Geometric torsion Galois representations and ramification theory

在这两年的学习期间，我研究了混合特征（0,p）局部场的扭力伽罗瓦表示的分支理论。众所周知，在这样一个局部域的整数环上，由p杀死的有限平面群方案的范畴存在一个包涵到相等特征p的相似范畴中。在本研究项目中，我证明了这个包涵保留了两边的分枝。这使我们能够把分枝的研究简化到更容易的等特征的情况。作为这一结果的应用，我还证明了一个新的阿贝尔变体正则子群的存在性定理，该定理是p进Siegel模形式理论的关键组成部分之一。

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.