arXiv subjects

Vladimir I. Chernousov

Publications and source records attributed to Vladimir I. Chernousov.

6 recordsLinked to original sources

arXiv2023

Simple algebraic groups with the same maximal tori, weakly commensurable Zariski-dense subgroups, and good reduction

We provide a new condition for an absolutely almost simple algebraic group to have good reduction with respect to a discrete valuation of the base field which is formulated in terms of the existence of maximal tori with special properties. This characterization, in particular, shows that the Finiteness Conjecture for forms of an absolutely almost simple algebraic group over a finitely generated field that have good reduction at a divisorial set of places of the field would imply the finiteness of the genus of the group at hand. It also leads to a new phenomenon that we refer to as "killing the genus by a purely transcendental extension." Yet another application deals with the investigation of "eigenvalue rigidity" of Zariski-dense subgroups, which in turn is related to the analysis of length-commensurable Riemann surfaces and general locally symmetric spaces. Finally, we analyze the Finiteness Conjecture and the genus problem for simple algebraic groups of type $\textsf{F}_4$.

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arXiv2019

The finiteness of the genus of a finite-dimensional division algebra, and some generalizations

We prove that the genus of a finite-dimensional division algebra is finite whenever the center is a finitely generated field of any characteristic. We also discuss potential applications of our method to other problems, including the finiteness of the genus of simple algebraic groups of type $\textsf{G}_2$. These applications involve the double cosets of adele groups of algebraic groups over arbitrary finitely generated fields: while over number fields these double cosets are associated with the class numbers of algebraic groups and hence have been actively analyzed, similar question over more general fields seem to come up for the first time. In the Appendix, we link the double cosets with $\check{\rm C}$ech cohomology and indicate connections between certain finiteness properties involving double cosets (Condition (T)) and Bass's finiteness conjecture in $K$-theory.

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arXiv2018

Spinor Groups with Good Reduction

Let $K$ be a 2-dimensional global field of characteristic $\neq 2$, and let $V$ be a divisorial set of places of $K$. We show that for a given $n \geqslant 5$, the set of $K$-isomorphism classes of spinor groups $G = \mathrm{Spin}_n(q)$ of nondegenerate $n$-dimensional quadratic forms over $K$ that have good reduction at all $v \in V$, is finite. This result yields some other finiteness properties, such as the finiteness of the genus $\mathbf{gen}_K(G)$ and the properness of the global-to-local map in Galois cohomology. The proof relies on the finiteness of the unramified cohomology groups $H^i(K , μ_2)_V$ for $i \geqslant 1$ established in the paper. The results for spinor groups are then extended to some unitary groups and to groups of type $\textsf{G}_2$.

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arXiv2015

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.

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arXiv2015

Division algebras with the same maximal subfields

We give a survey of recent results related to the problem of characterizing finite-dimensional division algebras by the set of isomorphism classes of their maximal subfields. We also discuss various generalizations of this problem and some of its applications. In the last section, we extend the problem to the context of absolutely almost simple algebraic groups.

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arXiv2013

The genus of a division algebra and the unramified Brauer group

Let D be a finite-dimensional central division algebra over a field K. We define the genus gen(D) of D to be the collection of classes in the Brauer group of K represented by central division K-algebras D' having the same maximal subfields as D. In this paper, we describe a general approach to proving the finiteness of gen(D) and estimating its size that involves the unramified Brauer group with respect to an appropriate set of discrete valuations of K.

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