Branched Galois Covers of Curves: Lifting and Reduction

曲线的分支伽罗瓦覆盖：提升和归约

基本信息

批准号：
1900396
负责人：
Andrew Obus
金额：
$ 2.88万
依托单位：
CUNY Baruch College
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-09-01 至 2020-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1900396&HistoricalAwards=false
关键词：
Branched Galois Covers Curves Lifting

项目摘要

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.

该研究项目涉及代数几何，即多变量多项式方程组的研究。该项目的重点是针对某些素数 p 以特征 p 表示的方程。 “独特的 p 世界”是这样一个世界：你每向前迈出 p 一步，你就会回到起点。这些数学系统绝不是深奥的，它描述了一周中的每一天（p = 7）、计算机体系结构的基础知识（p = 2）以及重要密码系统的设置。特征 p 中的代数几何是对在此设置下多项式方程的解给出的几何对象（“种类”）的研究。该项目研究了特征 p 和特征 0（“标准”代数几何）中的簇之间的关系，目的是阐明这两个世界。研究生将主要参与该研究项目。这项研究将得到本科生数学俱乐部的教育活动的补充。该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 上升到特征零，并获取有关该特征的几何和算术的信息。电梯。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-adic 域上曲线的伽罗瓦覆盖积分模型。涉及变形数据的新工具揭示了具有复杂伽罗瓦群的曲线伽罗瓦覆盖的稳定模型。这些工具将被进一步利用，以便尽可能明确地理解伽罗瓦覆盖的稳定模型。