Shimura varieties, Galois modules and Galois representations

PI刻划了Shimura簇在非光滑约化素数下的积分模型，并研究了相关的p-进空间。特别是，他利用代数环群理论和p-进Hodge理论的方法研究了Shimura变种在这种素数下的奇点结构，并正在为这些变种发展一种“局部模型”理论。其动机是获得可用于计算它们的Hasse-Weil Zeta函数和其他数论应用的信息。他还研究了紧连通p-进空间，如局部Galois表示的形变空间、Rapoport-Zink空间及其p-进上同调。此外，他正在研究具有有限群作用的算术簇的上同调中的表示，并发展了相关的Riemann-Roch型定理。在过去，PI已经得到了用于计算整数上具有阿贝尔群作用的簇的等变欧拉特征的公式。他正在将他的工作扩展到非阿贝尔群作用的情况，并将应用他的技术来推导从模曲线的覆盖中获得的关于Galois模的信息。他还开发了Riemann-Roch型定理的精化版本，允许计算扭类，在某些情况下，他提供了Riemann-Roch恒等式同构的局部或非对称描述。这是算术代数几何领域的研究，这门学科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)描述的形状的几何，以及对数字的研究。这种学科的结合被证明是非常有成效的--解决了几代人都经历过的问题(例如费马最后定理)。该提案主要集中于研究具有多种对称性的特定多项式方程。一般的领域与物理学、纠错码的构造和密码学有关。

The PI is attempting to describe integral models for Shimura varieties at primes of non-smooth reduction and study related p-adic spaces. In particular, he investigates the structure of the singularities of Shimura varieties at such primes using methods from the theory of algebraic loop groups and p-adic Hodge theory and he is developing a theory of ``local models" for these varieties. The motivation is to obtain information that can be used in the calculation of their Hasse-Weil zeta functions and in other number theoretic applications. He is also studying closely connected p-adic spaces such as deformation spaces of local Galois representations, Rapoport-Zink spaces and their p-adic cohomology. In addition he is studying the representations that appear in the cohomology of arithmetic varieties with a finite group action and is developing related Riemann-Roch type theorems. In the past, the PI has obtained formulae useful for calculating equivariant Euler characteristics of varieties over the integers with an abelian group action. He is extending his work to the case of non-abelian group actions and he will apply his techniques to deduce information about Galois modules obtained from covers of modular curves. He is also developing refined versions of Riemann-Roch type theorems that allow for calculations of torsion classes, and, in some cases, he is providing local or adelic descriptions of the isomorphisms underlying the Riemann-Roch identities. This is research in the field of arithmetic algebraic geometry,a subject that blends two of the oldest areas of mathematics: the geometry of shapes that can be described by the simplest equations, namely polynomials, and the study of numbers. This combination of disciplines has proved extraordinarily fruitful - having solved problems that withstood generations (such as ``Fermat's last theorem"). The proposal 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.