Moduli of Galois Representations

Abstract

We develop a theory of moduli of Galois representations. More generally, for an object in a rather general class $\mathfrak{A}$ of noncommutative topological rings, we construct a moduli space of its absolutely irreducible representations of a fixed degree as a (so we call) "f-$\mathfrak{A}$ scheme". Various problems on Galois representations can be reformulated in terms of such moduli schemes. As an application, we show that the "difference" between the strong and weak versions of the finiteness conjecture of Fontaine–Mazur is filled in by the finiteness conjecture of Khare–Moon.