Arithmetic Algebraic Geometry

DMS 0400877 PI摘要：道格拉斯UlmerDr.乌尔默建议工作的三个项目最终有关thaarithmetic交换品种的功能领域。 在第一个项目中，他计划证明存在任意维数的阿贝尔簇，其L-函数在其临界带的中心处消失到任意高阶。 这推广了他以前关于椭圆曲线的工作。 在第二个和第三个项目中，他计划将他以前在模曲线上的通用椭圆曲线上的局部点的构造推广到函数域上的任意非isotrivial椭圆曲线的上下文中，并研究这种构造在多大程度上可用于构造全局点。 这项研究与Birch和Swinnerton-Dyer在函数场背景下的猜想密切相关。Ulmer博士从事算术代数几何的一般领域，这是一个以前被认为超出应用范围的基础数学领域。 然而，现在知道它对许多重要技术至关重要，如编码理论和密码学，这些技术影响着我们的日常生活。 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，数学最初是为了追求算术代数几何中的问题而创造的。 这个领域与其他科学领域也有着深刻而神秘的联系，比如量子场论。 Ulmer博士的工作似乎并没有立即应用，但他在这一领域的知识提供了一个广阔的视野，对现代数学和形式的基础上，许多教育，推广和培训活动，他是从事。其中包括西南中心的算术代数几何，培训和指导新的研究生的倡议，以及众多的本科生研究项目。

Abstract of DMS 0400877PI: Douglas UlmerDr. Ulmer proposes to work on three projects ultimately related to thearithmetic of abelian varieties over function fields. In the firstproject, he plans to prove that there exist abelian varieties ofarbitrary dimension whose L-functions vanish to arbitrarily high orderat the center of their critical strips. This generalizes his previouswork on elliptic curves. In the second and third projects, he plansto generalize his previous construction of local points on theuniversal elliptic curve over a modular curve to the context of anarbitrary non-isotrivial elliptic curve over a function field and tostudy the extent to which this construction may be used to constructglobal points. This research is intimately connected with theconjecture of Birch and Swinnerton-Dyer in the function field context.Dr. Ulmer works in the general area of Arithmetical AlgebraicGeometry, an area of fundamental mathematics that was previouslythought to be beyond application. However, it is now known to be ofcrucial importance to many important technologies, such as codingtheory and cryptography, which affect our everyday lives. CD and DVDplayers, mobile telephones, and secure internet communication all relyon mathematics originally created in the pursuit ofquestions in arithmetical algebraic geometry. The field also has deepand mysterious links to other areas of science such as quantum fieldtheory. Dr. Ulmer's work does not appear to have immediateapplication, but his knowledge of this area provides a broadperspective on modern mathematics and forms the basis for manyeducation, outreach, and training activities which he is engaged in.These include the Southwestern Center for Arithmetical AlgebraicGeometry, initiatives in training and mentoring of new graduatestudents, and numerous undergraduate research projects.