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.

目前的申请提出研究两个领域，一个是算术几何，另一个是与动机类别有关的。 首先，我们广泛的研究课题是非交换岩泽理论在希达家庭。科茨等人提出的非对易主猜想在建立特定情形方面有了新的进展。等人关于全真实的数域的平凡动机和全真实的p-adic Lie扩张. 我们提出在希达族的背景下研究非交换主猜想的形成.两个具体的问题，我们要集中在（一）研究的μ不变的CM和非CM希达家庭（B）制定一个非交换的主要猜想希达家庭和研究其关系的主要猜想的专业化沿着纤维。 沿着的方式，我们建议解决一个猜想制定的科茨，等。的结构上的对偶塞尔默群的椭圆曲线具有良好的普通约化在一个奇素数p，被认为是一个模块上的非交换岩泽代数上的可容许的p-adic李扩张。 在早期与B的合作中。Kahn的方法，我们构造了基域F上的双有理Chow动机范畴和双有理几何动机的三角范畴，并证明了这些范畴的一些结果。这些结果中有许多需要假设场F是特征零。当场F具有正特征时，有可能将某些结果推广到这种情况。方法将是完全不同的，特别是我们希望使用的结果德容决议的奇异性。我们还想从双理性动机的角度来研究与模式形式相关的动机。我们要完成我们的项目扩展Voevodsky的工作，以一个双理性的设置，通过构建一个双理性的三角形类motivic复合物。