Effective Diophantine Geometry over Function Fields

Kim 9701489这个奖项为M.Kim继续他过去在任意特征函数域上曲线的丢番图几何方面的研究提供了资金。在初等语言中，这是对f(x，y，t)=0(1)型方程的研究，它被认为是一个二元多项式方程，其系数是t的函数，我们对其求解(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(或有理函数)对。特别是，他将继续他先前关于函数域上的有效的Mordell猜想的工作，它允许人们通过给出解对(p(T)，q(T))的高度(在这种情况下是度)的先验界，从而找到(1)的所有解(对于高亏格的f)。在数域上进行这种排序的结果是本研究的长期最终目标，这是丢番图几何中的一个突出问题，它的解决将为寻找有理数上的二元丢番图方程的所有有理解提供一个粗略的算法。具体的项目沿着三个密切相关的方向进行：(1)找到更精确的任意特征的几何高度不等式，作为算法搜索的有效输入；(2)秉承Bogomolov关于有界亏格曲线的有界性的定理以及他与Shepherd-Barron的合作精神，将PI的先前工作嵌入到一般类型曲面上的曲线的研究中；(3)从更明确的算法观点来探讨几何高度不等式，希望找到比现有技术更容易转移到数域上的技术。这个项目属于算术几何的一般领域--一个融合了数论和几何这两个最古老的数学领域的学科。事实证明，这种结合非常有成效--最近解决了几代人经受住的问题。它的众多后果之一是新的纠错码。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.