Iwasawa Theory and p-adic Hodge Theory

拟议的项目分为以下三个大标题：*****1）精细Selmer群和Selmer群的Iwasawa理论：我将继续对Iwasawa理论的研究，并研究对偶Selmer群和对偶精细Selmer群的mu不变量，它们是某些Iwasawa代数上的有限生成模。这些模块已经在素数 p 处的伽罗瓦表示的情况下进行了广泛的研究。我们将把我们早期的研究扩展到在素数 p 处具有超奇异约简的伽罗瓦表示的情况。我们打算研究的两个基本伽罗瓦表示是由椭圆曲线和椭圆模形式产生的伽罗瓦表示。我们还将同时研究岩泽理论不变量（例如 mu 不变量和 lambda 不变量）对相关残差表示的依赖性。在数域上定义的椭圆曲线的情况下，固定研究 Iwasawa 模的奇素数 p，这自然会导致理解以素数 p 为模的椭圆曲线的 L 函数值的同余性质。******2) 修补和进数空间：我打算在完美空间和 p 进 Hodge 理论的背景下开始研究修补技术。修补技术允许我们将 p 进局部场上定义的曲线上的局部数据修补到曲线上的全局数据。例如，当我们在局部环上给出二次空间时，这些技术允许我们在整个曲线上构建二次丛，局部环是曲线函数场上的评估环并满足附加的可验证条件。我计划探索这些技术对其他情况的适应性，特别是在 p-adic Galois 表示理论和 Adic 空间理论中出现的环和域。我们期望这些在 p 进 Hodge 理论中具有有趣的应用，并计划研究 Peter Scholze 在 Perfectoid 空间上的工作的可能应用。 *****3) 实数域和有限域上的光滑射影曲面的 Witt 群：特征域的 Witt 环不同于 2 个研究域上二次形式的等价类。它具有与代数 K 理论和伽罗瓦上同调相关的丰富结构。代数簇的维特群涉及研究该簇上的向量丛，该簇在相关束上配备有二次空间结构。维特群是簇的稳定双有理不变量，并且与其他双有理不变量有有趣的联系，例如代数环的 Chow 群、布劳尔群和无分支上同调群。簇的维特群的结构取决于定义簇的基域。我们打算计算某些类型的曲面的显式结构，例如 K3 曲面、实数基域和有限域上的椭圆曲面。

The proposed project falls under the following three broad headings:******1) Iwasawa theory of the fine Selmer groups and Selmer groups: I will continue my investigations in Iwasawa theory and study the mu invariant of the dual Selmer group and the dual fine Selmer group, which are finitely generated modules over certain Iwasawa algebras. These modules have been studied extensively in the case of Galois representations that are ordinary at a prime p. We shall extend our earlier study to the case of Galois representations that have supersingular reduction at the prime p.Two fundamental Galois representations that we intend to study are those arising from elliptic curves and elliptic modular forms. We shall also simultaneously study the dependence of the Iwasawa theoretic invariants, such as the mu invariant and the lambda invariant, on the associated residual representation. In the case of an elliptic curve defined over a number field, fixing an odd prime p at which the Iwasawa modules are studied, this will naturally lead to understanding the congruence properties of the values of the L-functions of the elliptic curve modulo the prime p.******2) Patching and Adic spaces: I intend to initiate the study of patching techniques in the context of Perfectoid spaces and p-adic Hodge theory. The Patching techniques allow us to patch local data on a curve defined over a p-adic local field to a global one on the curve. As an example, these techniques allow us to construct a quadratic bundle on the whole curve when we are given quadratic spaces over the local rings which are valuation rings on the function field of the curve and satisfy additional verifiable conditions. I plan to explore the adaptability of these techniques to other situations, especially over rings and fields arising in the theory of p-adic Galois representations and those of Adic spaces. We expect these to have interesting applications in p-adic Hodge theory and plan to investigate possible applications to the work of Peter Scholze on Perfectoid spaces.******3) Witt groups of smooth projective surfaces over the reals and finite fields: The Witt ring of a field of characteristic different from 2 studies equivalence classes of quadratic forms over the field. It has a rich structure with connections to algebraic K-theory and Galois cohomology. The Witt group of an algebraic variety involves studying vector bundles on the variety which are equipped with a quadratic space structure on the associated sheaf. The Witt group is a stable birational invariant of the variety and has interesting connections to other birational invariants such as the Chow group of algebraic cycles, the Brauer group and the unramified cohomology groups.The structure of the Witt group of the variety depends on the base field over which the variety is defined. We intend to compute the explicit structure of certain classes of surfaces such as the K3 surfaces, elliptic surfaces over the base field of real numbers and finite fields.