Deformation Rings and Group Schemes

该提案的目标是更好地理解从伽罗瓦表示产生的（限制）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相关联是一个基本问题。目前，人们对这一点知之甚少。 与代数几何类似，我们能否定义一个合适的上同调群，它的消失对应于变形环的形式光滑性？ 一个有代表性的非平凡的例子，我们将研究的情况下，所产生的陈述有限，平面组计划高度分歧的基础。 这个例子应该能在更一般的情况下有所启发，它也应该能应用于伽罗瓦表示和椭圆曲线的模块化问题。本文研究的是数论，它是研究整数问题的数学分支，是一门古老的学科，但充满了困难的问题和重要的成果。许多基本问题在主题关心的行为或素数的整数解的多项式方程组的属性。 这类问题可以用大量的分析、代数和几何工具来研究。 这种数论思想在安全和精确的电子通信系统的开发中也有重要的现实应用。椭圆曲线分解算法是这种应用的一个示例。

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.