Shimura varieties, Galois representations and Riemann-Roch theorems

主要研究以下三个问题：(A)他试图描述非光滑约简素数下Shimura变量的积分模型。他特别研究了志村变体的“局部模型”及其与无限维群的仿射标志变体和伽罗瓦表示的变形空间的关系。其动机是获得可用于计算这些变量的Hasse-Weil zeta函数和其他数论应用的信息。(B)他正在发展Grothendieck-Riemann-Roch定理的精细化和泛函版本，这将允许计算扭转信息。(C)他正在研究有限群作用下算术变异上同调中出现的表示。特别是，他继续他的工作，发展不动点公式计算不变量的这种（积分）表示使用两个相互关联的主题：三次结构理论和代数环群的中心扩展理论。这位研究者的研究领域是算术代数几何，这是一个融合了数学中两个最古老领域的学科：可以用最简单的方程（即多项式）定义的图形几何，以及对数字的研究。事实证明，这种结合非常富有成效——解决了几代人经受住考验的问题（比如“费马大定理”）。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。它与物理学、纠错码的构造和密码学都有联系。

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.