Algebraic families of Galois representations and potentially semi-stable pseudodeformation rings

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

Pseudorepresentations of weight one are unramified

“…The idea behind this definition, as we shall see shortly below, is to capture the notion that the product (h − ψ(h))(φ − α) is identically zero, rather than just of the form 0 * 0 0 . There is presumably a close relationship between this definition and the definition of ordinary pseudo-characters in Wake, Wang-Erickson (see [WWE17] and §7.3 of [WE18]), although in our context it is important that we can work in non-p distinguished situations by choosing an eigenvalue of Frobenius, which amounts to a partial resolution of the corresponding deformation rings (presumably such modifications could also be adapted to [WE18]). On the other hand, we do exploit the crucial idea due to Wang-Erickson that the notion of ordinarity for pseudo-representations should be a global rather than local condition.…”

Ordinary pseudorepresentations and modular forms

“…A 2-dimensional pseudorepresentation of G Q with values in a ring A is the data of two functions {tr, det} that satisfy conditions as if they were the trace and determinant of a representation G Q → GL 2 (A). The (fine) moduli of pseudorepresentations may be thought of as the coarse moduli of Galois representations produced by geometric invariant theory [WE15,Thm. A].…”

“…For our purposes, the important properties of Cayley-Hamilton representations are the following (see [WE15,Prop. 3…”

Deformation Conditions for Pseudorepresentations

“…The main tool we use to bridge the gap between G-pseudorepresentations and G-modules is the category CH G of Cayley-Hamilton representations of G, which was introduced by Chenevier [Che14], building on work of Bellaïche-Chenevier [BC09], and further developed by second-named author [WE18a]. The data of a Cayley-Hamilton representation of G includes a Z p [G]-module, so it is possible to relate these objects to condition C. Moreover, CH G is closely related to both Rep G and PsR G ; the key properties can be summarized as follows:…”