Arakelov Theory, K-Theory, and Motives

吉列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。他打算利用动机上同调给出算术Chow群的一个纯粹的层理论构造。这将包括研究亚当斯的运算对格雷森的基元上同调复合体模型的作用，目的是证明算术簇的基元上同调可以用来计算它的Chow群(具有有理系数)。对于半稳定算术簇，他打算利用对法锥的变形来构造具有整系数的Chow群上的交积。(以前，只有当品种在底部光滑时，才知道如何做到这一点。)他还将研究用本地系统中的系数来定义算术Chow群的可能性。这将允许通过生成具有算术循环的系数的级数来构造新的模形式。他计划从事的另外两个课题是：第一，研究有限剩余域离散赋值环的Gersten猜想的Brauer提升证明与有限集的K-理论之间的关系；第二，给出Riemann-Roch定理的显式证明，这将导致有关第二类的信息的改善。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，即理解整系数方程的整数解集的性质。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.