On local Galois deformation rings: generalised tori

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.

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 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.

<p>The goal of this thesis is to study non-commutative deformation rings of representations of algebras. The main motivation is to provide a generalization of the deformation theory over commutative local rings studied by B. Mazur, M. Schlessinger and others. The latter deformation theory has played an important role in number theory, and in particular in the proof of Fermat's Last Theorem.</p> <p>The thesis is divided into two parts.</p> <p>In the first part, <em>A</em> is an arbitrary λ-algebra for a complete local commutative Noetherian ring λ with residue field <em>k</em>. A category Ĉ is defined whose objects are complete local λ-algebras <em>R</em> with residue field <em>k</em> such that <em>R</em> is a quotient ring of a power series algebra over λ in finitely many non-commuting variables. If <em>V</em> is a finite dimensional <em>k</em>-vector space that is also a left <em>A</em>-module and that satisfies a natural finiteness condition, it is proved that <em>V</em> has a so-called versal deformation ring <em>R(A,V)</em>. More precisely, <em>R(A,V)</em> is an object in Ĉ such that the isomorphism class of every lift of <em>V</em> over an object <em>R</em> in Ĉ arises from a morphism α : <em>R(A,V)</em>→ <em>R</em> in Ĉ and α is unique if <em>R</em> is the ring of dual numbers <em>k</em>[ϵ].</p> <p>In the second part, two particular examples of λ, <em>A</em> and <em>V</em> are studied and the versal deformation ring <em>R(A,V)</em> is determined in each of these cases. In the first example, λ=<em>k</em>, <em>A</em> is a series of non-commutative <em>k</em>-algebras depending on a parameter <em>r≥2</em>, and <em>V</em> is a particular quotient module of <em>A</em>; it is shown that <em>R(A,V)</em> is isomorphic to <em>A</em>. The second example builds on the first example when <em>r=2</em> and uses that, if additionally the characteristic of <em>k</em> is 2, then <em>A</em> is isomorphic to the group ring <em>k[D<sub>8</sub>]</em> of a dihedral group <em>D<sub>8</sub></em> of order 8.</p> <p>It is shown that if <em>k</em> is perfect and <em>W</em> is the ring of infinite Witt vectors over <em>k</em>, then <em>R(W[D<sub>8</sub>],V)</em> is isomorphic to <em>W[D<sub>8</sub>]</em>.</p>

We show that framed deformation rings of mod p representations of the absolute Galois group of a p-adic local field are complete intersections of expected dimension. We determine their irreducible components and show that they and their special fibres are normal and complete intersection. As an application, we prove density results of loci with prescribed p-adic Hodge theoretic properties.

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.

Given a continuous, odd, semi-simple <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn></mml:math>-dimensional representation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi> <mml:mrow><mml:mi>ℚ</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mi>p</mml:mi></mml:mrow> </mml:msub></mml:math> over a finite field of odd characteristic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> and a prime <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ℓ</mml:mi></mml:math> not dividing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:math>, we study the relation between the universal deformation rings of the corresponding pseudo-representations for the groups <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi> <mml:mrow><mml:mi>ℚ</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mi>ℓ</mml:mi><mml:mi>p</mml:mi></mml:mrow> </mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi> <mml:mrow><mml:mi>ℚ</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mi>p</mml:mi></mml:mrow> </mml:msub></mml:math>. As a related problem, we investigate when the universal pseudo-representation arises from an actual representation over the universal deformation ring. Under some hypotheses, we prove analogues of theorems of Boston and Böckle for the reduced pseudo-deformation rings. We improve these results when the pseudo-representation is unobstructed and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> does not divide <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>. When the pseudo-representation is unobstructed and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> divides <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>, we prove that the universal deformation rings in characteristic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> of the pseudo-representation for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi> <mml:mrow><mml:mi>ℚ</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mi>ℓ</mml:mi><mml:mi>p</mml:mi></mml:mrow> </mml:msub></mml:math> are not local complete intersection rings. As an application of our main results, we prove a big <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>R</mml:mi><mml:mo>=</mml:mo><mml:mi>𝕋</mml:mi></mml:mrow></mml:math> theorem.

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.

We compute the deformation rings of two dimensional mod l representations of Gal(Fbar/F) with fixed inertial type, for l an odd prime, p a prime distinct from p and F/Q_p a finite extension. We show that in this setting (when p is also odd) an analogue of the Breuil-M\'{e}zard conjecture holds, relating the special fibres of these deformation rings to the mod l reduction of certain irreducible representations of GL_2(O_F).

Density of Selmer ranks in families of even Galois representations, Wiles' formula, and global reciprocity

We study representations of profinite groups over some particular type of local rings. More specifically, suppose a profinite group and its continuous finite dimensional representation over a finite field k are given. Then we are interested in studying all possibilities of lifting this representation to a representation over a ring that is complete, local, noetherian and whose residue field is isomorphic to k. For each problem of the above described type, an associated deformation functor can be defined. If such a functor is representable then the object representing it is called the universal deformation ring of the given representation. The following inverse problem is central in the thesis: which rings do occur as universal deformation rings in the introduced setting? The main results of the thesis go in two directions. Firstly, we show that every complete noetherian local commutative ring R with residue field k can be realized as a universal deformation ring of a continuous representation of a profinite group. This way we completely answer the stated question in its general form. Secondly, we address its modification and provide a non-trivial necessary condition for characteristic zero universal deformation rings of representations of groups that are finite.

We use the Taylor-Wiles-Kisin patching method to investigate the\nmultiplicities with which Galois representations occur in the mod $\\ell$\ncohomology of Shimura curves over totally real number fields. Our method relies\non explicit computations of local deformation rings done by Shotton, which we\nuse to compute the Weil class group of various deformation rings. Exploiting\nthe natural self-duality of the cohomology groups, we use these class group\ncomputations to precisely determine the structure of a patched module in many\nnew cases in which the patched module is not free (and so multiplicity one\nfails).\n Our main result is a "multiplicity $2^k$" theorem in the minimal level case\n(which we prove under some mild technical hypotheses), where $k$ is a number\nthat depends only on local Galois theoretic information at the primes dividing\nthe discriminant of the Shimura curve. Our result generalizes Ribet's classical\nmultiplicity 2 result and the results of Cheng, and provides progress towards\nthe Buzzard-Diamond-Jarvis local-global compatibility conjecture. We also prove\na statement about the endomorphism rings of certain modules over the Hecke\nalgebra, which may have applications to the integral Eichler basis problem.\n

&lt;p&gt;The main objective of deformation theory is to study the behavior of mathematical objects, such as modules or group representations, under perturbations. This theory is useful in both pure and applied mathematics and has led to the solution of many long-standing problems. For example, in number theory, universal deformation rings of Galois representations played an important role in the proof of Fermat’s Last Theorem by Wiles and Taylor.&lt;/p&gt;&lt;p&gt;In this thesis, we consider the case when &lt;em&gt;SD&lt;sub&gt;n&lt;/sub&gt;&lt;/em&gt; is a semidihedral 2-group of order 2&lt;sup&gt;n+1&lt;/sup&gt; for n ≥ 3 and &lt;em&gt;k&lt;/em&gt; is an algebraically closed field of characteristic 2. The indecomposable &lt;em&gt;kSD&lt;sub&gt;n&lt;/sub&gt;&lt;/em&gt;-modules have been completely described by Bondarenko and Drozd, and Crawley-Boevey. We concentrate on so-called endo-trivial &lt;em&gt;kSD&lt;sub&gt;n&lt;/sub&gt;&lt;/em&gt;-modules, which possess a well-defined universal deformation ring by work of Bleher and Chinburg. Using the classification of Carlson and Thevenaz of all endo-trivial &lt;em&gt;kSD&lt;sub&gt;n&lt;/sub&gt;&lt;/em&gt;-modules, we show that the universal deformation ring of every endo-trivial &lt;em&gt;kSD&lt;sub&gt;n&lt;/sub&gt;&lt;/em&gt;-module is isomorphic to the group ring W [ℤ/2 x ℤ/2], where W = W (k) is the ring of infinite Witt vectors over k.&lt;/p&gt;

Universal deformation rings and fusion

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.