Iwasawa theory, Galois representations and motives

岩泽理论、伽罗瓦表示和动机

• 批准号: 402071-2011
• 负责人: Ramdorai, Sujatha
• 金额: $ 2.55万
• 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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=573150
• 关键词: Iwasawa; theory; Galois; representations; motives

The current application proposes to study two areas, one in arithmetic geometry and the other related to the category of motives. For the first, our broad topic of study is non-commutative Iwasawa theory in Hida families. There has been recent progress in establishing specific cases of the non-commutative main conjecture, proposed by Coates, et. al. for the trivial motive and totally real p-adic Lie extensions of a totally real number field. We propose to investigate the formulation of the non=commutative main conjecture in the context of Hida families. Two specific problems we then want to concentrate on are (a) the study of the mu-invariant in CM and non-CM Hida families (b) Formulation of a non-commutative main conjecture for Hida families and studying its relation to the main conjecture for the specialisation along a fibre. Along the way, we propose to tackle a conjecture formulated by Coates, et.al. on the structure of the dual Selmer group of elliptic curves with good ordinary reduction at an odd prime p, considered as a module over a non-commutative Iwasawa algebra over admissible p-adic Lie extensions. In earlier work with B. Kahn, we constructed the category of birational Chow motives and also the triangulated category of birational geometric motives over a base field F and proved some results about these categories. Many of these results require the hypothesis that the field F be of characteristic zero. It might be possible to extend some of these results to the case when the field F has positive characteristic. The methods would be entirely different, in particular we hope to use the results of de Jong on resolution of singularities. We would also like to study the motives associated to modular forms from the point of view of birational motives. We would like to complete our project of extending Voevodsky's work to a birational setting by constructing a birational triangulated category of motivic complexes.

当前的申请建议研究两个领域，一个是算术几何，另一个与动机类别相关。 首先，我们广泛的研究主题是飞騨族中的非交换岩泽理论。 Coates 等人提出的非交换主猜想的具体案例的建立最近取得了进展。等人。对于全实数域的微不足道的动机和全实 p 进李扩展。 我们建议研究 Hida 族背景下非交换主猜想的表述。我们接下来要关注的两个具体问题是（a）CM 和非 CM Hida 族中 mu 不变量的研究（b）Hida 族非交换主猜想的公式化并研究其与沿纤维特化的主猜想的关系。 在此过程中，我们建议解决 Coates 等人提出的猜想。椭圆曲线的对偶 Selmer 群结构，在奇素数 p 处具有良好的普通约简，被视为在容许 p 进李扩展上的非交换岩泽代数上的模。 在早期与 B. Kahn 的合作中，我们构建了双有理 Chow 动机的范畴以及在基域 F 上的双有理几何动机的三角范畴，并证明了关于这些类别的一些结果。其中许多结果需要假设场 F 的特征为零。或许可以将其中一些结果扩展到场 F 具有正特性的情况。方法会完全不同，特别是我们希望使用 de Jong 在解决奇点方面的结果。我们还想从双理性动机的角度研究与模块化形式相关的动机。我们希望通过构建动机复合体的双有理三角范畴来完成将 Voevodsky 的工作扩展到双有理环境的项目。