Iwasawa Theory and p-adic Hodge Theory

岩泽理论和 p-adic Hodge 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758567
• 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.

拟议的项目属于以下三个广泛的标题：1）精美的Selmer群体和Selmer群体的Iwasawa理论：我将继续对Iwasawa理论进行投资，并研究Dual Selmer群体的MU不变性和双重精细Selmer集团，最终在某些Iwasawa Algebras上产生了模块。对于在Prime p的Galois表示情况下，这些模块已被广泛研究。我们应将我们的早期研究扩展到GALOIS表示的情况，该案例在我们打算研究的Prime P. Two P. Two基本形式上降低了，这是由椭圆曲线和椭圆形模块化形式引起的。我们还将简单地研究硫磺理论不变性的依赖性，例如MU不变性和Lambda不变性，对相关的残留表示。如果在一个数字字段上定义的椭圆曲线，固定了研究iWasawa模块的奇数P，这自然会导致理解椭圆曲线模块的L型值的一致性属性。 2）修补和ADIC空间：我打算在完美的空间和P-Adic Hodge理论的背景下启动修补技术的研究。修补技术使我们能够在P-ADIC局部字段定义的曲线上修补本地数据，以在曲线上的全局磁场上。例如，这些技术使我们能够在整个曲线上构造一个二次束，当我们在本地环上给出了二次空间，这些二次空间是曲线功能场上的值环并满足其他可验证的条件。我计划探索这些技术对其他情况的适应性，尤其是在P-Adic Galois表示和ADIC空间的理论中产生的环和田地。我们希望这些在P-Adic Hodge理论中具有有趣的应用，并计划调查彼得·索尔兹（Peter Scholze）在Perfectoid空间上的工作的可能应用。 3）在真实和有限领域上的一组光滑的投射表面组：特征性领域的Witt环与两种研究等效类别的二次形式的等效类别不同。它具有丰富的结构，并与代数K理论和GALOIS共同学有联系。代数品种的WITT组涉及在该品种上研究载体束，该品种配备了相关捆的二次空间结构。 Witt Group是该品种的稳定亿万富翁不变的，与其他亿万富翁不变式建立了有趣的联系，例如代数周期的Chow Group，Brauer Group和Unlagiped Coolomology群体。该品种的结构取决于该品种的Witt Group的结构，取决于该品种所定义的基本场。我们打算计算某些类别表面的明确结构，例如K3表面，椭圆形表面在实数和有限场的基础场上。