Bounds on the Chabauty–Kim Locus of Hyperbolic Curves

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/Q of genus g ≥ 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.