Shimura varieties, Galois modules and Galois representations

PI试图描述Shimura簇在非光滑约化素数上的积分模型，并研究相关的p-adic空间。特别是，他调查结构的奇异性志村品种在这种素数使用的方法从理论的代数循环群和p进霍奇理论，他正在开发一个理论的“本地模型”这些品种。其动机是为了获得可用于计算其Hasse-Weil zeta函数和其他数论应用的信息。 他还研究密切联系的p进空间，如变形空间的局部伽罗瓦表示，拉波波特，津克空间和他们的p进上同调。此外，他正在研究的代表性，出现在上同调的算术品种与有限群的行动，并正在发展相关的黎曼-罗奇型定理。在过去，PI已经获得了公式，用于计算具有阿贝尔群作用的整数簇的等变欧拉特征。他正在扩大他的工作的情况下，非阿贝尔群行动，他将运用他的技术来推断信息伽罗瓦模块获得覆盖的模块化曲线。他还发展完善版本的黎曼-罗奇型定理，允许计算扭转类，并在某些情况下，他提供当地或adelic描述的同构基础的黎曼-罗奇身份。这是算术代数几何领域的研究，这是一个融合了两个最古老的数学领域的学科：可以用最简单的方程描述的形状的几何，即多项式，以及数字的研究。这种学科的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”）。该提案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 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.