Questions de corps de définition pour les variétés abéliennes en caractéristique positive

17 pagesDichotomies in various conjectures from algebraic geometry are in fact occurrences of the dichotomy among Zariski structures. This is what Hrushovski showed and which enabled him to solve, positively, the geometric Mordell Lang conjecture in positive characteristic. Are we able now to avoid this use of Zariski structures? Pillay and Ziegler have given a direct proof that works for abelian varieties they called ``very thin'', which include the ordinary abelian varieties. But it does not apply in all generality: We describe here an abelian variety which is not very thin. More generally, we consider from a model-theoretical point of view several questions about the fields of definition of semi-abelian varieties

Le principe de Hasse pour les espaces homog\`enes : r\'eduction au cas des stabilisateurs finis (The Hasse principle for homogeneous spaces: reduction to the case of finite stabilizers)

Nous montrons, pour une grande famille de propri\'et\'es $P$ des espaces homog\`enes, que $P$ vaut pour tout espace homog\`ene d'un groupe lin\'eaire connexe d\`es qu'elle vaut pour les espaces homog\`enes de $\mathrm{SL}_n$ \`a stabilisateur fini. Nous r\'eduisons notamment \`a ce cas particulier la v\'erification d'une importante conjecture de Colliot-Th\'el\`ene sur l'obstruction de Brauer-Manin au principe de Hasse et \`a l'approximation faible. Des travaux r\'ecents de Harpaz et Wittenberg montrent que le r\'esultat principal s'applique \'egalement \`a la conjecture analogue (dite conjecture (E)) pour les z\'ero-cycles. We prove, for a wide family of properties $P$ of homogeneous spaces, that if $P$ is satisfied for homogeneous spaces of $\mathrm{SL}_n$ with finite stabilizers, then $P$ is satisfied for all homogeneous spaces of linear connected groups. In particular, we reduce to this particular case the verification of an important conjecture by Colliot-Th\'el\`ene on the Brauer-Manin obstruction to the Hasse principle and to weak approximation. Recent work by Harpaz and Wittenberg show that our main result can also be applied to the analog conjecture on zero-cycles (known as conjecture (E)).Comment: 30 pages, in French. V4: generalized Sections 2, 3 and 4 to perfect fields assuming that all algebraic groups are smooth (with practically no changes!); other minor change

Groupe de Brauer et points entiers de deux familles de surfaces cubiques affines

Let a be a nonzero integer. If a is not congruent to 4 or 5 modulo 9 then there is no Brauer-Manin obstruction to the existence of integers x, y, z such that x^3+y^3+z^3=a. In addition, there is no Brauer-Manin obstruction to the existence of integers x, y, z such that x^3+y^3+2z^3=a.Comment: 24 pages; minor changes onl

La transformée de Fourier pour les espaces tordus sur un groupe réductif p-adique

le papier a été accepté dans la revue Astérisque— Let G be a connected reductive group defined over a non–Archimedean local field F. Put G = G(F). Let θ be an F –automorphism of G, and let ω be a smooth character of G. This paper is concerned with the smooth complex representations π of G such that π θ = π • θ is isomorphic to ωπ = ω ⊗ π. If π is admissible, in particular irreducible, the choice of an isomorphism A from ωπ to π θ (and of a Haar measure on G) defines a distribution Θ A π = tr(π • A) on G. The twisted Fourier transform associates to a compactly supported locally constant function f on G, the function (π, A) → Θ A π (f) on a suitable Grothendieck group. Here we describe its image (Paley– Wiener theorem) and its kernel (spectral density theorem).Soit G un groupe réductif connexe défini sur un corps local non ar-chimédien F. On pose G = G(F). Soit aussi θ un F –automorphisme de G, et ω un caractère lisse de G. On s'intéresse aux représentations complexes lisses π de G telles que π θ = π • θ est isomorphe à ωπ = ω ⊗ π. Si π est admissible, en particulier irréductible, le choix d'un isomorphisme A de ωπ sur π θ (et d'une mesure de Haar sur G) définit une distribution Θ A π = tr(π • A) sur G. La transformée de Fourier tordue associe à une fonction f sur G localement constante et à support compact, la fonction (π, A) → Θ A π (f) sur un groupe de Grothendieck adéquat. On décrit ici son image (théorème de Paley–Wiener) et son noyau (théorème de densité spectrale)