p进局部不变闭链定理的研究和应用

This project focuses on the p-adic Local Invariant Cycle Theorem(LICT for short, which is indeed a conjecture). This conjecture(together with P. Deligne's l-adic one), constructed by the applicant and M. Flach, is used in the reformulation of the Equivariant Tamagawa Number Conjecture(ETNC for short, which is an important approach to BSD conjecture), and also has important applications in p-adic Hodge Theory...In this project, first, the applicant wants to give a direct sheaf theoretic construction and geometric description of the specialization map based on recent work on the p-adic Hodge theory. This should give us some insights on the comparison morphisms, and may lead to the proof of p-adic LICT. Then we want to prove the compatibility between the trace maps defined by Berthelot etc. and the specialization map constructed by the applicant, this will lead to the proof of the p-adic LICT on the slope [0,1) part, and also enough for the required consequences in the reformulation of ETNC. Our idea is to consider the parallel trace maps in the derived category, we can prove that the required compatibility argument can be induced from the "trace map associativity" of our new trace morphisms via derived algebraic geometry. The applicant also wants to consider the p-adic LICT in some special cases by other methods.

本项目关注p进局部不变闭链定理（简称LICT，实际是猜想）的研究。此猜想由M. Flach和申请人提出。和P. Deligne的l-进的猜想一起用于等变Tamagawa数猜想（简称ETNC，是研究BSD猜想的重要途径之一）的重构需要，其本身在p进Hodge领域中也有重要应用。..在本项目中，首先，申请人希望基于最近p进Hodge理论的工作给出p进特化映射在sheaf层面的直接定义和部分几何解释，这将导出对比较态射的新认识，并可能导出p进LICT。其次，申请人希望证明Berthelot等人定义的一类迹映射和申请人构造的p进特化映射的相容性，这将导出限制在斜率[0,1]部分的p进LICT，而且将满足ETNC的重构需要。我们的证明思路是在导出范畴中构造对应的迹映射，利用导出代数几何的工具，将上述相容性转化为证明新迹映射的结合律。最后，我们还将利用其它方法研究一些特殊和具体情形下的p进LICT。