We describe the $J$-invariant of a semisimple algebraic group $G$ over a generic splitting field of a Tits algebra of $G$ in terms of the $J$-invariant over the base field. As a consequence we prove a 10-year-old conjecture of Quéguiner-Mathieu, Semenov, and Zainoulline on the $J$-invariant of groups of type $\mathrm {D}_n$. In the case of type $\mathrm {D}_n$ we also provide explicit formulas for the first component and in some cases for the second component of the $J$-invariant.

Let $G$ be a split semisimple algebraic group over a field and let $A^*$ be an oriented cohomology theory in the Levine–Morel sense. We provide a uniform approach to the $A^*$-motives of geometrically cellular smooth projective $G$-varieties based on the Hopf algebra structure of $A^*(G)$. Using this approach, we provide various applications to the structure of motives of twisted flag varieties.

In this article we establish the arithmetic purity of strong approximation for certain semisimple simply connected linear algebraic groups and their homogeneous spaces over a number field $k$. For instance, for any such group $G$ and for any open subset $U$ of $G$ with ${\mathrm {codim}}(G\setminus U, G)\geqslant 2$, we prove that (i) if $G$ is $k$-simple and $k$-isotropic, then $U$ satisfies strong approximation off any finite number of places; and (ii) if $G$ is the spin group of a non-degenerate quadratic form which is not compact over archimedean places, then $U$ satisfies strong approximation off all archimedean places. As a consequence, we prove that the same property holds for affine quadratic hypersurfaces. Our approach combines a fibration method with subgroup actions developed for induction on the codimension of $G\setminus U$, and an affine linear sieve which allows us to produce integral points with almost-prime polynomial values.

We solve the inverse differential Galois problem over differential fields with a large field of constants of infinite transcendence degree over $\mathbb{Q}$. More generally, we show that over such a field, every split differential embedding problem can be solved. In particular, we solve the inverse differential Galois problem and all split differential embedding problems over $\mathbb{Q}_{p}(x)$.

Given a locally finite leafless tree $T$, various algebraic groups over local fields might appear as closed subgroups of $\operatorname{Aut}(T)$. We show that the set of closed cocompact subgroups of $\operatorname{Aut}(T)$ that are isomorphic to a quasi-split simple algebraic group is a closed subset of the Chabauty space of $\operatorname{Aut}(T)$. This is done via a study of the integral Bruhat–Tits model of $\operatorname{SL}_{2}$ and $\operatorname{SU}_{3}^{L/K}$, that we carry on over arbitrary local fields, without any restriction on the (residue) characteristic. In particular, we show that in residue characteristic $2$, the Tits index of simple algebraic subgroups of $\operatorname{Aut}(T)$ is not always preserved under Chabauty limits.

Abstract Let G be a linear algebraic group over an algebraically closed field 𝕜 acting rationally on a G-module V with its null-cone. Let δ(G, V) and σ(G, V) denote the minimal number d such that for every and , respectively, there exists a homogeneous invariant f of positive degree at most d such that f(v) ≠ 0. Then δ(G) and σ(G) denote the supremum of these numbers taken over all G-modules V. For positive characteristics, we show that δ(G) = ∞ for any subgroup G of GL2(𝕜) that contains an infinite unipotent group, and σ(G) is finite if and only if G is finite. In characteristic zero, δ(G) = 1 for any group G, and we show that if σ(G) is finite, then G0 is unipotent. Our results also lead to a more elementary proof that βsep(G) is finite if and only if G is finite.

This article gives a complete classification of generically split projective homogeneous varieties. This project was begun in our previous article [PS10], but here we remove all restrictions on the characteristic of the base field, give a new uniform proof that works in all cases and in particular includes the case PGO2n+ which was missing in [PS10].

Let G be a complex linear algebraic group and ρ:G → GL(V) a finite dimensional rational representation. Assume that G is connected and reductive, and that V has an open G-orbit. Let f in C[V] be a non-zero relative invariant with character ϕ ∈ Hom (G, C$^×$), meaning that f ˆ ρ (g) =ϕ (g) f for all g in G. Choose a non-zero relative invariant f$^v$ in C[V$^v$], with character ϕ$^-1$, for the dual representation ρ$^v$:G → GL(V$^v$). Roughly, the fundamental theorem of the theory of prehomogeneous vector spaces due to M. Sato says that the Fourier transform of |f|$^s$equals |f$^v$|$^-s$ up to some factors. The purpose of the present paper is to study a finite field analogue of Sato‘s theorem and to give a completely explicit description of the Fourier transform assuming that the characteristic of the base field F$_q$ is large enough. Now |f|$^s$ is replaced by χ (f), with χ in Hom (F$_q$$^×$, C$^×$), and the factors involve Gauss sums, the Bernstein–Sato polynomial b(s) of f, and the parity of the split rank of the isotropy group at v$^v$∈ V$^v(F<math>_q$). We also express this parity in terms of the quadratic residue of the discriminant of the Hessian of log f$^v$ (v$^v$). Moreover we prove a conjecture of N. Kawanaka on the number of integer roots of b(s).

Let $F$ be a $p$-adic field, let $L$ be the completion of a maximal unramified extension of $F$, and let $\sigma$ be the Frobenius automorphism of $L$ over $F$. For any connected reductive group $G$ over $F$ one denotes by $B(G)$ the set of $\sigma$-conjugacy classes in $G(L)$ (elements $x,y$ in $G(L)$ are said to be $\sigma$-conjugate if there exists $g$ in $G(L)$ such that $g^-1 \kappa \sigma(g)=y$. One of the main results of this paper is a concrete description of the set $B(G)$ (previously this was known only in the quasi-split case).