Shimura Varieties, Galois Modules and the Determinant of Cohomology

研究者正在研究以下两个问题：（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PEL志村品种的“本地模型”，并试图制定关于一般志村品种的非平滑减少的准确预测。 其动机是获得可用于计算这些品种的Hasse-Weil zeta函数和其他算术应用的信息。(B)他正在研究的代表性，出现在上同调的算术品种与一组行动。他将继续他的工作描述之间的关系不变量描述这种表示和L-功能和发展理论的``非阿贝尔”立方结构的行列式上同调的一般束曲线。这是研究领域的算术代数几何，一个主题，融合了两个最古老的数学领域：可以用最简单的方程，即多项式来定义的图形的几何学，以及对数字的研究。 这种结合已经被证明是非常富有成效的--最近解决了几代人都无法解决的问题（比如“费马最后定理”）。 研究者的工作主要集中在研究具有许多对称性的特定多项式方程。一般领域与物理学、纠错码的构造和密码学有关。

The investigator is working on the following two 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 PEL Shimura varieties and tries to formulate exact predictions about the non-smooth reduction of a general Shimura variety. 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 arithmetic applications. (B) He is studying the representations that appear in the cohomology of arithmetic varieties with a group action. He will continue his work on describing relations between invariants describing such representations and L-functions and develop a theory of ``non-abelian" cubic structures for the determinant of cohomology of general bundles over curves.This is research 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 provedextraordinarily fruitful - having recently solved problems that withstood generations (such as ``Fermat's last theorem"). The investigator's work mainly concentrates on the study of specific polynomial equations that have many symmetries. The general field has connections with physics, the construction of error correcting codes and cryptography.