Arithmetic Geometry of Diophantine Problems

丢番图问题的算术几何

• 批准号: 0071921
• 负责人: Lucien Szpiro
• 金额: $ 18.3万
• 依托单位: Research Foundation of the City University of New York
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0071921&HistoricalAwards=false
• 关键词: Arithmetic; Geometry; Diophantine; Problems

Abstract for 0071921- SzpiroTechnical description: This is a project in the arithmetic algebraic geometry of diophantine problems. The PI is continuing his work which was successfully used by Faltings in the solution of the Mordell Conjecture and by E. Ullmo, S. Zhang and the PI in the solution of the Bogomolov Conjecture. The use of the modern theory of heights has been very effective. The PI views the Equidistribution Theorem as the solution to the problem of finding lower bounds for solutions of algebraic equations. The question of finding effective upper bounds for solutions of algebraic equations leads to many conjectures. Some are very well known and may be unattainable (the abc conjecture, or the discriminant conjecture for elliptic curves) but the PI has always believed that attacking difficult problems is the secret of success in doing high-level mathematics. The project will concentrate on: a) The study of the degree of Belyi maps (these are coverings of the Riemann sphere ramified in only 3 points and they characterize curves defined over the field of algebraic numbers). b) The study of the Zariski closure of the non-zero p-division points (for p big enough) in an abelian variety as a finite scheme. c) The consequences for the Tate-Shafarevich group of recent results of the PI and J. Pesenti on the discriminant inequality for potential good reduction. d) Dynamical Systems (first on the Sphere then on towers of Shimura varieties): The canonical height associated to these objects should lead to equidistribution statements, for example for CM points (cf the work of Duke).Non technical description: The PI and his collaborators are studying a subject first investigated by Diophantus in ancient Greece: find the solutions in integers of algebraic equations. The modern attack uses algebraic geometry, analysis, and geometry. Many problems in the natural world (asking: How many times? How to decipher?) require a solution in integers. This no doubt explains why number theory, like physics, has been a constant motivation for the development of mathematics.

摘要 0071921- Szpiro 技术描述：这是丢番图问题的算术代数几何中的一个项目。 PI 正在继续他的工作，该工作已被 Faltings 成功用于解决 Mordell 猜想，并被 E. Ullmo、S. Zhang 和 PI 成功用于解决 Bogomolov 猜想。 现代高度理论的运用非常有效。 PI 将均匀分布定理视为寻找代数方程解下界问题的解决方案。 寻找代数方程解的有效上限的问题导致了许多猜想。 有些是众所周知的，可能是无法实现的（abc猜想，或者椭圆曲线的判别猜想），但PI始终相信，攻克难题是做高级数学成功的秘诀。 该项目将集中于： a) Belyi 映射次数的研究（这些是仅由 3 个点分支的黎曼球面的覆盖，它们描述了在代数数域上定义的曲线）。 b) 研究阿贝尔簇中非零 p 划分点（对于 p 足够大）的 Zariski 闭包作为有限方案。 c) PI 和 J. Pesenti 关于潜在良好减少的判别不平等的最新结果对 Tate-Shafarevich 小组的影响。 d) 动力系统（首先在球体上，然后在 Shimura 品种的塔上）：与这些物体相关的规范高度应导致均匀分布陈述，例如 CM 点（参见 Duke 的工作）。非技术描述：PI 和他的合作者正在研究由古希腊丢番图首先研究的一个课题：求代数方程整数的解。 现代攻击使用代数几何、分析和几何。 自然界中的许多问题（问：多少次？如何破译？）需要整数解。 这无疑解释了为什么数论像物理学一样一直是数学发展的持续动力。