Arithmetic Algebraic Geometry

DMS 0400877PI摘要：Douglas UlmerDr。Ulmer建议在三个项目上工作，最终与函数域上的阿贝尔变量的算法有关。在第一个项目中，他计划证明存在任意维的阿贝尔变体，其l函数在其临界带的中心消失到任意高阶。这概括了他以前关于椭圆曲线的工作。在第二和第三个项目中，他计划将他之前在模曲线上的泛椭圆曲线上的局部点构造推广到函数场上的任意非等平凡椭圆曲线上，并研究这种构造在多大程度上可以用于构造全局点。本研究与Birch和Swinnerton-Dyer在功能场背景下的猜想密切相关。乌尔默在算术代数几何的一般领域工作，这是一个以前被认为无法应用的基础数学领域。然而，现在已知它对许多重要技术至关重要，例如影响我们日常生活的编码理论和密码学。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，这些数学最初是为了追求算术代数几何中的问题而创造的。该领域还与量子场论等其他科学领域有着深刻而神秘的联系。乌尔默博士的工作似乎没有立即应用，但他在这一领域的知识为现代数学提供了一个广阔的视角，并为他从事的许多教育、推广和培训活动奠定了基础。其中包括西南算术代数几何中心，新研究生培训和指导的倡议，以及众多本科生研究项目。

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.