Shimura varieties, Galois representations and Riemann-Roch theorems

主要研究者正在研究以下三个问题：（A）他试图描述Shimura簇在非光滑约化素数下的积分模型。特别是，他研究了“当地模式”， 志村簇及其与无限维群的仿射旗簇和伽罗瓦表示的变形空间的关系。其动机是为了获得信息，可以用于计算这些品种的Hasse-Weil zeta函数和其他数论应用。(B)他正在开发精细和函子版本的格罗滕迪克-黎曼-罗克定理，这将允许计算扭转信息。(C)他正在研究出现在有限群作用的算术变种的上同调中的表示。特别是，他继续他的工作，发展不动点公式，用于计算这样的不变量。（积分）表示使用两个相互关联的主题：立方结构理论和代数圈群的中心扩展理论。研究者的研究领域是算术代数几何，一门融合了两个最古老的数学领域的学科：可以由最简单的方程（即多项式）定义的图形几何学和对数字的研究。这种结合已经证明是非常富有成效的-解决了几代人的问题（如“费马最后定理”）。研究者的工作主要集中在研究某些具有许多对称性的多项式方程。这与物理学、纠错码的构造和密码学有关。

The principal investigator is working on the following three problems:(A) He is attempting to describe integral models for Shimura varieties at primes of non-smooth reduction. In particular, he studies ``local models" for Shimura varieties and their relation with affine flag varieties for infinite dimensional groups and with deformation spaces of Galois representations. The motivation is to obtain information that can be used in the calculation of the Hasse-Weil zeta function of these varieties and in other number theoretic applications.(B) He is developing refined and functorial versions of the Grothendieck-Riemann-Roch theorem that would allow for the calculation of torsion information.(C) He is studying the representations that appear in the cohomology of arithmetic varieties with a finite group action.In particular, he continues his work on developing fixed point formulas for calculating invariants of such (integral) representations using two interconnected themes: the theory of cubic structures and the theory of central extensions of algebraic loop groups.The investigator's research is in the field of arithmetic algebraic geometry, a subject that blends two of the oldest areas of mathematics: the geometry of figures that can be defined by the simplest equations, namely polynomials, and the study of numbers. This combination has proved extraordinarily fruitful - having solved problems that withstood generations (such as ``Fermat's last theorem"). The investigator's work mainly concentrates on the study of certain polynomial equations that have many symmetries. There are connections with physics, the construction of error correcting codes and cryptography.