Deformations and the rigidity method

Let 𝒜 be the ring ℤ p [[t 0, t 1, t ∞]]/((t 0 + 1)(t 1 + 1)(t ∞ + 1) − 1)equipped with the non-trivial action of G ℚ ≔ Gal(ℚ¯/ℚ) described in the introduction. In Ihara (1986b), Ihara constructs a universal cocycle arising from the action of Gal(ℚ¯/ℚ) on certain quotients of the Jacobians of the Fermat curves for each n ≥ 1. This paper gives a different construction of part of Ihara's cocycle by considering the universal deformation of certain two-dimensional representations of Πℚ¯, the algebraic fundamental group of ℙ1(ℚ¯)\\{0, 1, ∞}. More precisely, we determine the universal deformation ring subject to certain deformation conditions arising from a residual representation Belyĭ's Rigidity Theorem is then used to extend each such universal deformation to a representation of Π K , where K is a finite cyclotomic extension of ℚ(μ p ∞ ). When ρ¯ is the representation arising from the p-division points of the Legendre family of elliptic curves, we give a geometric construction of one such extended universal deformation ρ, and show that part of Ihara's cocycle can be recovered by specializing ρ at infinity.

The coefficient space is a kind of resolution of singularities of the universal flat deformation space for a given Galois representation of some local field. It parametrizes (in some sense) the finite flat models for the Galois representation. The aim of this note is to determine the image of the coefficient space in the universal deformation space.

This article deals with universal deformations of dihedral representations with a particular focus on the question when the universal deformation is dihedral. Results are obtained in three settings: (1) representation theory, (2) algebraic number theory, (3) modularity. As to (1), we prove that the universal deformation is dihedral if all infinitesimal deformations are dihedral. Concerning (2) in the setting of Galois representations of number fields, we give sufficient conditions to ensure that the universal deformation relatively unramified outside a finite set of primes is dihedral, and discuss in how far these conditions are necessary. As side-results, we obtain cases of the unramified Fontaine–Mazur conjecture, and in many cases positively answer a question of Greenberg and Coleman on the splitting behaviour at p of p-adic Galois representations attached to newforms. As to (3), we prove a modularity theorem of the form ‘ $$R=\mathbb {T}$$ ’ for parallel weight one Hilbert modular forms for cases when the minimal universal deformation is dihedral.

Analysis of the condition for universal deformations in compressible finite elasticity leads to the identification of two classes of strain energies that support universal irrotational deformations. These classes represent a significant generalization of the classes of strain energies for which universal deformation solutions have been hitherto available. Closed form solutions are presented for cylindrically and spherically radial irrotational deformation for all of these strain energies. Some questions about maximality of some classes of strain energies that support cylindrically radial universal deformations are resolved.

We prove that the universal unramified deformation ring [math] of a continuous Galois representation [math] (for a totally real field [math] and finite field [math] ) is finite over [math] in many cases. We also prove (under similar hypotheses) that the universal deformation ring [math] is finite over the local deformation ring [math] .

We investigate the case of deformations of even Galois representations. Our methods are the group theoretic ones mainly developed by Nigel Boston to study odd representations. We present conditions for Borel and tame cases under which the universal deformation ring is isomorphic to ℤp[[T]] and where we compute the universal deformation explicitly. Furthermore we produce a family of examples of totally real S3 extensions which satisfy the above conditions in the tame case and we give examples in the Borel case. Finally we study the change of the deformation space under enlarging the ramification and thus give an example of an even representation that is not twist‐finite.

Openness of the Galois image for τ-modules of dimension 1

We determine the universal deformation ring, in the sense of Mazur, of a residual representation $\bar \rho :G_K\to {\rm GL}_2(k)$ , where k is a finite field of characteristic p and K is a local field of residue characteristic p . As one might hope for, but is not proven in the global case, the deformation ring is a complete intersection, flat over W(k) , with the exact number of equations given by the dimension of $H^2(G_K,{\rm ad}_{\bar \rho})$ . We then go on to determine the ordinary locus inside the deformation space and, using ideas of Mazur, apply this to compare the universal and the universal ordinary deformation spaces. Provided that the universal ring for ordinary deformations with fixed determinant is finite flat over W(k) , as was shown in many cases by Diamond, Fujiwara, Taylor–Wiles and Wiles, we show that the corresponding universal deformation ring – with no restriction of ordinariness or fixed determinant – is a complete intersection, finite flat over W(k) of the dimension conjectured by Mazur, provided that the restriction of $\det (\bar \rho)$ to the inertia subgroup is different from the inverse cyclotomic character.

We study G-valued Galois deformation rings with prescribed properties, where G is an arbitrary (not necessarily connected) reductive group over an extension of Z_l for some prime l. In particular, for the Galois groups of p-adic local fields (with p possibly equal to l) we prove that these rings are generically smooth, compute their dimensions, and show that functorial operations on Galois representations give rise to well-defined maps between the sets of irreducible components of the corresponding deformation rings. We use these local results to prove lower bounds on the dimension of global deformation rings with prescribed local properties. Applying our results to unitary groups, we improve results in the literature on the existence of lifts of mod l Galois representations, and on the weight part of Serre's conjecture.

For a fixed mod $p$ automorphic Galois representation, $p$-adic automorphic Galois representations lifting it determine points in universal deformation space. In the case of modular forms and under some technical conditions, Bockle showed that every component of deformation space contains a smooth modular point, which then implies their Zariski density when coupled with the infinite fern of Gouvea-Mazur. We generalize Bockle's result to the context of polarized Galois representations for CM fields, and to two dimensional Galois representations for totally real fields. More specifically, under assumptions necessary to apply a small $R = \mathbb{T}$ theorem and an assumption on the local mod $p$ representation, we prove that every irreducible component of the universal polarized deformation space contains an automorphic point. When combined with work of Chenevier, this implies new results on the Zariski density of automorphic points in polarized deformation space in dimension three.

Deformations of Pseudorepresentations

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.

We prove the vanishing of the geometric Bloch-Kato Selmer group for the adjoint representation of a Galois representation associated to regular algebraic polarized cuspidal automorphic representations under an assumption on the residual image. Using this, we deduce that the localization and completion of a certain universal deformation ring for the residual representation at the characteristic zero point induced from the automorphic representation is formally smooth of the correct dimension. We do this by employing the Taylor-Wiles-Kisin patching method together with Kisin's technique of analyzing the generic fibre of universal deformation rings. Along the way we give a characterization of smooth closed points on the generic fibre of Kisin's potentially semistable local deformation rings in terms of their Weil-Deligne representations.

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.

Universal deformation rings, endo-trivial modules, and semidihedral and generalized quaternion 2-groups