Uniformity in Mordell–Lang for curves

Consider a smooth, geometrically irreducible, projective curve of genus $g\ge 2$ defined over number field degree $d \ge 1$. It has at most finitely many rational points by the Mordell Conjecture, theorem Faltings. We show that is bounded only in terms $g$, $d$ and Mordell–Weil rank curve's Jacobian, thereby answering affirmative question Mazur. In addition we obtain uniform bounds, $g$ $d$, for geometric torsion Jacobian which lie image an Abel–Jacobi map. Both estimates generalize our previous work one-parameter families. Our proof uses Vojta's approach to key new ingredient generalization height inequality due second- third-named authors.