Study of the moduli of Galois representations

构造了伽罗瓦表示的模的基本理论，并将其应用于数论中的一些问题。特别地，我们定义了具有受限分支的截断离散估值环的扩展范畴，并证明了它不依赖于任何选择，等价于具有受限分支的完全离散估值域的有限扩展范畴。另外，针对Serre的实数二次域的模性猜想，我们得到了一些二次域模2伽罗瓦表示的不存在性和有限性的有趣结果。

We constructed a foundational theory of moduli of Galois representations and applied it to some problems in number theory. In particular, we defined a category of extensions of truncated discrete valuation ring with restricted ramification and proved that it does not depend on any choices, and that it is equivalent to the category of finite extensions of a complete discrete valuation field with restricted ramification. Also, in relation to Serre's modularity conjecture for real quadratic fields, we obtained interesting results on the non-existence and finiteness of mod 2 Galois representations of some quadratic fields.