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.

1)岩泽精细Selmer群和Selmer群的岩泽理论：我将继续我在岩泽理论方面的研究，研究对偶Selmer群和对偶精细Selmer群的Mu不变量，它们是某些岩泽代数上的有限生成模。这些模已经在素数p上常见的Galois表示的情况下得到了广泛的研究。我们将把前面的研究扩展到素数具有超奇异约化的Galois表示的情况。我们打算研究的两个基本Galois表示是由椭圆曲线和椭圆模形式产生的。我们还将同时研究岩泽理论不变量，如u不变量和lambda不变量对相关剩余表示的依赖性。对于定义在数域上的椭圆曲线，通过固定研究岩泽模的奇素数p，这将自然地导致理解以素数p为模的椭圆曲线的L函数的值与ADIC空间的同余性质：我打算在完备空间和p-adadHodge理论的背景下开始研究拼接技术。补丁技术允许我们将定义在p-adi局部场上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，这些技巧允许我们在整个曲线上构造二次丛，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。我计划探索这些技巧对其他情况的适应性，特别是在p-进Galois表示理论和ADIC空间的环和域上。我们期望它们在p-进Hodge理论中有有趣的应用，并计划研究在Peter Scholze关于Perfetoid空间的工作中的可能应用。3)实数域和有限域上光滑射影曲面的Witt群：特征不同于2的域的Witt环研究域上二次型的等价类。它有一个丰富的结构，与代数K-理论和Galois上同调有关。代数簇的Witt群涉及研究簇上的具有二次空间结构的簇上的向量丛。Witt群是该簇的一个稳定的双调不变量，它与代数圈的Chow群、Brauer群和未分支上同调群等其他双调不变量有着有趣的联系。我们打算计算某些曲面的显式结构，例如实数基域上的K3曲面、椭圆曲面和有限域上的椭圆曲面。