123 research outputs found
Centrality of the congruence kernel for elementary subgroups of Chevalley groups of rank >1 over noetherian rings
We prove the centrality of the congruence kernel for the elementary subgroup
of a Chevalley group G of rank >1 over an arbitrary noetherian ring R (under
some minor restrictions on R if G is of type C_n or G_2)
Linear algebraic groups with good reduction
This article is a survey of conjectures and results on reductive algebraic
groups having good reduction at a suitable set of discrete valuations of the
base field. Until recently, this subject has received relatively little
attention, but now it appears to be developing into one of the central topics
in the emerging arithmetic theory of (linear) algebraic groups over
higher-dimensional fields. The focus of this article is on the Main Conjecture
asserting the finiteness of the number of isomorphism classes of forms of a
given reductive group over a finitely generated field that have good reduction
at a divisorial set of places of the field. Various connections between this
conjecture and other problems in the theory of algebraic groups (such as the
analysis of the global-to-local map in Galois cohomology, the genus problem,
etc.) are discussed in detail. The article also includes a brief review of the
required facts about discrete valuations, forms of algebraic groups, and Galois
cohomology
On the size of the genus of a division algebra
Let D be a central division algebra of degree n over a field K. One defines
the genus gen(D) of D as the set of classes [D'] in the Brauer group Br(K) of K
represented by central division algebras D' of degree n over K having the same
maximal subfields as D. We prove that if the field K is finitely generated and
n is prime to its characteristic, then gen(D) is finite, and give explicit
estimations of its size in certain situations
Spinor Groups with Good Reduction
Let be a 2-dimensional global field of characteristic , and let
be a divisorial set of places of . We show that for a given , the set of -isomorphism classes of spinor groups of nondegenerate -dimensional quadratic forms over
that have good reduction at all , is finite. This result yields some
other finiteness properties, such as the finiteness of the genus
and the properness of the global-to-local map in Galois
cohomology. The proof relies on the finiteness of the unramified cohomology
groups for established in the paper. The
results for spinor groups are then extended to some unitary groups and to
groups of type .Comment: Added dedication and made minor stylistic changes. To appear in
Compositio Mathematic
Abstract homomorphisms of algebraic groups and applications
This paper is an overview of my recent work on abstract homomorphisms of
algebraic groups. It is based on a talk given at the Conference on Group
Actions and Applications in Geometry, Topology, and Analysis held in Kunming in
July 2012.Comment: arXiv admin note: substantial text overlap with arXiv:1005.0422,
arXiv:1111.629
On abstract homomorphisms of Chevalley groups over the coordinate rings of affine curves
The goal of this paper is to establish a general rigidity statement for
abstract representations of elementary subgroups of Chevalley groups of rank at
least 2 over a class of commutative rings that includes the localizations of
1-generated rings and the coordinate rings of affine curves. This is achieved
by developing the approach introduced in our previous work, and in particular
by verifying condition (Z) over the class of rings at hand. Our main result
implies, for example, that any finite-dimensional representation of SL_n(Z[X])
(for n at least 3) over an algebraically closed field of characteristic 0 has a
standard description, yielding thereby the first unconditional rigidity
statement for finitely generated linear groups other than arithmetic
groups/lattices.Comment: Dedication adde
A generalization of Serre's condition (F) with applications to the finiteness of unramified cohomology
In this paper, we introduce a condition on a field
, for a positive integer , that generalizes Serre's condition (F) and
which still implies the finiteness of the Galois cohomology of finite Galois
modules annihilated by and algebraic -tori that split over an extension
of degree dividing , as well as certain groups of \'etale and unramified
cohomology. Various examples of fields satisfying ,
including those that do not satisfy (F), are given.Comment: Added Remark 1.2, corrected several typos. arXiv admin note: text
overlap with arXiv:1602.0451
On the character varieties of finitely generated groups
We establish three results dealing with the character varieties of finitely
generated groups. The first two are concerned with the behavior of as a function of , and the third addresses the problem of
realizing a -defined complex affine algebraic variety as a
character variety
On some finiteness results for unramified cohomology
We obtain several finiteness results for the unramified cohomology of
function fields of algebraic varieties defined over fields of type (F'_m), a
class that includes algebraically closed fields, finite fields, local fields,
and some higher local fields of characteristic 0
Developments on the congruence subgroup problem after the work of Bass, Milnor and Serre
In this survey article we give an overview of the developments on the
congruence subgroup and the metaplectic problems after the work of Bass, Milnor
and Serre.Comment: 22 page
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