Euler Characteristics and Lifting Problems in Arithmetic Geometry

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This proposal concerns research by Prof. Ted Chinburg on arithmetic geometryand Galois theory. The first goal of the proposal is to study equivariant Riemann Roch theorems. This subject concerns Euler characteristics associated to actions of groups on objects arising from arithmetic geometry. New methods for determining coherent Euler characteristics will be developed and applied to study modular forms and conjectures about L-series. In particular, some predictions about the Mordell-Weil groups and Tate-Shafarevich groups ofJacobians of modular curves will be considered. The second goal of the proposalis to study equivariant Weil-etale cohomology and its generalizations, alongwith invariants in derived categories arising from group actions on schemes.The third goal of the proposal is to continue work on a fundamental finitenessproblem concerning how finite groups of automorphisms act on the homogeneouscoordinate rings of varieties in positive characteristic. The final goal ofthe proposal is to study the liftability of group actions on curves in positivecharacteristic. In particular, a conjecture about those groups for which lifts to characteristic 0 always exist will be considered.The broader context of this proposal is the study of symmetry as it pertains to algebra and number theory. Symmetry has been a guiding principle in algebrasince the work of Galois on the solvability of equations by radicals in the 1830's.By studying the symmetries which solutions of algebraic problems must have it they exist, one can in many cases show that no such solutions exist, or that the solutions are strongly constrained. This proposal will develop this principle in some new directions. The goal is to apply the principle to prove some central conjectures about L-series and the relation between being able to solve systems of equations modulo prime numbers and being able to solve them exactly with integers or with whole numbers. The study of solutionsof equations in integers and modulo prime numbers has been of practical significance to cryptography and to the construction of error correcting codes.