Abstract

We construct and study the moduli of continuous representations of a profinite group with integral p-adic coefficients. We present this moduli space over the moduli space of continuous pseudorepresentations and show that this morphism is algebraizable. When this profinite group is the absolute Galois group of a p-adic local field, we show that these moduli spaces admit Zariski-closed loci cutting out Galois representations that are potentially semi-stable with bounded Hodge–Tate weights and a given Hodge and Galois type. As a consequence, we show that these loci descend to the universal deformation ring of the corresponding pseudorepresentation.

Highlights

  • Mazur [35] initiated the systematic study of the moduli of representations of a Galois group G in terms of complete local deformation rings

  • The main idea pursued is that the adjoint action of GLd on the scheme of framed representations RepR,d, whose associated quotient stack is RepdR, has geometric invariant theoretic (GIT) quotient nearly equal to PsRdR

  • We establish that the GIT quotient and PsRdR naturally have identical geometric points because each set of geometric points naturally corresponds to isomorphism classes of semi-simple representations

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Summary

Overview

Mazur [35] initiated the systematic study of the moduli of representations of a Galois group G in terms of complete local deformation rings. For a fixed residual representation ρwith coefficients in the finite residue field F, which admits a universal deformation ring Rρ, the resulting moduli space Spf Rρis “purely formal” in the sense that the underlying algebraic scheme Spec F is 0-dimensional. The fact that ψ is algebraic of finite type and universally closed can be used to produce a potentially semi-stable pseudodeformation ring. We remark that the correct notion of “a global pseudorepresentation that is locally potentially semi-stable” is more restrictive than “a global pseudorepresentation such that its restriction to each decomposition group over p is potentially semi-stable.” This is well-illustrated through the explicit example of a 2-dimensional global ordinary pseudodeformation ring, which we discuss in Sect. We emphasize that Theorem A is based on a study of the moduli of representations of a finitely generated associative algebra over a Noetherian ring in Sect. The conclusions of Theorem A may be viewed as generalizations, allowing for the profinite topology and non-zero characteristic, of parts of the investigations of Le Bruyn [30,31] (building on [39]) in non-commutative algebraic geometry

Summary outline
Moduli of representations of a finitely generated group or algebra
Moduli spaces of representations and pseudorepresentations
Cayley–Hamilton algebras are polynomial identity rings
Invariant theory
Generalized matrix algebras
Algebraic families of Galois representations
Consequences of formal GAGA for ψ
Families of Étale φ-modules and Kisin modules
Algebraic families of Étale φ-modules
Functors of lattices and affine grassmannians
A universal family of Kisin modules in characteristic 0
Background and notation
Period maps in families
Algebraic families of potentially semi-stable Galois representations
Families of semi-stable Galois representations with bounded Hodge–Tate weight
Galois type
Potentially semi-stable pseudorepresentations
Global potentially semi-stable pseudodeformation rings
C Dof for any finite extension
Full Text
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