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 |$X/{\mathbb{Q}}$| 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. [ABSTRACT FROM AUTHOR]