Arithmetical Algebraic Geometry

乌尔默博士DMS-0701053奖摘要。乌尔默提出了三个与阿贝尔变种等级和功能域塔上的Birch和Swinnerton-Dyer猜想有关的项目。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小的零点；这是建立在他最近在阿贝尔塔中演示类似结果的工作的基础上的。在第二个项目中，他计划研究保证曲线的雅可比在函数域的每一层上满足Birch和Swinnerton-Dyer猜想的一般准则；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，这些结果允许人们证明某些阿贝尔变种在功能域的塔层中具有有界的等级。所有这三个项目目前都涉及学生或博士后，他们有足够的空间继续贡献。乌尔默从事的是算术代数几何，这是一个基础数学领域，其激励问题是关于求解具有整数或有理数的多项式方程组。该领域是好奇心驱动的，曾被认为是没有应用的。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播放机、移动电话和安全的互联网通信都依赖于最初在数学代数几何中寻找问题时创造的数学。该领域与其他数学领域如代数、几何、分析和拓扑学有着深刻的联系，也与其他科学领域如量子场论有着神秘的联系。乌尔默博士希望通过他对椭圆曲线和L函数的研究来阐明数字、形状和微积分之间的联系。他在这方面的工作也为他所从事的许多教育、外展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.