Euler Characteristics,Qquadratic Invariants, Arithmetic Groups and Lifting Problems

本提案是关于算术几何中的几个主题。将研究相干欧拉特征的黎曼洛克公式，并将其应用于Iwasawa理论和容量理论。该建议还涉及到相干束和单位群的二次不变量。数论和代数几何中的各种方法将用于证明算术群可以由小高度的元素或小秩的子群生成。最后，考虑了变形环的逆问题，以及群作用在正特征曲线上的可升性。这个提议是关于对称及其在数论和算术几何中的应用。利用对称性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，在许多情况下，由数论和几何产生的复杂代数系统可以用简单而紧凑的方式来描述。对称性还将用于区分这些系统何时彼此之间存在根本差异，以及它们何时可以或不能扩展到其原始范围之外。

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.