Towards Uniform Chabauty--Kim

Conditionally on the Tate-Shafarevich and Bloch-Kato Conjectures, we give an explicit upper bound on the number of rational points on a smooth projective curve $X/\mathbb{Q}$ of genus $g\geq2$ in terms of $g$, the Mordell-Weil rank $r$ of its Jacobian, and the reduction types of $X$ at bad primes. This is achieved using the effective Chabauty-Kim method, generalising bounds found by Coleman and Balakrishnan-Dogra using the abelian and quadratic Chabauty methods.