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 Ш(T,V)=ker(H1(K,T)→∏v∈VH1(Kv,T)). 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 Ш(T,V) is finite in the following situations: (1) k is finitely generated and X(k)≠∅; (2) k is a number field.

# Publications

## 2024

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 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 short proof of the theorem of Raghunathan–Ramanathan about *G*-torsors over the affine line.

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 F_{4}.

## 2023

This is a companion paper to [29], 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 f ields 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 ∈ 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.

The first edition of this book provided the first systematic exposition of the arithmetic theory of algebraic groups. This revised second edition, now published in two volumes, retains the same goals, while incorporating corrections and improvements, as well as new material covering more recent developments. Volume I begins with chapters covering background material on number theory, algebraic groups, and cohomology (both abelian and non-abelian), and then turns to algebraic groups over locally compact fields. The remaining two chapters provide a detailed treatment of arithmetic subgroups and reduction theory in both the real and adelic settings. Volume I includes new material on groups with bounded generation and abstract arithmetic groups. With minimal prerequisites and complete proofs given whenever possible, this book is suitable for self-study for graduate students wishing to learn the subject as well as a reference for researchers in number theory, algebraic geometry, and related areas.

## 2022

We study the partial ordering on isomorphism classes of central simple algebras over a given field *F*, defined by setting A1≤A2 if degA1=degA2 and every étale subalgebra of A1 is isomorphic to a subalgebra of A2, and generalizations of this notion to algebras with involution. In particular, we show that this partial ordering is invariant under passing to the completion of the base field with respect to a discrete valuation, and we explore how this partial ordering relates to the exponents of algebras.

We prove that if a linear group Γ⊂GLn(K) over a field *K* of characteristic zero is boundedly generated by semi-simple (diagonalizable) elements then it is virtually solvable. As a consequence, one obtains that infinite *S*-arithmetic subgroups of absolutely almost simple anisotropic algebraic groups over number fields are *never* boundedly generated. Our proof relies on Laurent’s theorem from Diophantine geometry and properties of generic elements.