Sharp bounds for the number of rational points on algebraic curves and dimension growth, over all global fields

Let (Formula presented.) be an algebraic curve over a number field (Formula presented.), and denote by (Formula presented.) the degree of (Formula presented.) over (Formula presented.). We prove that the number of (Formula presented.) -rational points of height at most (Formula presented.) in (Formula presented.) is bounded by (Formula presented.) where (Formula presented.) are absolute constants. We also prove analogous results for global fields in positive characteristic, and, for higher dimensional varieties. The quadratic dependence on (Formula presented.) in the bound as well as the exponent of (Formula presented.) are optimal; the novel aspect is the quadratic dependence on (Formula presented.) which answers a question raised by Salberger. We derive new results on Heath-Brown and Serre's dimension growth conjecture for global fields, which generalize in particular the results by the first two authors and Novikov from the case (Formula presented.). The proofs however are of a completely different nature, replacing the real analytic approach previously used by the (Formula presented.) -adic determinant method. The optimal dependence on (Formula presented.) is achieved by a key improvement in the treatment of high multiplicity points on mod (Formula presented.) reductions of algebraic curves.