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.

拟议的项目福尔斯属于以下三个大标题：1）岩泽理论的罚款塞尔默群体和塞尔默群体：我将继续我的调查岩泽理论和研究的亩不变的双重塞尔默群体和双重罚款塞尔默群体，这是一个生成模块在某些岩泽代数。这些模块已被广泛研究的情况下，伽罗瓦表示是普通的素数p。我们将我们以前的研究扩展到伽罗瓦表示的情况下，有supersingular减少在素数p。两个基本的伽罗瓦表示，我们打算研究的是那些所产生的椭圆曲线和椭圆模的形式。我们还将同时研究岩泽理论不变量的依赖性，如μ不变量和λ不变量，相关的剩余表示。在定义于数域上的椭圆曲线的情况下，固定一个奇素数p，研究岩泽模，这自然会导致理解椭圆曲线模素数p的L函数值的同余性质。2）修补和Adic空间：我打算在Perfectoid空间和p-adic Hodge理论的背景下开始研究修补技术。Patching技术允许我们将定义在p-adic局部域上的曲线上的局部数据修补到曲线上的全局数据。作为一个例子，这些技术允许我们在整个曲线上构造二次丛，当我们在局部环上给出二次空间时，这些局部环是曲线的函数域上的赋值环，并且满足额外的可验证条件。我计划探索这些技术的适应性，以其他情况下，特别是在环和领域所产生的理论的p-adic伽罗瓦表示和Adic空间。我们希望这些有有趣的应用在p-adic霍奇理论，并计划调查可能的应用程序的工作彼得Scholze的Perfectoid空间。3)实数域和有限域上光滑射影曲面的维特群：特征不等于2的域的维特环研究域上二次型的等价类。它有一个丰富的结构与连接代数K理论和伽罗瓦上同调。维特群的代数簇涉及研究向量丛的品种配备了二次空间结构的相关层。Witt群是簇的一个稳定的双有理不变量，并且与其他双有理不变量有着有趣的联系，如代数圈的Chow群，Brauer群和非分歧上同调群。簇的Witt群的结构取决于簇定义的基域。我们打算计算某些类型的曲面，如K3曲面，椭圆曲面上的基域的真实的数字和有限域的显式结构。