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Let G be a connected linear algebraic group over a field k. We say that G is toric-friendly if for any field extension K/k and any maximal K-torus T in G the group G(K) has only one orbit in (G/T)(K). Our main result is a classification of semisimple (and under certain assumptions on k, of connected) toric-friendly groups., Comment: 16 pages. Shortened

For a homogeneous space X (not necessarily principal) of a connected algebraic group G (not necessarily linear) over a number field k, we prove a theorem of strong approximation for the adelic points of X in the Brauer-Manin set. Namely, for an adelic point x of X orthogonal to a certain subgroup (which may contain transcendental elements) of the Brauer group Br(X) of X with respect to the Manin pairing, we prove a strong approximation property for x away from a finite set S of places of k. Our result extends a result of Harari for torsors of semiabelian varieties and a result of Colliot-Th\'el\`ene and Xu for homogeneous spaces of simply connected semisimple groups, and our proof uses those results., Comment: 52 pages

Let X be a homogeneous space of a connected linear algebraic group G' over a number field k, containing a k-point x. Assume that the stabilizer of x in G' is connected. Using the notion of a quasi-trivial group, recently introduced by Colliot-Th\'el\`ene, we can represent X in the form X=G/H, where G is a quasi-trivial k-group and H is a connected k-subgroup of G. Let S be a finite set of places of k. Applying results of [B2], we compute the defect of weak approximation for X with respect to S in terms of the biggest toric quotient T of H. In particular, we show that if T splits over a metacyclic extension of k, then X has the weak approximation property. We show also that any homogeneous space X with connected stabilizer (without assumptions on T) has the real approximation property., Comment: 10 pages. A section on metacyclic extensions is added

For a smooth geometrically integral variety $X$ over a field $k$ of characteristic 0, we introduce and investigate the extended Picard complex $UPic(X)$. It is a certain complex of Galois modules of length 2, whose zeroth cohomology is $\bar{k}[X]^*/ \bar{k}^*$ and whose first cohomology is $Pic(\bar{X})$, where $\bar{k}$ is a fixed algebraic closure of $k$ and $\bar{X}$ is obtained from $X$ by extension of scalars to $\bar{k}$. When $X$ is a $k$-torsor of a connected linear $k$-group $G$, we compute $UPic(X)=UPic(G)$ (in the derived category) in terms of the algebraic fundamental group $\pi_1(G)$. As an application we compute the elementary obstruction for such $X$., Comment: 22 pages

A classical theorem of R. Baer describes the nilpotent radical of a finite group G as the set of all Engel elements, i.e. elements y in G such that for any x in G the n-th commutator [x,y,...,y] equals 1 for n big enough. We obtain a characterization of the solvable radical of a finite dimensional Lie algebra defined over a field of characteristic zero in similar terms. We suggest a conjectural description of the solvable radical of a finite group as the set of Engel-like elements and reduce this conjecture to the case of a finite simple group., Comment: 17 pages

Let $f$ be an indefinite ternary quadratic form, and let $q$ be an integer such that $-q det(f)$ is not a square. Let $N(T,f,q)$ denote the number of integral solutions of the equation $f(x)=q$ where $x$ lies in the ball of radius $T$ centered at the origin. We are interested in the asymptotic behavior of $N(T,f,q)$ as $T$ tends to infinity. We deduce from the results of our joint paper with Z. Rudnick that $N(T,f,q)$ grows like cE(T,f,q)$ as $T$ tends to infinity, where $E(T,f,q)$ is the Hardy-Littlewood expectation (the product of local densities) and $0 \le c \le 2$. We give examples of $f$ and $q$ such that $c$ takes the values 0, 1, 2., Comment: AMSTeX, 10 pages