Bounds on the Chabauty--Kim Locus of Hyperbolic Curves

TL;DR

This paper provides explicit bounds on the number of rational points on hyperbolic curves over rationals, conditional on major conjectures, using an advanced Chabauty--Kim method that generalizes previous bounds.

Contribution

It introduces an explicit upper bound on the Chabauty--Kim locus size for hyperbolic curves, extending prior bounds by incorporating the effective Chabauty--Kim method.

Findings

Conditional bounds depend on Tate--Shafarevich and Bloch--Kato conjectures.

Generalizes Coleman and Balakrishnan--Dogra bounds.

Relates bounds to genus, rank, and reduction types of the curve.

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 $g\geq2$ 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, generalising bounds found by Coleman and Balakrishnan--Dogra using the abelian and quadratic Chabauty methods.