Iwasawa theory, Galois representations and motives

目前的申请建议研究两个领域，一个是算术几何领域，另一个是与动机类别有关的领域。首先，我们的主要研究主题是Hida家族中的非对易岩泽理论。最近，Coates等人提出的非对易主要猜想的具体情形的建立取得了进展。艾尔关于全实数域的平凡动机和全实p-进Lie扩张。我们建议在Hida族的背景下研究非对易主要猜想的提法。我们接下来要集中讨论的两个具体问题是：(A)研究CM和非CM Hida族中的Mu不变量；(B)给出Hida族的一个非对易主要猜想，并研究它与沿纤维专门化的主要猜想的关系。在此过程中，我们建议解决Coates等人提出的一个猜想。关于在奇素数p处具有良好普通约化的椭圆曲线的对偶Selmer群的结构，被认为是可容许的p-进Lie扩张上的非交换Iwa awa代数上的模.在之前与B.Kahn的工作中，我们构造了基域F上的二元Chow动机范畴和二元几何动机三角范畴，并证明了关于这些范畴的一些结果.这些结果中的许多都要求假设场F的特征为零。有可能将这些结果中的一些推广到场F具有正特征的情况。方法将是完全不同的，特别是我们希望使用德容的结果来解决奇点。我们还想从双生动机的角度来研究与模块形式相关的动机。我们想要通过构建一个动机复合体的二元三角范畴来完成我们的项目，将沃沃茨基的工作扩展到一个二元背景下。

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.