Effective Diophantine Geometry over Function Fields

这个奖为M. Kim继续他过去关于任意特征函数场上曲线的丢番图几何的研究提供了资金。在初等语言中，这是对f(x,y,t)=0(1)型方程的研究，将其视为一个双变量多项式方程，其系数是t的函数，我们寻求解(x,y)=(p(t), q(t))，也就是说，多项式对（或有理函数）满足f(p(t),q(t),t)=0作为t的函数。特别是，他将继续他之前关于函数域上的“有效”莫德尔猜想的工作，它允许我们通过给出解对(p（t) q(t)）的“高度”（在这种情况下是度）的先验界来找到(1)（对于高属f）的所有解。这是丢图图几何中的一个突出问题，它的解决将为在有理数上寻找两个变量的丢图图方程的所有有理数解提供一个粗略的算法。具体项目有三个密切相关的方向：(1)在任意特征中找到更精细的几何高度不等式，作为算法搜索解决方案的有效输入；(2)以Bogomolov关于有界属曲线有界性的定理及其与Shepherd-Barron的合作成果的精神，将PI先前的工作嵌入到一般曲面上曲线的研究中；(3)从更明确的算法角度研究几何高度不等式，希望找到比现有技术更容易转移到数字领域的技术。这个项目属于算术几何的一般领域，它融合了数学中两个最古老的领域：数论和几何。事实证明，这一组合非常富有成效——最近解决了几代人经受住了考验的问题。其诸多后果之一是新的纠错代码。这些代码对现代计算机（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.