On isomorphisms between deformation rings and Hecke rings

Inventiones mathematicae -     

Singularities of ordinary deformation rings

Mathematische Zeitschrift - Let $$R^{\mathrm {univ}}$$ be the universal deformation ring of a residual representation of a local Galois group. Kisin showed that many loci in...     

Discrete semi-stable distributions

The purpose of this paper is to introduce and study the concepts of discrete semi-stability and geometric semi-stability for distributions with support inZ ₊. We offer several properties, including characterizations, of discrete semi-stable distributions. We establish that these distributions posses the property of infinite divisibility and that their probability generating functions admit canonical representations that are analogous to those of their continuous counterparts. Properties of discrete geometric semi-stable distributions are deduced from the results obtained for discrete semi-stability. Several limit theorems are established and some examples are constructed.     

Universal deformation rings and cyclic blocks

In this paper we determine the universal deformation rings of certain modular representations of finite groups which belong to cyclic blocks. The representations we consider are those for which every endomorphism is stably equivalent to multiplication by a scalar. We then apply our results to study the counterparts for universal deformation rings of conjectures about embedding problems in Galois theory. Received July 19, 1999 / Revised May 13, 2000 / Published online October 30, 2000     

Temperature deformation of laminated composite rings

The results of dilatometric measurements on wound circular glass-reinforced plastic rings composed of glass tape and epoxy-phenolic resin are presented. The changes in inside and outside diameter were measured on the temperature interval from 20 to 100°C at various diameter ratios. The theoretical conclusion [1] that anisotropy has a strong influence on the temperature deformation of circular cylinders and rings of composite material is confirmed.Moscow Power Engineering Institute. Translated from Mekhanika Polimerov, No. 6, pp. 1131–1132, November–December, 1969.     

Galois deformation theory for norm fields and flat deformation rings

Let K be a finite extension of Qp, and choose a uniformizer π∈K, and put . We introduce a new technique using restriction to to study flat deformation rings. We show the existence of deformation rings for -representations “of height ≤h” for any positive integer h, and prove that when h=1 they are isomorphic to “flat deformation rings”. This -deformation theory has a good positive characteristics analogue of crystalline representations in the sense of Genestier-Lafforgue. In particular, we obtain a positive characteristic analogue of crystalline deformation rings, and can analyze their local structure.     

Universal deformation rings and dihedral 2-groups

Let k be an algebraically closed field of characteristic 2,and let W be the ring of infinite Witt vectors over k. Supposethat D is a dihedral 2-group. We prove that the universal deformationring R(D, V) of an endo-trivial kD-module V is always isomorphicto W [/2x/2]. As a consequence, we obtain a similar result formodules V with stable endomorphism ring k belonging to an arbitrarynilpotent block with defect group D. This confirms, for suchV, conjectures on the ring structure of the universal deformationring of V that had previously been shown for V belonging tocyclic blocks or to blocks with Klein four defect groups.     

Rank determines semi-stable conductor

Suppose that E₁ and E₂ are elliptic curves over the rational field, , such that for all quadratic fields . We prove that their conductors N(E₁), and N(E₂) are equal up to squares. If for all quadratic fields , then the same conclusion holds, provided the 2-parts of their Tate-Shafarevich groups are finite.     

Universal deformation rings and self-injective Nakayama algebras

Let k be a field and let Λ be an indecomposable finite dimensional k-algebra such that there is a stable equivalence of Morita type between Λ and a self-injective split basic Nakayama algebra over k. We show that every indecomposable finitely generated Λ-module V has a universal deformation ring $R(\mathrm{\Lambda },V)$ and we describe $R(\mathrm{\Lambda },V)$ explicitly as a quotient ring of a power series ring over k in finitely many variables. This result applies in particular to Brauer tree algebras, and hence to p-modular blocks of finite groups with cyclic defect groups.     

Stable and semi-stable unicyclic graphs

We show by a constructive proof, that if a unicyclic graph has a transposition in its automorphism group, then it is stable. Using a similar technique, we also determine which unicyclic graphs are not semi-stable.     

Universal deformation rings and generalized quaternion defect groups

We determine the universal deformation rings R(G,V) of certain mod 2 representations V of a finite group G which belong to a 2-modular block of G whose defect groups are isomorphic to a generalized quaternion group D. We show that for these V, a question raised by the author and Chinburg concerning the relation of R(G,V) to D has an affirmative answer. We also show that R(G,V) is a complete intersection even though R(G/N,V) need not be for certain normal subgroups N of G which act trivially on V.     

The deformation theory of sheaves of commutative rings

We define a sheaf of abelian groups whose cohomology is represented by the cotangent complex, permitting a rapid introduction to the theory of the cotangent complex in the same generality as it was defined by Illusie, but avoiding simplicial methods. We show how obstructions to some standard deformation problems arise as the classes of torsors under and gerbes banded by this sheaf. This generalizes results of Quillen, Rim, and Gaitsgory.     

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. The first author was supported in part by NSF Grant DMS01-39737 and NSA Grant H98230-06-1-0021. The second author was supported in part by NSF Grants DMS00-70433 and DMS05-00106.     

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 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. To cite this article: F.M. Bleher, T. Chinburg, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Universal deformation rings and Klein four defect groups

In this paper, the universal deformation rings of certain modular representations of a finite group are determined. The representations under consideration are those which are associated to blocks with Klein four defect groups and whose stable endomorphisms are given by scalars. It turns out that these universal deformation rings are always subquotient rings of the group ring of a Klein four group over the ring of Witt vectors.

     

Two applications of semi-stable graphs

A Michigan graph G on a vertex set V is called semi-stable if for some υ?V, Γ(G_υ) = Γ(G)_υ. It can be shown that all regular graphs are semi-stable and this fact is used to show (i) that if Γ(G) is doubly transitive then G = K_n or K?_n, and (ii) that Γ(G) can be recovered from Γ(G_υ). The second result is extended to the case of stable graphs.     

p-Rank and semi-stable reduction of curves

Let R be a discrete complete valuation ring, with field of fractions K, and with algebraically closed residue field k of characteristic p > 0. Let X be a germ of an R-curve at an ordinary double point. Consider a finite Galois covering f: Y → X, whose Galois group G is a p-group, such that Y is normal, and which is étale above X_k≔ x × _rk. Asume that Y has a semi-stable model :→ Y over R, and let y be a closed point of Y. If the inertia subgroup I(y) at y is cyclic of order pⁿ, we compute the p-rank of tf⁻¹ (y) by using a result of Raynaud. In particular, we prove that this p-rank is bounded by pⁿ −1.     

A note on lattices in semi-stable representations

Let p be a prime, K a finite extension over \mathbb Q_p{{\mathbb Q}_p} and G = Gal([`(K)] /K){G = {\rm Gal}(\overline K /K)} . We extend Kisin’s theory on j{\varphi} -modules of finite E(u)-height to give a new classification of G-stable \mathbb Z1_p{{\mathbb Z}1_p} -lattices in semi-stable representations.