Euler Characteristics,Qquadratic Invariants, Arithmetic Groups and Lifting Problems

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

This proposal is about several topics in arithmetic geometry. Riemann Roch formulas for coherent Euler characteristics will be 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 and algebraic geometry will be used to show that arithmetic groups can 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 positive characteristic. This proposal is about symmetry and its applications in number theory and arithmetic geometry. The exploitation of symmetry to understand the solutions of equations and the geometry of objects has been a basic theme in mathematics. In this proposal, 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.