Extending p-divisible Groups and Barsotti–Tate Deformation Ring in the Relative Case

Let $k$ be a perfect field of characteristic $p> 2$, and let $K$ be a finite totally ramified extension of $W(k)\big[\frac{1}{p}\big]$ of ramification degree $e$. We consider an unramified base ring $R_0$ over $W(k)$ satisfying certain conditions, and let $R = R_0\otimes _{W(k)}\mathcal{O}_K$. Examples of such $R$ include $R = \mathcal{O}_K[\![s_1, \ldots , s_d]\!]$ and $R = \mathcal{O}_K\langle t_1^{\pm 1}, \ldots , t_d^{\pm 1}\rangle $. We show that the generalization of Raynaud’s theorem on extending $p$-divisible groups holds over the base ring $R$ when $e < p-1$, whereas it does not hold when $R = \mathcal{O}_K[\![s]\!]$ with $e \geq p$. As an application, we prove that if $R$ has Krull dimension $2$ and $e < p-1$, then the locus of Barsotti–Tate representations of $\textrm{Gal}(\overline{R}\big[\frac{1}{p}\big]/R\big[\frac{1}{p}\big])$ cuts out a closed subscheme of the universal deformation scheme. If $R = \mathcal{O}_K[\![s]\!]$ with $e \geq p$, we prove that such a locus is not $p$-adically closed.

Smoothness of definite unitary eigenvarieties at critical points

We compute an upper bound for the dimension of the tangent spaces at classical points of certain eigenvarieties associated with definite unitary groups, especially including the so-called critically refined cases. Our bound is given in terms of “critical types” and when our bound is minimized it matches the dimension of the eigenvariety. In those cases, which we explicitly determine, the eigenvariety is necessarily smooth and our proof also shows that the completed local ring on the eigenvariety is naturally a certain universal Galois deformation ring.

ON THE IRREDUCIBLE COMPONENTS OF SOME CRYSTALLINE DEFORMATION RINGS

We adapt a technique of Kisin to construct and study crystalline deformation rings of $G_{K}$ for a finite extension $K/\mathbb{Q}_{p}$ . This is done by considering a moduli space of Breuil–Kisin modules, satisfying an additional Galois condition, over the unrestricted deformation ring. For $K$ unramified over $\mathbb{Q}_{p}$ and Hodge–Tate weights in $[0,p]$ , we study the geometry of this space. As a consequence, we prove that, under a mild cyclotomic-freeness assumption, all crystalline representations of an unramified extension of $\mathbb{Q}_{p}$ , with Hodge–Tate weights in $[0,p]$ , are potentially diagonalizable.

The inverse deformation problem

Given a commutative complete local noetherian ring $A$ with finite residue field $\boldsymbol{k}$, we show that there is a topologically finitely generated profinite group $\unicode[STIX]{x1D6E4}$ and an absolutely irreducible continuous representation $\overline{\unicode[STIX]{x1D70C}}:\unicode[STIX]{x1D6E4}\rightarrow \text{GL}_{n}(\boldsymbol{k})$ such that $A$ is a universal deformation ring for $\unicode[STIX]{x1D6E4},\overline{\unicode[STIX]{x1D70C}}$.

LIFTING TORSION GALOIS REPRESENTATIONS

Let $p\geqslant 5$ be a prime, and let ${\mathcal{O}}$ be the ring of integers of a finite extension $K$ of $\mathbb{Q}_{p}$ with uniformizer ${\it\pi}$. Let ${\it\rho}_{n}:G_{\mathbb{Q}}\rightarrow \mathit{GL}_{2}\left({\mathcal{O}}/({\it\pi}^{n})\right)$ have modular mod-${\it\pi}$ reduction $\bar{{\it\rho}}$, be ordinary at $p$, and satisfy some mild technical conditions. We show that ${\it\rho}_{n}$ can be lifted to an ${\mathcal{O}}$-valued characteristic-zero geometric representation which arises from a newform. This is new in the case when $K$ is a ramified extension of $\mathbb{Q}_{p}$. We also show that a prescribed ramified complete discrete valuation ring ${\mathcal{O}}$ is the weight-$2$ deformation ring for $\bar{{\it\rho}}$ for a suitable choice of auxiliary level. This implies that the field of Fourier coefficients of newforms of weight 2, square-free level, and trivial nebentype that give rise to semistable $\bar{{\it\rho}}$ of weight 2 can have arbitrarily large ramification index at $p$.

A deformation problem for Galois representations over imaginary quadratic fields

We prove the modularity of minimally ramified ordinary residually reducible p-adic Galois representations of an imaginary quadratic field F under certain assumptions. We first exhibit conditions under which the residual representation is unique up to isomorphism. Then we prove the existence of deformations arising from cuspforms on GL2(AF) via the Galois representations constructed by Taylor et al. We establish a sufficient condition (in terms of the non-existence of certain field extensions which in many cases can be reduced to a condition on an L-value) for the universal deformation ring to be a discrete valuation ring and in that case we prove an R=T theorem. We also study reducible deformations and show that no minimal characteristic 0 reducible deformation exists.

Representations related to CM elliptic curves

In [10], Mazur showed that the p-adic lifts of a given absolutely irreducible representation are parametrized by a universal deformation ξ:Gℚ, S → GL2() where has the form . (Here Gℚ, S is the Galois group over ℚ of a maximal algebraic extension unramified outside a finite set S of rational primes.) In [1, 3, 10], situations were investigated where the universal deformation ring turned out to be ℚp[[T1T2, T3]] (i.e. r = 3, I = (0)). In [2], the tame relation of algebraic number theory led to more complicated universal deformation rings, ones whose prime spectra consist essentially of four-dimensional sheets.