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. DMS-0701053 奖摘要乌尔默提议开展三个与阿贝尔簇的等级以及函数域塔上的伯奇和斯温纳顿-戴尔猜想相关的项目。 在第一个项目中，他计划展示函数域的非阿贝尔塔，其中某些 L 函数在临界点具有任意大阶的零点；这是建立在他最近在阿贝尔塔中展示类似结果的工作的基础上的。 在第二个项目中，他计划研究一般标准，以保证函数场塔的每一层中曲线的雅可比行列式满足 Birch 和 Swinnerton-Dyer 的猜想；这再次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近的结果，使人们能够证明某些阿贝尔变体在功能域塔的各层中具有界限等级。 所有这三个项目目前都有学生或博士后参与，他们的持续贡献还有很大的空间。乌尔默从事算术代数几何研究，这是一个基础数学领域，其激发问题是关于求解具有整数或有理数的多项式方程组。该领域是由好奇心驱动的，一度被认为没有应用价值。 然而，现在人们知道它对于影响我们日常生活的许多现代技术至关重要，例如编码理论和密码学。 CD 和 DVD 播放器、移动电话和安全的互联网通信都依赖于最初为解决算术代数几何问题而创建的数学。 该领域与代数、几何、分析和拓扑等其他数学领域有着深厚的联系，也与量子场论等其他科学领域有着神秘的联系。 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从事的许多教育、推广和培训活动奠定了基础。