Euler Characteristics and Lifting Problems in Arithmetic Geometry

这一建议涉及研究教授特德Chinburg算术几何和伽罗瓦理论。 该提案的第一个目标是研究等变Riemann Roch定理。 这门课涉及到欧拉特征线，这些特征线与由算术几何产生的物体上的群的作用有关。 发展了确定拟序欧拉特征线的新方法，并将其应用于L-级数的模形式和模结构的研究。 特别地，将考虑关于模曲线的Jacobian的Mordell-Weil群和Tate-Shafarevich群的一些预测。 第二个目标是研究等变Weil-etale上同调及其推广，以及由群作用于概型而产生的导出范畴中的不变量;第三个目标是继续研究一个基本的有限性问题，即有限自同构群如何作用于正特征簇的广义坐标环。 该方案的最终目标是研究曲线上正特征群作用的提升性。 特别是，一个猜想，这些团体的电梯的特征0总是存在将被考虑。更广泛的背景下，这一建议是研究对称性，因为它涉及到代数和数论。 对称性一直是一个指导原则algebrassince工作伽罗瓦的可解性方程的自由基在19世纪30年代。通过研究对称性的解决方案的代数问题必须有它的存在，人们可以在许多情况下表明，没有这样的解决方案存在，或解决方案是强烈的约束。 这一建议将在一些新的方向上发展这一原则。 目标是应用该原理证明一些关于L-级数的中心定理，以及能够解模素数方程组与能够用整数或整数精确求解方程组之间的关系。 整数和模素数方程解的研究对密码学和纠错码的构造具有实际意义。

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.