Weil-Etale Cohomology and the Tamagawa number conjecture

Matthias Flach 的 DMS-0701029 奖摘要该项目的目的是定义和研究一种新的 Grothendieck 拓扑（“Weil-etale 拓扑”），与数域上簇的 Hasse-Weil L 函数的特殊值（“Tamakawa 数猜想”）相关。这种拓扑的想法以及第一个定义是由 Lichtenbaum 提出的，他还展示了在零维变换的最简单情况下与 L 函数的预期关系。然而，复杂计算 Weil-etale 上同调的截断是必要的，因为在大于 2 的偶数度上，上同调群是非零的且具有无限秩。该项目的目标之一是重新定义数域整数环的Weiletale拓扑，使其具有有界上同调以及许多其他属性，例如到etale拓扑和实数分类拓扑的映射。第二个目标是找到特征零的高维算术方案的 Weil etaletopos 的定义（由于 Lichtenbaum 和 Geisser 的工作，有限特征的情况已经得到很好的理解）。第三个更具推测性的目标是使用 Weil-etale 拓扑重新证明解析类数公式，目的是将其推广到 ArtinL 函数。该项目的一个有趣的方面是拓扑理论和逻辑与更经典和完善的数论（例如解析类数公式）的相互作用。几个世纪以来，丢番图方程及其解一直占据着对数学感兴趣的人们的想象力，但他们也在编码理论中找到了现实世界的应用。现代数学提供了一系列令人眼花缭乱的技术，从简单到高度抽象，用于研究丢番图方程。将解决方案视为“空间”的几何视角特别有用。该项目旨在通过定义和研究丢番图方程的新上同调理论来促进这一思路。

Abstract for award DMS-0701029 of Matthias FlachThe aim of the project is to define and study a new Grothendiecktopology (the "Weil-etale topology") in connection with specialvalues of Hasse-Weil L-functions of varieties over number fields(the "Tamagawa Number conjecture"). The idea of such a topology, aswell as a first definition, is due to Lichtenbaum who also shows theexpected relationship to the L-function in the simplest case of azero-dimensional variety. However, truncation of the complexcomputing Weil-etale cohomology is necessary because in even degreesgreater than two the cohomology group is nonvanishing and ofinfinite rank. One goal of the project would be to redefine the Weiletale topos of the ring of integers of a number field so that it hasbounded cohomology as well as a number of other properties such as amap to the etale topos and to the classifying topos of the realnumbers. A second goal is to find a definition of the Weil etaletopos for higher dimensional arithmetic schemes of characteristiczero (the case of finite characteristic is already well understooddue to work of Lichtenbaum and Geisser). A third, more speculativegoal would be to reprove the analytic class number formula using theWeil-etale topology with the aim of generalizing it to ArtinL-functions. One interesting aspect of this project is theinteraction of topos theory and logic with more classical and wellestablished number theory, such as the analytic class numberformula.Diophantine equations and their solutions have occupied theimagination of mathematically interested people for centuries butthey also have found real world applications in coding theory.Modern mathematics provides a bewildering array of techniques,ranging from the disarmingly simple to the highly abstract, to studydiophantine equations. The geometric perspective, viewing thesolution set as a "space", has been particularly useful. The projectaims to contribute to this line of thought by defining and studyinga new cohomology theory for diophantine equations.