Moduli of Galois Representations

Abstract

The theme of this thesis is the study of moduli stacks of representations of an associative algebra, with an eye toward continuous representations of profinite groups such as Galois groups. The central object of study is the geometry of the map \(\bar{\psi}\) from the moduli stack of representations to the moduli scheme of pseudorepresentations. The first chapter culminates in showing that \(\bar{\psi}\) is very close to an adequate moduli space of Alper. In particular, \(\bar{\psi}\) is universally closed. The second chapter refines the results of the first chapter. In particular, certain projective subschemes of the fibers of \(\bar{\psi}\) are identified, generalizing a suggestion of Kisin. The third chapter applies the results of the first two chapters to moduli groupoids of continuous representations and pseudorepresentations of profinite algebras. In this context, the moduli formal scheme of pseudorepresentations is semi-local, with each component Spf \(B_\bar{D}\) being the moduli of deformations of a given finite field-valued pseudorepresentation \(\bar{D}\). Under a finiteness condition, it is shown that \(\bar{\psi}\) is not only formally finite type over Spf \(B_\bar{D}\), but arises as the completion of a finite type algebraic stack over Spec \(B_\bar{D}\). Finally, the fourth chapter extends Kisin's construction of loci of coefficient spaces for p-adic local Galois representations cut out by conditions from p-adic Hodge theory. The result is extended from the case that the coefficient ring is a complete Noetherian local ring to the more general case that the coefficient space is a Noetherian formal scheme.