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Giancarlo Lucchini Arteche

Publications and source records attributed to Giancarlo Lucchini Arteche.

31 records · Page 2Linked to original sources
arXiv2019

Homogeneous spaces, algebraic $K$-theory and cohomological dimension of fields

Let $q$ be a non-negative integer. We prove that a perfect field $K$ has cohomological dimension at most $q+1$ if, and only if, for any finite extension $L$ of $K$ and for any homogeneous space $Z$ under a smooth linear connected algebraic group over $L$, the $q$-th Milnor $K$-theory group of $L$ is spanned by the images of the norms coming from finite extensions of $L$ over which $Z$ has a rational point. We also prove a variant of this result for imperfect fields.

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arXiv2019

Le principe de Hasse pour les espaces homogènes : réduction 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 propriétés $P$ des espaces homogènes, que $P$ vaut pour tout espace homogène d'un groupe linéaire connexe dès qu'elle vaut pour les espaces homogènes de $\mathrm{SL}_n$ à stabilisateur fini. Nous réduisons notamment à ce cas particulier la vérification d'une importante conjecture de Colliot-Thélène sur l'obstruction de Brauer-Manin au principe de Hasse et à l'approximation faible. Des travaux récents de Harpaz et Wittenberg montrent que le résultat principal s'applique également à la conjecture analogue (dite conjecture (E)) pour les zéro-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-Thélène 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)).

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arXiv2019

The Hasse principle for homogeneous spaces with nilpotent stabilizer (Le principe de Hasse pour les espaces homogènes à stabilisateur fini)

We prove that the Brauer-Manin obstruction is the only obstruction to the Hasse principle for homogeneous spaces with nilpotent stabilizer. We thus generalize recent results by Harpaz and Wittenberg on finite "hyper-solvable" stabilizers. In particular, this result is true for abelian stabilizers without any hypotheses on the ambient group, which generalizes classic results by Borovoi on the subject. We prove moreover that these spaces have the real approximation property. On montre que l'obstruction de Brauer-Manin est la seule obstruction au principe de Hasse pour les espaces homogènes à stabilisateur nilpotent. On généralise ainsi les résultats récents de Harpaz et Wittenberg sur les stabilisateurs finis "hyper-résolubles". En particulier, ce résultat vaut pour les espaces homogènes à stabilisateur abélien sans hypothèse sur le groupe ambiant, ce qui généralise des résultats déjà classiques de Borovoi sur le sujet. On démontre au passage que ces espaces possèdent la propriété d'approximation réelle.

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arXiv2018

Smooth quotients of abelian varieties by finite groups

We give a complete classification of smooth quotients of abelian varieties by finite groups that fix the origin. In the particular case where the action of the group $G$ on the tangent space at the origin of the abelian variety $A$ is irreducible, we prove that $A$ is isomorphic to the self-product of an elliptic curve and $A/G\simeq \mathbb P^n$. In the general case, assuming $\dim(A^G)=0$, we prove that $A/G$ is isomorphic to a direct product of projective spaces.

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arXiv2018

The unramified Brauer group of homogeneous spaces with finite stabilizer

We give formulas for calculating the unramified Brauer group of a homogeneous space $X$ of a semisimple simply connected group $G$ with finite geometric stabilizer $\bar F$ over a wide family of fields of characteristic 0. When $k$ is a number field, we use these formulas in order to study the Brauer-Manin obstruction to the Hasse principle and weak approximation. We prove in particular that the Brauer-Manin pairing is constant on $X(k_v)$ for every $v$ outside from an explicit finite set of non archimedean places of $k$.

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arXiv2017

Extensions of algebraic groups with finite quotient and nonabelian 2-cohomology

For a finite smooth algebraic group $F$ over a field $k$ and a smooth algebraic group $\bar G$ over the separable closure of $k$, we define the notion of $F$-kernel in $\bar G$ and we associate to it a set of nonabelian 2-cohomology. We use this to study extensions of $F$ by an arbitrary smooth $k$-group $G$. We show in particular that any such extension comes from an extension of finite $k$-groups when $k$ is perfect and we give explicit bounds on the order of these finite groups when $G$ is linear. We prove moreover some finiteness results on these sets.

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arXiv2016

Un résultat de recherche obtenu grâce à des lycéens

We give an elementary proof of a question asked by Fouvry, Kowalski and Michel on the solutions of the equation $x+\frac{1}{x}+y+\frac{1}{y}+t=0$ over a finite field, using the results obtained by highschool students during their participation at the maths tournament "Tournoi Français des Jeunes Mathématiciennes et Mathématiciens". Nous présentons une démonstration élémentaire d'une question posée par Fouvry, Kowalski et Michel sur les solutions de l'équation $x+\frac{1}{x}+y+\frac{1}{y}+t=0$ sur un corps fini, en utilisant des résultats obtenus par des lycéens lors de leur participation au Tournoi Français des Jeunes Mathématiciennes et Mathématiciens.

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arXiv2016

The Grunwald problem and approximation properties for homogeneous spaces

Given a group $G$ and a number field $K$, the Grunwald problem asks whether given field extensions of completions of $K$ at finitely many places can be approximated by a single field extension of $K$ with Galois group G. This can be viewed as the case of constant groups $G$ in the more general problem of determining for which $K$-groups $G$ the variety $\mathrm{SL}_n/G$ has weak approximation. We show that away from an explicit set of bad places both problems have an affirmative answer for iterated semidirect products with abelian kernel. Furthermore, we give counterexamples to both assertions at bad places. These turn out to be the first examples of transcendental Brauer-Manin obstructions to weak approximation for homogeneous spaces.

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arXiv2015

Groupe de Brauer non ramifié algébrique des espaces homogènes (The unramified algebraic Brauer group of homogeneous spaces)

Using a reduction of the Galois cohomology of a linear algebraic group $G$ to that of a certain finite subquotient, we give different formulas allowing the calculation of the unramified algebraic Brauer group of a homogeneous space $V=G\backslash G'$ with $G'$ semisimple and simply connected, both over a finite field and over an arbitrary field of characteristic $0$.

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arXiv2015

Approximation faible et principe de Hasse pour des espaces homogènes à stabilisateur fini résoluble (Weak Approximation and Hasse Principle for homogeneous spaces with finite solvable stabilizer)

Soit $K$ un corps global et $G$ un $K$-groupe fini résoluble. Sous certaines hypothèses sur une extension déployant $G$, on démontre que l'espace homogène $V:=G'/G$ avec $G'$ un $K$-groupe semi-simple simplement connexe vérifie l'approximation faible. On utilise une version plus précise de ce résultat pour démontrer le principe de Hasse pour des espaces homogènes $X$ sous un $K$-groupe $G'$ semi-simple simplement connexe à stabilisateur géométrique $\bar G$ fini et résoluble, sous certaines hypothèses sur le $K$-lien $(\bar G,κ)$ défini par $X$. ----- Let $K$ be a global field and $G$ a finite solvable $K$-group. Under certain hypotheses concerning the extension splitting $G$, we show that the homogeneous space $V=G'/G$ with $G'$ a semi-simple simply connected $K$-group has weak approximation. We use a more precise version of this result to prove the Hasse principle for homogeneous spaces $X$ under a semi-simple simply connected $K$-group $G'$ with finite solvable geometric stabilizer $\bar G$, under certain hypotheses concerning the $K$-kernel (or $K$-lien) $(\bar G,κ)$ defined by $X$.

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arXiv2014

Groupe de Brauer non ramifié des espaces homogènes à stabilisateur fini (Unramified Brauer group of homogeneous spaces with finite stabilizer)

We develop different formulas of algebraic and/or arithmetic nature allowing an explicit calculation, both over a finite field and over a field of characteristic 0, of the algebraic part of the unramified Brauer group of a homogeneous space G\G' under a semisiple simply connected linear group G' with finite stabilizer G. We also give examples of the calculations that can be done with these formulas.

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arXiv2013

Le groupe de Brauer non ramifié sur un corps global de caractéristique positive (The unramified Brauer group over a global field of positive characteristic)

En utilisant un théorème de Gabber sur les altérations, on démontre un résultat décrivant la partie de torsion première à $p$ du groupe de Brauer non ramifié d'une variété $V$ lisse et géométriquement intègre sur un corps global de caractéristique $p$ au moyen de l'évaluation des éléments de $\Br V$ sur ses points locaux. Using a theorem of Gabber on alterations, we prove a result describing the prime-to-$p$ torsion part of the unramified Brauer group of a smooth and geometrically integral variety $V$ over a global field of characteristic $p$ by evaluating the elements of $\Br V$ at its local points.

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