We prove Manin’s conjecture for bi-equivariant compactifications of unipotent groups.

Consider a smooth connected solvable group G over a field k. If k is algebraically closed then G = T n Ru(G) for any maximal torus T of G [Bo, 10.6(4)]. Over more general k, an analogous such structure can fail to exist. For example, consider an imperfect field k of characteristic p > 0 and an element a ∈ k − kp, so k′ := k(a1/p) is a degree-p purely inseparable extension of k. Note that k′ s :...

We should think of the coefficients aij of Q as real numbers (not necessarily rational or integer). One can still ask what will happen if one substitutes integers for the xi. It is easy to see that if Q is a multiple of a form with rational coefficients, then the set of values Q(Z) is a discrete subset of R. Much deeper is the following conjecture: Conjecture 1.1 (Oppenheim, 1929). Suppose Q is...

Linearization of the nonlinear equations and iterative solution is the most well-known scheme in many engineering problems. For geodetic applications of the LEO satellites, e.g. the Earth’s gravity field recovery, one needs to provide an initial guess of the satellite location or the so-called reference orbit. Numerical integration can be utilized for generating the reference orbit if a satelli...

0. Introduction. Let Qp be the field of p-adic numbers, and let Q∞ = R. Let Gp be a connected semisimpleQp-algebraic group. The unipotent radical of a proper parabolic Qp-subgroup of Gp is called a horospherical subgroup. Two horospherical subgroups are called opposite if they are the unipotent radicals of two opposite parabolic subgroups. In [5] and [6], we studied discrete subgroups generated...

Résumé : Cet article fait suite à [2], où nous étudions le radical unipotent Ru(G) du groupe de Galois différentiel d’un produit de deux opérateurs différentiels irréductibles. Nous passons ici au cas général, où nous donnons divers critères pour que ce radical unipotent soit ‘aussi gros que possible’. À l’inverse, nous explicitons une situation fortement dégénérée, qui apparâıt en présence d’o...

In 1980, Lusztig posed the problem of showing the existence of a unipotent support for the irreducible characters of a finite group of Lie type. This problem was solved by Lusztig in the case where the characteristic of the field over which the group is defined is large enough. The first named author extended this to the case where the characteristic is good. It is the purpose of this paper to ...