Arithmetic Geometry of Diophantine Problems

技术说明：这是算术代数几何中丢番图问题的一个课题。PI正在继续他的工作，这一工作曾被Faltings成功地用于解决莫德尔猜想，并被E. Ullmo， S. Zhang和PI成功地用于解决Bogomolov猜想。现代高度理论的运用非常有效。PI将等分布定理视为求解代数方程解的下界问题的解。求代数方程解的有效上界的问题引起许多猜想。有些是非常著名的，可能是无法实现的（abc猜想，或椭圆曲线的判别猜想），但PI一直认为，攻克难题是在高级数学中取得成功的秘诀。该项目将集中于：a) Belyi映射度的研究（这些映射是黎曼球的覆盖，仅在3个点上分叉，它们表征在代数数域上定义的曲线）。b)有限格式下的非零p-分点（对于p足够大）的Zariski闭包的研究。c) PI和J. Pesenti关于歧视性不平等的潜在好的减少的最新结果对Tate-Shafarevich组的影响。d)动力系统（首先在球体上，然后在志村变种的塔上）：与这些对象相关的规范高度应该导致等分布陈述，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古希腊丢番图（Diophantus）最先研究的课题：找到代数方程的整数解。现代攻击使用代数几何、分析和几何。自然界的许多问题(问：多少次？如何解密？)需要一个整数形式的解。这无疑解释了为什么数论，像物理学一样，一直是数学发展的动力。

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.