The Algebra and Arithmetic of Splitting Fields

该项目将探索在数学和物理的许多领域的发展中发挥重要作用的代数和几何对象，称为中心简单代数及其分裂场。四元数代数理论，这些结构的例子，在整个19世纪和20世纪被应用于经典力学、量子力学和相对论中，目前在航空航天建模、计算机图形学、机器人和无线通信等领域也有应用。在数学中，中心简单代数和相关的Brauer群为不同研究领域之间的技术交流提供了一座桥梁，从数论和算术几何到拓扑和代数几何。这个项目的目的是在中心简单代数和椭圆曲线算术的分裂领域之间拓宽一个相对新的边界，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、公众宣传和参与的机会、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。更准确地说，这个项目围绕三个主要问题：Brauer群的周期指数问题，用一属曲线分裂中心简单代数的问题，以及椭圆曲线模空间的显式表示。所使用的技术——环面几何、变形理论和希尔伯特方案——在代数几何中被广泛应用，但在纯代数中却很少系统地应用。虽然Brauer群的周期指数问题最近的许多进展是通过利用代数几何工具来控制分支分裂，但该项目采用了互易序列等新工具来实现改进的结果，并指出了长期的推测目标。布劳尔类是否被一条曲线的函数域分割的问题，虽然在过去的十年里才引起人们的注意，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而深刻的联系。该项目启动了一个广泛的项目，利用这个问题，素数次中心简单代数的环性的经典开放问题，阿贝尔变量的周期指数问题，模曲线的算法，以及椭圆曲线模空间的显式构造问题之间的联系。该项目由代数和数论项目和促进竞争研究的既定项目（EPSCoR）共同资助。该奖项反映了美国国家科学基金会的法定使命，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。

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.