Geometric torsion Galois representations and ramification theory

在这两年的研究中，我研究了具有混合特征(0，p)的局域场的挠伽罗瓦表示的一个分支理论。众所周知，在这样一个局部域的整数环上被p扼杀的有限平坦群方案范畴中存在一个包含到相同特征p的相似范畴中。在这个研究项目中，我证明了这个包含保持两边的分支。这使我们能够将对分支的研究简化为特征相等的更简单的情况。作为这一结果的一个应用，我还证明了一个新的Abelian簇的标准子群的存在定理，它是p-进Siegel模形式理论的重要组成部分之一。

In this two-year study period, I had investigated a ramification theory of torsion Galois representations of local fields of mixed characteristic (0,p). It is well-known that there exists an inclusion from the category of finite flat group schemes killed by p over the ring of integers of such a local field into a similar category for equal characteristic p. In this research project, I proved that this inclusion preserves ramification of both sides. This enables us to reduce the study of ramification to the much easier case of equal characteristic. As an application of this result, I also proved a new existence theorem of canonical subgroups of Abelian varieties, which is one of the key ingredients of the theory of p-adic Siegel modular forms.