Igor A. Rapinchuk

activity

20092026

most citedOn linear representations of Chevalley groups over commutative rings

16 citations · 42 across the 21 of their papers we have counts for

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10 papers · 1 filter

math.NT2026

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

Igor A. Rapinchuk, Avinash Roy

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 valuat…

math.NT2023

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

Andrei S. Rapinchuk, Igor A. Rapinchuk

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 \i…

math.NT2021★ 1 cited

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

Vladimir I. Chernousov, Andrei S. Rapinchuk, Igor A. Rapinchuk

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 term…

math.NT2021

Recent developments in the theory of linear algebraic groups: Good reduction and finiteness properties

Andrei S. Rapinchuk, Igor A. Rapinchuk

This is a survey article on some recent developments in the arithmetic theory of linear algebraic groups over higher-dimensional fields, written for the Notices of the AMS.

math.NT2020

Linear algebraic groups with good reduction

Andrei S. Rapinchuk, Igor A. Rapinchuk

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,…

math.NT2020

Some finiteness results for algebraic groups and unramified cohomology over higher-dimensional fields

Andrei S. Rapinchuk, Igor A. Rapinchuk

We formulate and analyze several finiteness conjectures for linear algebraic groups over higher-dimensional fields. In fact, we prove all of these conjectures for algebraic tori as…

math.NT2018

On residue maps for affine curves

Igor A. Rapinchuk

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 re…

math.NT2017★ 1 cited

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

Igor A. Rapinchuk

In this paper, we introduce a condition $(F_{m}^{'})$ on a field $K$ , for a positive integer $m$ , that generalizes Serre's condition (F) and which still implies the fi…

math.NT2017★ 8 cited

Spinor Groups with Good Reduction

Vladimir I. Chernousov, Andrei S. Rapinchuk, Igor A. Rapinchuk

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 $n ⩾ 5$ , the set of $K$ -isomo…

math.NT2016★ 1 cited

On some finiteness results for unramified cohomology

Igor A. Rapinchuk

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…