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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635521
• 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.

目前的应用建议研究两个领域，一个是算术几何领域，另一个是与动机范畴有关的领域。首先，我们广泛的研究主题是Hida家族的非交换Iwasawa理论。最近在建立由Coates等人提出的非交换主猜想的具体情况方面取得了进展，该猜想是针对全实数域的平凡动机和全实数p进李扩展提出的。我们提出在希达族的背景下研究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族中的模不变量的研究(b) Hida族的非交换主猜想的公式以及研究它与沿纤维专一化的主猜想的关系。在此过程中，我们建议解决Coates等人提出的一个猜想。在可容许的p进Lie扩展上，作为非交换Iwasawa代数上的模，讨论了在奇素数p处具有良好常约化的椭圆曲线的对偶Selmer群的结构。在早期与B. Kahn的合作中，我们在基域F上构造了出生周动机范畴和出生几何动机的三角化范畴，并证明了这些范畴的一些结果。这些结果中的许多都需要假设场F的特征为零。当场F具有正特征时，可能将这些结果中的一些推广到这种情况。方法将会完全不同，特别是我们希望使用de Jong在奇点解析上的结果。我们还想从出生动机的角度来研究与模形式相关的动机。我们想完成我们的项目，将Voevodsky的工作扩展到一个双民族背景下，通过构建一个双民族三角化的动机复合体类别。