Branched Galois Covers of Curves: Lifting and Reduction

这项研究项目涉及代数几何，即研究多变量多项式方程组。这个项目的重点是一些素数p用特征p表示的方程。“特征p世界”是这样一个世界，在这个世界里，你每向前走p步，你就会回到你开始的地方。这样的数学系统并不深奥，它们描述了一周中几天的进程(p=7)，计算机体系结构的基本原理(p=2)，以及重要密码系统的设置。特征p中的代数几何是研究在这种背景下由多项式方程的解所给出的几何对象(“簇”)。这个项目研究了特征p和特征零(“标准”代数几何)中的变化之间的关系，目的是揭示这两个世界。研究生将在这个研究项目中发挥重要作用。这项研究将由本科生数学俱乐部的教育活动来补充。本研究项目的第一部分致力于了解曲线的分支伽罗瓦覆盖何时从特征p提升到特征零，并获得关于提升的几何和算法的信息。特别是，该项目旨在获得所谓的“局部奥尔特群”和“弱局部奥尔特群”的分类。该项目的第二部分涉及分析p-adad场上曲线的伽罗瓦覆盖的积分模型。涉及变形数据的新工具揭示了具有复杂伽罗瓦群的曲线的伽罗瓦覆盖的稳定模型。将进一步利用这些工具，以便尽可能明确地理解伽罗瓦覆盖的稳定模型。

This research project concerns algebraic geometry, the study of systems of polynomial equations in many variables. The project focuses on equations formulated in characteristic p for some prime number p. The "characteristic p world" is one where, every time you take p steps forward, you wind up back where you start. Far from being esoteric, such mathematical systems describe the procession of the days of the week (p = 7), the fundamentals of computer architecture (p = 2), and also the setting for important cryptosystems. Algebraic geometry in characteristic p is the study of geometric objects ("varieties") given by solutions to polynomial equations in this setting. This project investigates the relationship between varieties in characteristic p and in characteristic zero ("standard" algebraic geometry), with the goal of shedding light on both worlds. Graduate students will take a major part in the research project. The research will be complemented by educational activities involving the undergraduate math club.The first part of this research project is devoted to understanding when branched Galois covers of curves lift from characteristic p to characteristic zero, and to obtaining information about the geometry and arithmetic of the lifts. In particular, the project aims to obtain a classification of the so-called "local Oort groups" and "weak local Oort groups". The second part of the project involves analyzing integral models of Galois covers of curves over p-adic fields. New tools involving deformation data shed light on stable models of Galois covers of curves with complicated Galois groups. These tools will be exploited further in order to understand stable models of Galois covers as explicitly as possible.