Iwasawa Theory and p-adic Hodge Theory

岩泽理论和p进霍奇理论

• 批准号: RGPIN-2019-03987
• 负责人: ramdorai, sujatha
• 金额: $ 2.33万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=737323
• 关键词: Iwasawa; Theory; adic; Hodge

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.

本课题分为以下三大类：1)Iwasawa fine Selmer群和Selmer群的Iwasawa理论：我将继续我在Iwasawa理论的研究，并研究对偶Selmer群和对偶fine Selmer群的mu不变量，它们是某些Iwasawa代数上有限生成的模。这些模已经在素数p处普通的伽罗瓦表示的情况下进行了广泛的研究。我们将把我们先前的研究扩展到在素数p处具有超奇异化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。我们也将同时研究Iwasawa理论的不变量，例如mu不变量和lambda不变量，对相关残差表示的依赖性。对于定义在数域上的椭圆曲线，确定一个奇素数p作为Iwasawa模的研究点，这将自然地导致对椭圆曲线模素数p的l函数值的同余性质的理解。2)补进空间：我打算在完美曲面空间和p进Hodge理论的背景下开始补进技术的研究。修补技术允许我们将在p进局部域上定义的曲线上的局部数据修补为曲线上的全局数据。作为一个例子，当给定曲线函数域上的赋值环上的二次空间并满足附加可验证条件时，这些技术允许我们在整个曲线上构造一个二次束。我计划探索这些技术在其他情况下的适应性，特别是在p进伽罗瓦表示理论和进空间理论中出现的环和场。我们期望这些在p进Hodge理论中有有趣的应用，并计划研究Peter Scholze关于完美曲面空间的工作的可能应用。3)实域和有限域上光滑投影曲面的Witt群：特征不同于2的域上的Witt环研究了该域上二次型的等价类。它具有丰富的结构，与代数k理论和伽罗瓦上同调有联系。代数变种的Witt群涉及研究变种上的向量束，这些向量束在关联束上具有二次空间结构。Witt群是一类稳定的双分不变量，它与其它双分不变量如代数循环的Chow群、Brauer群和未分支上同调群有着有趣的联系。该品种的威特群的结构取决于该品种被定义的基域。我们打算在实数基域和有限域上计算某些曲面（如K3曲面、椭圆曲面）的显式结构。