Arakelov Theory, K-Theory, and Motives

Arakelov 理论、K 理论和动机

• 批准号: 0500762
• 负责人: Henri Gillet
• 金额: --
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500762&HistoricalAwards=false
• 关键词: Arakelov; Theory; Motives

Professor Gillet will be studying a number of problems related to Arakelov Theory, Motives, and K-theory. He intends to use motivic cohomology to give a purely sheaf theoretic construction of the arithmetic Chow groups. This will include studying the action of Adams' operations on Grayson's model of the motivic cohomology complexes, with the goal of showing that the motivic cohomology of an arithmetic variety can be used to compute its Chow groups (with rational coefficients). For semi-stable arithmetic varieties, he intends to use deformation to the normal cone, techniques to construct an intersection product on the Chow groups with integer coefficients. (Previously it was only known how to do this when the variety is smooth over the base.) He will also study the possibility of defining arithmetic Chow groups with coefficients in a local system. This would allow the construction of new modular forms via generating series with coefficients that are arithmetic cycles. Two other topics that he plans to pursue are first to study the relationship between the proof, using Brauer lifting, of Gersten's conjecture for discrete valuation rings with finite residue fields and the K-theory of finite sets, and secondly, giving an explicit proof of the Riemann-Roch theorem, which will lead to improved information about secondary classes.The overall goal of these areas of research is to improve our understanding of Diophantine equations, that is understanding the properties of the set of integer solutions of equations with integer coefficients. Diophantine equations have applications in particular to problems in cryptography and coding theory.

吉莱教授将研究一些与阿拉克洛夫理论，动机和K理论有关的问题。 他打算使用动机上同调给一个纯粹的层理论建设的算术周群。 这将包括研究行动的亚当斯'行动格雷森的模型motivic上同调复合物，与目标表明，motivic上同调的算术品种可以用来计算其周群（合理的系数）。对于半稳定的算术变种，他打算使用变形的正常锥，技术来构建一个交叉产品的周群与整数系数。 （以前只知道如何做到这一点时，品种是顺利的基础。他还将研究的可能性，定义算术周组的系数在当地的制度。这将允许通过生成具有算术循环的系数的系列来构造新的模形式。另外两个主题，他计划追求的是，首先研究之间的关系的证明，使用布劳尔提升，Gersten的猜想离散赋值环有限剩余领域和K-理论的有限集，其次，给出一个明确的证明黎曼-罗克定理，这将导致改进有关中学课程的信息。这些研究领域的总体目标是提高我们对丢番图方程的理解，也就是理解整系数方程组的整数解的性质。 丢番图方程在密码学和编码理论中有特别的应用。