Effective Diophantine Geometry over Function Fields

Kim 9701489 该奖项为M提供资金。金继续他过去的研究丢番图几何曲线的功能领域的任意特征。 用初等语言来说，这是对f（x，y，t）=0（1）类型方程的研究，将其视为二元多项式方程，其系数是t的函数，我们寻求其解（x，y）=（p（t），q（t）），即满足f（p（t），q（t），t）=0作为t的函数的多项式对（或有理函数）。 特别是，他将继续他以前的工作'有效'莫德尔代数的功能领域，这使得人们找到所有的解决方案（1）（为f的高属）给予先验界的'高度'（在这种情况下是程度）的解决方案对（p（t），q（t））。 这是丢番图几何中的一个突出问题，它的解决将提供一个粗略的算法来寻找有理数上的二元丢番图方程的所有有理解。 具体的项目有三个密切相关的方向：（1）发现更精细的几何高度不等式在任意特征，作为有效的输入到算法搜索的解决方案;（2）嵌入PI的以前的工作到一般类型的曲面上的曲线的研究在Bogomolov的有界性定理的精神，有界亏格曲线的有界性，以及他与Shepherd-Barron的联合工作;（3）从更显式的算法观点来处理几何高度不等式，希望能找到比现有方法更容易转移到数域的方法。 这个项目福尔斯属于算术几何的一般领域-一个融合了两个最古老的数学领域：数论和几何的主题。 事实证明，这种结合非常富有成效--最近解决了几代人都无法解决的问题。 在它的许多后果是新的纠错码。 这种代码对于现代计算机（硬盘）和光盘都是必不可少的。

Kim 9701489 This award provides funds for M. Kim to continue his past research on the Diophantine geometry of curves over function fields of arbitrary characteristic. In elementary language, this is the study of equations of type f(x,y,t)=0 (1) viewed as a two-variable polynomial equation whose coefficients are functions of t to which we seek solutions (x,y)=(p(t), q(t)), that is, pairs of polynomials (or rational functions) which satisfy f(p(t),q(t),t)=0 as a function of t. In particular, he will continue his previous work on `effective' Mordell conjectures over function fields, which allows one to find all solutions to (1) (for f's of high genus) by giving a priori bounds on the `height' (which in this case is the degree) of solution pairs (p(t),q(t)). A result of this sort over number fields is the long-term eventual goal of this research This is an outstanding problem in Diophantine geometry whose resolution would provide a crude algorithm for finding all rational solutions to Diophantine equations in two variables over the rational numbers. The specific projects go in three closely related directions: (1) finding more refined geometric height inequalities in arbitrary characteristic to serve as efficient input into algorithmic search for solutions; (2) embedding the PI's previous work into the study of curves on surfaces of general type in the spirit of Bogomolov's theorem on boundedness of curves of bounded genus, and that of his joint work with Shepherd-Barron; (3) approaching geometric height inequalities from a more explicitly algorithmic viewpoint in the hope of finding techniques which would transfer more readily to number fields than the existing ones. This project falls into the general area of arithmetic geometry - a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful - having recently solved problems that withstood generations. Among its many consequences are new error corre cting codes. Such codes are essential for both modern computers (hard disks) and compact disks.