves we prove $\log Δ\ll N$. Our methods build on the theory developed by the author for approaching the $abc$ conjecture via Shimura curves.