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The Algebra and Arithmetic of Splitting Fields

分裂域的代数和算术

基本信息

批准号：
2200845
负责人：
Asher Auel
金额：
$ 36万
依托单位：
Dartmouth College
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-08-01 至 2025-07-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200845&HistoricalAwards=false
关键词：
Algebra Arithmetic Splitting Fields

项目摘要

This project will explore algebraic and geometric objects that have played an important role in the development of many areas of mathematics and physics, known as central simple algebras and their splitting fields. The theory of quaternion algebras, examples of these structures, was utilized throughout the nineteenth and twentieth centuries in classical mechanics, quantum mechanics, and the theory of relativity, with current applications in fields such as aerospace modeling, computer graphics, robotics, and wireless communication. In mathematics, central simple algebras and the associated Brauer group have provided a bridge for techniques to be exchanged between diverse areas of research, from number theory and arithmetic geometry, to topology and algebraic geometry. This project aims to broaden a relatively new frontier between the splitting fields of central simple algebras and the arithmetic of elliptic curves, which are themselves important mathematical objects. The project will also support new pedagogical tools, opportunities for public outreach and engagement, early career mentoring and training opportunities for undergraduate and graduate students, and cross-disciplinary collaboration.More precisely, this project centers around three main problems: the period-index problem for the Brauer group, the problem of splitting central simple algebras by genus one curves, and explicit presentations of moduli spaces of elliptic curves. The techniques utilized---toroidal geometry, deformation theory, and Hilbert schemes---are widely employed in algebraic geometry, but less systematically so in pure algebra. While much recent progress in the period-index problem for the Brauer group has been through leveraging tools from algebraic geometry to control ramification splitting, this project adopts novel tools such as reciprocity sequences to achieve improved results as well as indicate long-term conjectural targets. The problem of whether any Brauer class is split by the function field of a genus one curve, while having only gained attention in the past decade, turns out to have surprising and deep connections to other important problems in algebra, number theory, and algebraic geometry. The project initiates a wide-ranging program exploiting connections between this problem, the classical open problem of cyclicity of central simple algebras in prime degree, the period-index problem for abelian variety torsors, the arithmetic of modular curves, and the problem of explicit constructions of moduli spaces of elliptic curves.This project is jointly funded by the Algebra and Number Theory Program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

这个项目将探索在数学和物理的许多领域的发展中发挥了重要作用的代数和几何对象，称为中心简单代数及其分裂领域。四元数代数的理论，这些结构的例子，在整个19世纪和20世纪被用于经典力学，量子力学和相对论，目前应用于航空航天建模，计算机图形学，机器人和无线通信等领域。在数学中，中心单代数和相关的布劳尔群为不同研究领域之间的技术交流提供了桥梁，从数论和算术几何到拓扑学和代数几何。本项目旨在拓宽中心单代数的分裂域与椭圆曲线算术之间的一个相对较新的前沿，这两个领域本身就是重要的数学对象。该项目还将支持新的教学工具、公共宣传和参与机会、本科生和研究生的早期职业指导和培训机会以及跨学科合作。更确切地说，该项目围绕三个主要问题：Brauer群的周期指数问题，用亏格为1的曲线分裂中心单代数的问题，以及椭圆曲线模空间的显式表示。所采用的技术-环面几何，变形理论，希尔伯特计划-被广泛应用于代数几何，但较少系统地在纯代数。虽然最近在Brauer组的周期指数问题的进展一直是通过利用代数几何工具来控制分歧分裂，但该项目采用了新的工具，如互易序列，以实现改进的结果，并指出长期的目标。任何布劳尔类是否被一个亏格为一的曲线的函数场分裂的问题，虽然在过去的十年里才得到关注，但事实证明，它与代数、数论和代数几何中的其他重要问题有着惊人的深刻联系。该项目启动了一个广泛的计划，利用这个问题之间的联系，经典的开放问题的循环中心简单代数在素数的程度，周期指数问题的交换品种torsors，算术的模曲线，以及椭圆曲线模空间的显式构造问题。该项目由代数与数论计划和既定刺激计划联合资助竞争性研究（EPSCoR）。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。