Local Galois representations for higher genus curves

本研究的目的是描述局部域上超椭圆曲线的伽罗瓦表示。这个问题的部分结果存在：特别是椭圆曲线，它是已知的如何计算惯性图像，在某些情况下，甚至整个伽罗瓦表示。这个项目的第一步是完成所有剩余的情况下，给出一个明确的解决方案，可以在计算机程序中实现。对于更高的亏格曲线，已知的较少，因此最终的目标是理解特殊曲线族的伽罗瓦表示，对于这些曲线族，可以应用曲线模型理论的最新发展，并再次提供明确的描述。

The aim of this research project is to describe the Galois representation attached to a hyperelliptic curve over a local field. Partial results on this problem exist: in particular, for elliptic curves it is known how to compute the inertia image and, in some cases, even the whole Galois representation. The first step of this project is to complete all the remaining cases, giving an explicit solution which can be implemented in a computer program. For higher genus curves, less is known, so the final objective is to understand the Galois representation for special families of curves, for which recent developments in the theory of models of curves can be applied, and again provide an explicit description of it.