Study of the moduli of Galois representations

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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.