Arakelov Theory, K-Theory, and Motives

Gillet教授将研究与Arakelov理论、动机和k理论相关的一些问题。他打算用动机上同给出算术周群的纯束理论构造。这将包括研究Adams的运算对Grayson的动机上同调复合体模型的作用，目的是表明算术变量的动机上同调可以用来计算它的Chow群（具有有理系数）。对于半稳定的算术变种，他打算利用法向锥的变形技术，在具有整数系数的Chow群上构造一个交积。（以前，人们只知道如何在品种光滑的基础上做到这一点。）他还将研究在局部系统中定义带系数的算术周群的可能性。这将允许通过生成具有等差循环系数的级数来构造新的模形式。他计划追求的另外两个主题是，首先研究使用Brauer举升的Gersten猜想对有限剩余域的离散评估环的证明与有限集的k理论之间的关系，其次，给出Riemann-Roch定理的显式证明，这将导致关于次要类的信息的改进。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的整解集的性质。丢番图方程特别适用于密码学和编码理论中的问题。

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.