The image of the coefficient space in the universal deformation space of a flat Galois representation of a p-adic field

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.

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 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] .

Based on comparison theorems for Hecke algebras and universal deformation rings with strong restrictions at the critical prime l , as provided by the results of Wiles, Taylor, Diamond, et al., we prove under rather general conditions that the corresponding universal deformation spaces with no restrictions at l can be identified with certain Hecke algebras of l -adic modular forms as conjectured by Gouvêa, thus generalizing previous work of Gouvêa and Mazur. Along the way, we show that the universal deformation spaces we consider are complete intersections, flat over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] l of relative dimension three, in which the modular points form a Zariski dense subset. Furthermore the fibers above [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] l of these spaces are generically smooth.

Deformations and the rigidity method

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.

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

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.

Let X be an irreducible symplectic manifold and L a divisor on X. Assume that L is isotropic with respect to the Beauville-Bogomolov quadratic form. We define the rational Lagrangian locus and the movable locus on the universal deformation space of the pair (X, L). We prove that the rational Lagrangian locus is empty or coincide with the movable locus.

Deformations of Pseudorepresentations

Universal deformation rings need not be complete intersections

We answer a question of M. Flach by showing that there is a linear representation of a profinite group whose (unrestricted) universal deformation ring is not a complete intersection. We show that such examples arise in arithmetic in the following way. There are infinitely many real quadratic fields F for which there is a mod 2 representation of the Galois group of the maximal unramified extension of F whose universal deformation ring is not a complete intersection. Finally, we discuss bounds on the singularities of universal deformation rings of representations of finite groups in terms of the nilpotency of the associated defect groups.

Singular equivalences of Morita type with level, Gorenstein algebras, and universal deformation rings

<p>This thesis is on the representation theory of finite groups. Specifically, it is about finding connections between fusion and universal deformation rings.</p> <p>Two elements of a subgroup <em>N</em> of a finite group Γ are said to be fused if they are conjugate in Γ, but not in <em>N</em>. The study of fusion arises in trying to relate the local structure of Γ (for example, its subgroups and their embeddings) to the global structure of Γ (for example, its normal subgroups, quotient groups, conjugacy classes). Fusion is also important to understand the representation theory of Γ (for example, through the formula for the induction of a character from <em>N</em> to Γ).</p> <p>Universal deformation rings of irreducible mod <em>p</em> representations of Γcan be viewed as providing a universal generalization of the Brauer character theory of these mod <em>p</em> representations of Γ.</p> <p>It is the aim of this thesis to connect fusion to this universal generalization by considering the case when Γ is an extension of a finite group <em>G</em> of order prime to <em>p</em> by an elementary abelian <em>p</em>-group <em>N</em> of rank 2. We obtain a complete answer in the case when <em>G</em> is a dihedral group, and we also consider the case when <em>G</em> is abelian. On the way, we compute for many absolutely irreducible <strong>F</strong><sub>p</sub>Γ-modules <em>V</em>, the cohomology groups H<sup>2</sup>(Γ,Hom<sub><strong>F</strong>p</sub>(V,V) for <em>i</em> = 1, 2, and also the universal deformation rings <em>R</em>(Γ,V).</p>

Let [math] be a CM field and let [math] be the compatible system of residual [math] -valued representations of [math] attached to a regular algebraic conjugate self-dual cuspidal (RACSDC) automorphic representation [math] of [math] , as studied by Clozel, Harris and Taylor (2008) and others. Under mild assumptions, we prove that the fixed-determinant universal deformation rings attached to [math] are unobstructed for all places [math] in a subset of Dirichlet density [math] , continuing the investigations of Mazur, Weston and Gamzon. During the proof, we develop a general framework for proving unobstructedness (with future applications in mind) and an [math] -theorem, relating the universal crystalline deformation ring of [math] and a certain unitary fixed-type Hecke algebra.