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Igor A. Rapinchuk

Publications and source records attributed to Igor A. Rapinchuk.

At least 19 recordsLinked to original sources
arXiv2026

Products of trees and ${\rm PGL}_2$-torsors over the punctured affine line

The goal of this article is to present a computation of the Galois cohomology of the group $G = \mathrm{PGL}_2$ over rings of Laurent polynomials. This computation recovers the main result of \cite{CGP} in this case by a new method, based on the analysis of actions on appropriate geometric objects (products of trees and, in general, of affine buildings), which was already used in \cite{ARR} to give a new proof of the theorem of Raghunathan-Ramanathan \cite{RR} concerning Galois cohomology over polynomial rings. Going beyond Galois cohomology, this method also enables one to determine the finite subgroups in the group of points over relevant polynomial rings. In view of these and other potential applications of the method in different situations (in particular, in the study of algebraic groups over the coordinate rings of general affine curves), we have attempted to make our exposition largely self-contained and accessible to broad mathematical audience.

math.GR↗
arXiv2026

Finiteness of the Tate-Shafarevich group over function fields for groups of multiplicative type

Let $K = k(X)$ be the function field of a smooth geometrically integral variety $X$ of dimension $\geq 2$ over a field $k$ of characteristic 0 and $V$ be the set of discrete valuations of $K$ associated with the prime divisors on $X$. We show that if $D$ is a $k$-defined group of multiplicative type, then the corresponding Tate-Shafarevich group $Sha(D,V) = \ker \left(H^1(K,D) \to \prod_{v \in V} H^1(K_v, D) \right)$ is finite in the following situations: (1) $k$ is finitely generated and $X(k) \neq \emptyset$; (2) $k$ is a number field. This complements previous work of Harari and Szamuely, which considered the case where $X$ is a curve.

math.NT↗
arXiv2024

Residue maps, Azumaya algebras, and buildings

The goal of this note is to give an explicit formula for the residues of twists of the matrix algebra in terms of the twisting cocycle. Combined with the Fixed Point Theorem for actions of finite groups on affine buildings, this leads to a quick proof of the well-known characterization of unramified algebras in terms of Azumaya algebras.

math.RA↗
arXiv2023

Properness of the global-to-local map for algebraic groups with toric connected component and other finiteness properties

This is a companion paper to our previous work, where we proved the finiteness of the Tate-Shafarevich group for an arbitrary torus $T$ over a finitely generated field $K$ with respect to any divisorial set $V$ of places of $K$. Here, we extend this result to any $K$-group $D$ whose connected component is a torus (for the same $V$), and as a consequence obtain a finiteness result for the local-to-global conjugacy of maximal tori in reductive groups over finitely generated fields. Moreover, we prove the finiteness of the Tate-Shafarevich group for tori over function fields $K$ of normal varieties defined over base fields of characteristic zero and satisfying Serre's condition (F), in which case $V$ consists of the discrete valuations associated with the prime divisors on the variety (geometric places). In this situation, we also establish the finiteness of the number of $K$-isomorphism classes of algebraic $K$-tori of a given dimension having good reduction at all $v \in V$, and then discuss ways of extending this result to positive characteristic. Finally, we prove the finiteness of the number of isomorphism classes of forms of an absolutely almost simple group defined over the function field of a complex surface that have good reduction at all geometric places.

math.AG↗
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 $\textsf{F}_4$.

math.NT↗
arXiv2023

Applications of the Fixed Point Theorem for group actions on buildings to algebraic groups over polynomial rings

We apply the Fixed Point Theorem for the actions of finite groups on Bruhat-Tits buildings and their products to establish two results concerning the groups of points of reductive algebraic groups over polynomial rings in one variable, assuming that the base field is of characteristic zero. First, we prove that for a reductive $k$-group $G$, every finite subgroup of $G(k[t])$ is conjugate to a subgroup of $G(k)$. This, in particular, implies that if $k$ is a finite extension of the $p$-adic field $\mathbb{Q}_p$, then the group $G(k[t])$ has finitely many conjugacy classes of finite subgroups, which is a well-known property for arithmetic groups. Second, we give a give a short proof of the theorem of Raghunathan-Ramanathan about $G$-torsors over the affine line.

math.GR↗
arXiv2023

The finiteness of the Tate-Shafarevich group over function fields for algebraic tori defined over the base field

Let $K$ be a field and $V$ be a set of rank one valuations of $K$. The corresponding Tate-Shafarevich group of a $K$-torus $T$ is $Sha(T , V) = \ker\left(H^1(K , T) \to \prod_{v \in V} H^1(K_v , T)\right)$. We prove that if $K = k(X)$ is the function field of a smooth geometrically integral quasi-projective variety over a field $k$ of characteristic 0 and $V$ is the set of discrete valuations of $K$ associated with prime divisors on $X$, then for any torus $T$ defined over the base field $k$, the group $Sha(T , V)$ is finite in the following situations: (1) $k$ is finitely generated and $X(k) \neq \emptyset$; (2) $k$ is a number field.

math.NT↗
arXiv2022

Algebraic groups with good reduction and the genus problem

We first provide an overview of several results dealing with the genus of a division algebra and highlight the role of ramification in its analysis. We then give a survey of recent developments on the genus problem for simple algebraic groups and its connections to the analysis of groups with good reduction.

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arXiv2022

On abstract homomorphisms of some special unitary groups

We analyze the abstract representations of the groups of rational points of even-dimensional quasi-split special unitary groups associated with quadratic field extensions. We show that, under certain assumptions, such representations have a standard description, as predicted by a conjecture of Borel and Tits. Our method extends the approach introduced by the first author to study abstract representations of Chevalley groups and is based on the construction and analysis of a certain algebraic ring associated to a given abstract representation.

math.GR↗
arXiv2020

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.

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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.

math.RA↗
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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arXiv2018

A generalization of Serre's condition (F) with applications to the finiteness of unramified cohomology

In this paper, we introduce a condition $\mathrm{(F}_m'\mathrm{)}$ on a field $K$, for a positive integer $m$, that generalizes Serre's condition (F) and which still implies the finiteness of the Galois cohomology of finite Galois modules annihilated by $m$ and algebraic $K$-tori that split over an extension of degree dividing $m$, as well as certain groups of étale and unramified cohomology. Various examples of fields satisfying $\mathrm{(F}_m'\mathrm{)}$, including those that do not satisfy (F), are given.

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arXiv2018

On residue maps for affine curves

We establish several compatibility results between residue maps in étale and Galois cohomology that arise naturally in the analysis of smooth affine algebraic curves having good reduction over discretely valued fields. These results are needed, and in fact have already been used, for the study of finiteness properties of the unramified cohomology of function fields of affine curves over number fields.

math.NT↗
arXiv2016

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.

math.GR↗