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.

当前的应用建议研究两个领域，一个领域，算术几何形状，另一个与动机类别有关。 首先，我们的广泛研究主题是HIDA家庭中的非共同iWasawa理论。 Coates等人提出的非交通性主要猜想的特定案例最近取得了进展。 al。对于一个完全真实的数字字段的琐碎动机和完全真实的P-Adic谎言扩展。 我们建议在HIDA家族的背景下研究非交换性主要猜想的制定。然后，我们要集中精力的两个具体问题是（a）在CM和非CM HIDA家族中进行MU不变的研究（b）HIDA家族的非交通性主要猜想的制定，并研究其与沿纤维专业化的主要猜想的关系。 一路上，我们建议解决Coates等人提出的猜想。关于在奇数p上的椭圆形曲线的双重椭圆形曲线的结构，在奇数prime p上良好，被视为一个非交换性iwasawa代数的模块，而不是可允许的p adic lie扩展。 在与B. Kahn的早期合作中，我们构建了Birational Chow动机的类别，也建立了基本场上F的三角分类类别，并证明了这些类别的一些结果。这些结果中的许多结果都需要假设F场F为零。当场F具有积极特征时，可能会将其中一些结果扩展到案例。这些方法将完全不同，特别是我们希望使用De Jong的结果来解决奇异之处。我们还想从异性动机的角度研究与模块化形式相关的动机。我们想通过构建一个典范的三角态类别的动机综合体来完成将Voevodsky的作品扩展到男子式环境的项目。