Arithmetical Algebraic Geometry

授予UlmerDr. DMS-0701053的摘要。乌尔默提议开展三个项目，涉及阿贝尔变体的等级，以及函数场塔上的伯奇和斯温纳顿-戴尔猜想。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函数在临界点处具有任意大阶的零；这是建立在他最近的工作上，证明了阿贝尔塔的类似结果。在第二个项目中，他计划研究保证曲线的雅可比矩阵在函数域塔的每一层上满足Birch和Swinnerton-Dyer猜想的一般准则；这再次延伸了他最近的工作。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体在函数域塔的层中具有有界的秩。这三个项目目前都有学生或博士后参与，他们继续贡献的空间很大。乌尔默在算术代数几何领域工作，这是基础数学的一个领域，其激励问题是关于用整数或有理数解决多项式方程系统。这个领域是由好奇心驱动的，曾经被认为没有应用价值。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、移动电话和安全的互联网通信都依赖于数学，这些数学最初是在追求算术代数几何问题时创造的。该领域与代数、几何、分析和拓扑学等其他数学领域有着深刻的联系，也与量子场论等其他科学领域有着神秘的联系。乌尔默博士希望通过他对椭圆曲线和l函数的研究，揭示数字、形状和微积分之间的联系。他在这个领域的工作也为他从事的许多教育、推广和培训活动提供了基础。

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.