Arithmetical Algebraic Geometry

算术代数几何

• 批准号: 0701053
• 负责人: Douglas Ulmer
• 金额: $ 12万
• 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: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701053&HistoricalAwards=false
• 关键词: Arithmetical; Algebraic; Geometry

Abstract for award DMS-0701053 of UlmerDr. Ulmer proposes to work on three projects related to ranks of abelian varieties and the Birch and Swinnerton-Dyer conjecture over towers of function fields. In the first project he plans to exhibit non-abelian towers of function fields over which certain L-functions have zeroes of arbitrarily large order at the critical point; this builds on his recent work demonstrating the analogous result in abelian towers. In the second project, he plans to investigate general criteria which guarantee that Jacobians of curves satisfy the conjecture of Birch and Swinnerton-Dyer in every layer of a tower of function fields; again this extends his recent work. In the third project, Dr. Ulmer will try to extend recent results which allow one to show that certain abelian varieties have bounded ranks in the layers of a tower of function fields. All three of these projects currently involve students or post-docs and there is ample scope for their continued contribution.Dr. Ulmer works in Arithmetical Algebraic Geometry, an area of fundamental mathematics whose motivating questions are about solving systems of polynomial equations with integers or rational numbers.The field is curiosity-driven and was once thought to be without application. However, it is now known to be crucial to many modern technologies which affect our everyday lives, such as coding theory and cryptography. CD and DVD players, mobile telephones, and secure internet communication all rely on mathematics originally created in the pursuit of questions in arithmetical algebraic geometry. The field has deep connections to other areas of mathematics such as algebra, geometry, analysis, and topology, as well as mysterious links to other areas of science such as quantum field theory. Dr. Ulmer hopes to shed light on the connections between numbers, shapes, and calculus through his research on elliptic curves and L-functions. His work in this area also provides the basis for many education, outreach, and training activities in which he is engaged.

Ulmerdr的DMS-0701053奖励摘要。乌尔默（Ulmer）建议在三个与阿贝利亚品种等级有关的项目中，以及在功能领域的塔塔上的桦木和Swinnerton-Dyer猜想。 在第一个项目中，他计划展示某些L功能在关键点的任意大量较大的功能领域的非亚伯塔塔；这是基于他最近的工作，证明了阿贝利安塔楼的类似结果。 在第二个项目中，他计划调查一般标准，以确保曲线的雅各布人满足桦木和Swinnerton-dyer在功能场塔的每一层中的猜想。再次，这扩展了他最近的工作。 在第三个项目中，乌尔默博士将尝试扩展最新的结果，这使人们可以表明某些Abelian品种在功能场塔层的层中有限制的等级。 目前，这三个项目均涉及学生或毕业后，并且持续贡献的范围充足。乌尔默（Ulmer）在算术代数几何形状中工作，这是一个基本数学的领域，其激励性问题是关于用整数或理性数字求解多项式方程的系统。该领域是好奇心驱动的，曾经被认为是没有应用的。 但是，现在众所周知，这对于许多影响我们日常生活的现代技术至关重要，例如编码理论和密码学。 CD和DVD播放器，移动电话和安全的Internet通信都取决于最初在追求算术代数几何的问题中创建的数学。 该领域与数学的其他领域有着深厚的联系，例如代数，几何，分析和拓扑，以及与量子场理论等其他科学领域的神秘联系。 乌尔默博士希望通过他对椭圆曲线和L功能的研究来阐明数字，形状和微积分之间的联系。 他在这一领域的工作还为他参与的许多教育，外展和培训活动提供了基础。