Euler Characteristics,Qquadratic Invariants, Arithmetic Groups and Lifting Problems

欧拉特性、Q二次不变量、算术群和提升问题

• 批准号: 1100355
• 负责人: Ted Chinburg
• 金额: $ 15万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1100355&HistoricalAwards=false
• 关键词: Euler; Characteristics; Qquadratic; Invariants; Arithmetic

This proposal is about several topics in arithmetic geometry.Riemann Roch formulas for coherent Euler characteristics willbe studied, with applications to Iwasawa theory and capacity theory.The proposal also has to do with quadratic invariants of coherent sheaves and of unit groups. Various methods from number theory andalgebraic geometry will be used to show that arithmetic groupscan be generated by elements of small height or by subgroups of small rank. Finally,the inverse problem for deformation rings will be considered,as well as the liftability of group actions on curves in positivecharacteristic.This proposal is about symmetry and its applications in numbertheory and arithmetic geometry. The exploitation of symmetryto understand the solutions of equations and the geometryof objects has been a basic theme in mathematics. In thisproposal, symmetry will be used to show that complicated algebraic systems arising from number theory and geometry can in many cases be described in simple and compact ways. Symmetries will also be used to distinguish when such systems are fundamentally different from one another and when they can or cannot be extended beyond their original scope.

本文是关于算术几何中的几个问题的建议.研究相干欧拉特征线的Riemann Roch公式，并应用于Iwasawa理论和容量理论.建议还涉及相干层和单位群的二次不变量. 从数论和代数几何的各种方法将被用来表明算术群可以由小高度的元素或由小秩的子群生成。 最后，讨论了变形环的反问题，以及正特征曲线上的群作用的可升性，这是关于对称性及其在数论和算术几何中的应用。 利用数学知识来理解方程的解和物体的几何形状一直是数学中的一个基本主题。 在这个建议中，对称性将被用来表明，从数论和几何中产生的复杂代数系统在许多情况下可以用简单而紧凑的方式来描述。对称性也将被用来区分这些系统何时从根本上不同于另一个系统，以及它们何时可以或不可以扩展到其原始范围之外。