Deformation Rings and Group Schemes

该提议的目标是更好地理解由伽罗瓦表示产生的（受限）变形环的结构。例如，将变形环的代数性质与被分类的表示的性质联系起来是一个基本问题。目前，人们对这一点知之甚少。从代数几何的类比出发，能否定义一个合适的上同群，其消失对应于变形环的形式光滑性？我们将研究的一个代表性的非平凡的例子是在高度分叉的基上由有限的平面群方案产生的表示。这个例子应该对更一般的情况有一些启发，它也应该应用于伽罗瓦表示和椭圆曲线的模块化问题。这项研究是在数论领域，这是数学的一个分支，与整数有关的问题。数论是一门非常古老的学科，但它充满了难题和重大猜想。这门学科中的许多基本问题涉及素数的行为或多项式方程组的整数解的性质。这类问题可以用大量的解析、代数和几何工具来研究。这样的数论思想在开发安全和精确的电子通信系统方面也有重要的实际应用。椭圆曲线分解算法就是这种应用的一个例子。

The goal of the proposal is to better understand the structure of (restricted) deformation rings arising from Galois representations. For example, it is a fundamental problem to relate algebraic properties of deformation rings to properties of the representations being classified. This is, at present, very poorly understood. Out of analogy with algebraic geometry, can one define a suitable cohomology group whose vanishing corresponds to formal smoothness of the deformation ring? A representative non-trivial example which we will study is the case of representations arising from finite, flat group schemes over highly ramified bases. This example should shed some light on more general cases, and it should also have applications to modularity questions for Galois representations and elliptic curves. This research is in the area of number theory, which is the branchof mathematics that is concerned with questions about the integers.Number theory is a very old subject, but is full of difficult problemsand significant conjectures. Many of the fundamental problems in the subject concern either the behavior of prime numbers or properties of the integer solutions to systems of polynomial equations. These kinds of problems can be studied with a vast range ofanalytic, algebraic, and geometric tools. Such number-theoretic ideas also have important real-world applications in the development of secure and accurate electronic communications systems. The elliptic curve factorization algorithm is one example of such an application.