Euler Characteristics,Qquadratic Invariants, Arithmetic Groups and Lifting Problems

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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.