Weil-Etale Cohomology and the Tamagawa number conjecture

本课题的目的是定义和研究一种与数域上的Hasse-Weil l-函数的特殊值（“Tamagawa数猜想”）有关的新的grothendieck拓扑（“Weil-etale拓扑”）。这种拓扑的想法，以及第一个定义，是由Lichtenbaum提出的，他还展示了在最简单的零维变化情况下与l函数的期望关系。然而，由于在大于2的偶数次上同调群是不消失的且秩是有限的，所以需要截断复计算的韦尔-埃塔莱上同调。该项目的一个目标是重新定义数字域的整数环的Weiletale拓扑，使其具有有界上同调以及许多其他属性，例如映射到eletale拓扑和实数的分类拓扑。第二个目标是找到特征为零的高维算法格式的Weil ettalopos的定义（由于Lichtenbaum和Geisser的工作，有限特征的情况已经得到了很好的理解）。第三个，更具推测性的目标是使用weil -etale拓扑来重新证明解析类数公式，目的是将其推广到artinl函数。这个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的相互作用，如解析类数公式。几个世纪以来，丢芬图方程及其解一直占据着对数学感兴趣的人的想象力，但它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，到研究丢番图方程。将解集视为一个“空间”的几何视角尤其有用。通过定义和研究丢番图方程的一种新的上同调理论，对这一思路作出贡献。

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.