Euler Characteristics,Qquadratic Invariants, Arithmetic Groups and Lifting Problems

这项建议是关于算术几何的几个主题。Riemann Roch公式的相干欧拉特征将被研究，并应用于岩泽理论和容量理论。该建议也涉及相干波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被用来证明算术群可以由小高度的元素或由小秩子群生成。最后，将考虑变形环的反问题，以及正特征曲线上群作用的可升性。这一建议是关于对称性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一直是数学中的一个基本主题。在这个方案中，对称性将被用来表明，在许多情况下，从数论和几何中产生的复杂的代数系统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彼此根本不同，以及它们何时可以或不能扩展到超出其原始范围。

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.