Euler Characteristics and Lifting Problems in Arithmetic Geometry

这项建议涉及Ted Chinburg教授在算术几何和伽罗瓦理论方面的研究。该提议的第一个目标是研究等变Riemann Roch定理。这个主题涉及与群对由算术几何产生的对象的作用相关的欧拉特征。我们将发展确定相干欧拉特征的新方法，并将其应用于L级数的模形式和猜想的研究。特别地，将考虑模曲线的雅可比的Mordell-Weil群和Tate-Shafarevich群的一些预测。第二个目标是研究等变Weiletale上同调及其推广，以及由模式上的群作用产生的派生范畴中的不变量。第三个目标是继续研究有限自同构群如何作用于具有正特征的变种的齐次坐标环上的一个基本有限问题。该方案的最终目的是研究曲线上具有正特征的群动作的可升降性。特别是，将考虑关于特征提升到0的群总是存在的猜想。这一提议的更广泛的背景是对称性的研究，因为它与代数和数论有关。自从19世纪30年代伽罗瓦关于方程的根可解性的工作以来，对称性一直是代数的指导原则。通过研究代数问题的解必须存在的对称性，在许多情况下可以证明不存在这样的解，或者解是强约束的。这项提议将在一些新的方向上发展这一原则。目的是应用这一原理证明关于L级数的一些核心猜想，以及能解模素数方程组与能用整数或用整数精确解它们之间的关系。研究整数和模素数方程的解对于密码学和纠错码的构造都具有实际意义。

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.