Bounds on the Chabauty–Kim Locus of Hyperbolic Curves

Abstract

Abstract Conditionally on the Tate–Shafarevich and Bloch–Kato Conjectures, we give an explicit upper bound on the size of the $p$-adic Chabauty–Kim locus, and hence on the number of rational points, of a smooth projective curve $X/{\mathbb{Q}}$ of genus $g\geq 2$ in terms of $p$, $g$, 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, generalizing bounds found by Coleman and Balakrishnan–Dogra using the abelian and quadratic Chabauty methods.