Let X be a curve of genus g≥ 2 over a number field F of degree d=[F: Q]. The conjectural
existence of a uniform bound N (g, d) on the number# X (F) of F-rational points of X is an
outstanding open problem in arithmetic geometry, known by the work of Caporaso, Harris,
and Mazur to follow from the Bombieri–Lang conjecture. A related conjecture posits the
existence of a uniform bound N tors,†(g, d) on the number of geometric torsion points of the
Jacobian J of X which lie on the image of X under an Abel–Jacobi map. For fixed X, the …

We show that there is a bound depending only on g, r and [K: Q] for the number of K-rational
points on a hyperelliptic curve C of genus g over a number field K such that the Mordell–Weil
rank r of its Jacobian is at most g− 3. If K= Q, an explicit bound is 8rg+ 33 (g− 1)+ 1. The
proof is based on Chabauty's method; the new ingredient is an estimate for the number of
zeros of an abelian logarithm on a p-adic 'annulus' on the curve, which generalizes the
standard bound on disks. The key observation is that for a p-adic field k, the set of k-points …