Arithmetic Algebraic Geometry

算术代数几何

• 批准号: 0400877
• 负责人: Douglas Ulmer
• 金额: --
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400877&HistoricalAwards=false
• 关键词: Arithmetic; Algebraic; Geometry

Abstract of DMS 0400877PI: Douglas UlmerDr. Ulmer proposes to work on three projects ultimately related to thearithmetic of abelian varieties over function fields. In the firstproject, he plans to prove that there exist abelian varieties ofarbitrary dimension whose L-functions vanish to arbitrarily high orderat the center of their critical strips. This generalizes his previouswork on elliptic curves. In the second and third projects, he plansto generalize his previous construction of local points on theuniversal elliptic curve over a modular curve to the context of anarbitrary non-isotrivial elliptic curve over a function field and tostudy the extent to which this construction may be used to constructglobal points. This research is intimately connected with theconjecture of Birch and Swinnerton-Dyer in the function field context.Dr. Ulmer works in the general area of Arithmetical AlgebraicGeometry, an area of fundamental mathematics that was previouslythought to be beyond application. However, it is now known to be ofcrucial importance to many important technologies, such as codingtheory and cryptography, which affect our everyday lives. CD and DVDplayers, mobile telephones, and secure internet communication all relyon mathematics originally created in the pursuit ofquestions in arithmetical algebraic geometry. The field also has deepand mysterious links to other areas of science such as quantum fieldtheory. Dr. Ulmer's work does not appear to have immediateapplication, but his knowledge of this area provides a broadperspective on modern mathematics and forms the basis for manyeducation, outreach, and training activities which he is engaged in.These include the Southwestern Center for Arithmetical AlgebraicGeometry, initiatives in training and mentoring of new graduatestudents, and numerous undergraduate research projects.

DMS摘要0400877PI：Douglas Ulmerdr。乌尔默（Ulmer）建议在三个最终与Abelian品种有关功能领域的棘齿相关的三个项目。 在第一个项目中，他计划证明存在Abelian品种的左侧尺寸，其L功能消失至任意高秩序的关键条件中心。 这概括了他先前在椭圆曲线上的工作。 在第二个和第三个项目中，他计划将自己先前在模块化曲线上的椭圆曲线上的本地点构建到北极不清的非异位椭圆曲线的背景，并在功能场上和tostudy范围内使用该构建可用于构建构造点的程度。 这项研究与功能字段中的桦木和swinnerton-dyer的核心注射密切相关。乌尔默（Ulmer）在算术代数测定法的一般领域工作，这是一个基本数学的领域，以前是无法应用的。 但是，现在众所周知，这对于许多重要的技术（例如影响我们日常生活的编码理论和密码学）至关重要。 CD和DVDPlayers，移动电话和安全的Internet通信都是最初在算术代数几何的追求问题中创建的。 该领域还与其他科学领域（例如量子现场理论）建立了深厚的神秘联系。 乌尔默博士的工作似乎没有立即施加，但是他对这一领域的了解为现代数学和培训活动提供了广泛的观点，构成了他从事的许多教育，外展和培训活动的基础。这些包括西南部算术代数分析中心，对培训和研究毕业生的培训和研究的计划，以及许多新毕业生的培训和众多研究。