EUDML

Let $B$ be a quaternion algebra over a number field $k$ . To a pair of Hilbert symbols ${a, b}$ and ${c, d}$ for $B$ we associate an invariant $ρ = ρ_{R} ([𝒟 (a, b)], [𝒟 (c, d)])$ in a quotient of the narrow ideal class group of $k$ . This invariant arises from the study of finite subgroups of maximal arithmetic kleinian groups. It measures the distance between orders $𝒟 (a, b)$ and $𝒟 (c, d)$ in $B$ associated to ${a, b}$ and ${c, d} .$ If $a = c$ , we compute $ρ_{R} ([𝒟 (a, b)], [𝒟 (c, d)])$ by means of arithmetic in the field $k (\sqrt{a}) .$ The problem of extending this algorithm to the general case leads to studying a finite graph associated...

Sums of k-th powers in the ring of polynomials with integer coefficients

Ted Chinburg; Melvin Henriksen — 1976

Acta Arithmetica

The finite subgroups of maximal arithmetic kleinian groups

Ted Chinburg; Eduardo Friedman — 2000

Annales de l'institut Fourier

Given a maximal arithmetic Kleinian group $Γ \subset PGL (2, ℂ)$ , we compute its finite subgroups in terms of the arithmetic data associated to $Γ$ by Borel. This has applications to the study of arithmetic hyperbolic 3-manifolds.

The local lifting problem for actions of finite groups on curves

Ted Chinburg; Robert Guralnick; David Harbater — 2011

Annales scientifiques de l'École Normale Supérieure

Let $k$ be an algebraically closed field of characteristic $p > 0$ . We study obstructions to lifting to characteristic $0$ the faithful continuous action $φ$ of a finite group $G$ on $k [[t]]$ . To each such $φ$ a theorem of Katz and Gabber associates an action of $G$ on a smooth projective curve $Y$ over $k$ . We say that the KGB obstruction of $φ$ vanishes if $G$ acts on a smooth projective curve $X$ in characteristic $0$ in such a way that $X / H$ and $Y / H$ have the same genus for all subgroups $H \subset G$ . We determine for which $G$ the KGB obstruction...

Obstructions for deformations of complexes

Frauke M. Bleher; Ted Chinburg — 2013

Annales de l’institut Fourier

We develop two approaches to obstruction theory for deformations of derived isomorphism classes of complexes of modules for a profinite group $G$ over a complete local Noetherian ring $A$ of positive residue characteristic.

Deformations and derived categories

Frauke M. Bleher; Ted Chinburg — 2005

Annales de l'institut Fourier

In this paper we generalize the deformation theory of representations of a profinite group developed by Schlessinger and Mazur to deformations of objects of the derived category of bounded complexes of pseudocompact modules for such a group. We show that such objects have versal deformations under certain natural conditions, and we find a sufficient condition for these versal deformations to be universal. Moreover, we consider applications to deforming Galois cohomology classes and the étale hypercohomology...

Finiteness Theorems for Deformations of Complexes

Frauke M. Bleher; Ted Chinburg — 2013

Annales de l’institut Fourier

We consider deformations of bounded complexes of modules for a profinite group $G$ over a field of positive characteristic. We prove a finiteness theorem which provides some sufficient conditions for the versal deformation of such a complex to be represented by a complex of $G$ -modules that is strictly perfect over the associated versal deformation ring.