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

考虑定义在度数为 $d\ge 1$ 的数域上的属 $g\ge 2$ 的平滑的、几何上不可约的投影曲线。根据莫德尔猜想（法尔廷斯定理），它最多具有有限多个有理点。我们证明有理点的数量仅在 $g$、$d$ 和曲线雅可比的 Mordell-Weil 秩方面有界，从而肯定地回答了 Mazur 的问题。此外，对于位于 Abel-Jacobi 映射图像中的雅可比几何扭转点的数量，我们获得了统一的边界，以 $g$ 和 $d$ 表示。两种估计都概括了我们之前对单参数族的工作。我们的证明使用了 Vojta 的 Mordell 猜想方法，关键的新成分是由第二和第三名作者引起的高度不等式的推广。