Arithmetical Algebraic Geometry

算术代数几何

• 批准号: 1004141
• 负责人: Douglas Ulmer
• 金额: $ 1.27万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-15 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1004141&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奖项摘要。乌尔默建议工作的三个项目有关的行列的阿贝尔品种和桦树和Swinnerton-Dyer猜想塔的功能领域。 在第一个项目中，他计划展示函数场的非阿贝尔塔，其中某些L函数在临界点处具有任意大阶的零点;这建立在他最近的工作的基础上，证明了阿贝尔塔中的类似结果。 在第二个项目中，他计划调查的一般标准，保证雅可比曲线满足猜想的伯奇和斯温纳顿戴尔在每一层塔的功能领域;再次这扩展了他最近的工作。 在第三个项目中，Ulmer博士将尝试扩展最近的结果，这些结果允许人们证明某些阿贝尔变种在函数场塔的层中具有有界秩。 这三个项目目前都有学生或博士后参与，他们有足够的空间继续做出贡献。Ulmer博士从事算术代数几何，这是一个基础数学领域，其激发性问题是关于用整数或有理数求解多项式方程组。该领域是好奇心驱动的，曾经被认为没有应用。 然而，现在已知它对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、移动的电话和安全的互联网通信都依赖于数学，而数学最初是为了解决算术代数几何问题而创建的。 该领域与其他数学领域有着深刻的联系，如代数，几何，分析和拓扑学，以及与其他科学领域的神秘联系，如量子场论。 Ulmer博士希望通过他对椭圆曲线和L函数的研究来阐明数字，形状和微积分之间的联系。 他在这一领域的工作也为他所从事的许多教育、外联和培训活动提供了基础。