论文标题
Mordell-Lang的均匀性
Uniformity in Mordell-Lang for curves
论文作者
论文摘要
考虑一个平稳的，几何不可约的，投射曲线的属$ g \ ge 2 $在数量的$ d \ ge 1 $的字段上定义的。 Mordell猜想是fal的定理，它最多有许多理性的观点。我们表明，理性点的数量仅以$ g $，$ d $和曲线的雅各比式的mordell-weil等级而有限，从而在Mazur的肯定问题中回答。此外，我们以$ g $和$ d $的数量获得了统一的界限，用于雅各比式的几何扭转点的数量，这是Abel-Jacobi地图的图像。两种估计都将我们以前的工作概括为$ 1美元的参数家庭。我们的证明使用VOJTA对Mordell猜想的方法，关键的新成分是由于第二名和第三名作者而导致高度不等式的概括。
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 bounded, 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 $1$-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.
