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Cyril Demarche

Publications and source records attributed to Cyril Demarche.

At least 19 recordsLinked to original sources
arXiv2026

A note on complex Lie Algebras isomorphic to their conjugate

A real Lie algebra defines by extension of scalars a complex Lie algebra that is isomorphic to its Galois conjugate. In this paper, we are interested in the converse property: given a complex Lie algebra that is isomorphic to its conjugate, is it defined over the real numbers? We prove the existence of a $10$-dimensional nilpotent complex Lie algebra for which the answer is negative, disproving a recent conjecture by Der\'e. In addition, we compute the generic obstruction to this descent problem in terms of Brauer groups.

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arXiv2025

Degree 4 cohomological invariants of algebraic tori

In this paper, we determine the motive of the classifying torsor of an algebraic torus. As a result, we give an exact sequence describing the degree 4 cohomological invariants of algebraic tori. Using results by Blinstein and Merkurjev, this provides a formula for the degree 4 unramified cohomology group of an algebraic torus, via a flasque resolution.

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arXiv2024

Lifting Galois representations via Kummer flags

Let $\Gamma$ be either i) the absolute Galois group of a local field $F$, or ii) the topological fundamental group of a closed connected orientable surface of genus $g$. In case i), assume that $\mu_{p^2} \subset F$. We give an elementary and unified proof that every representation $\rho_1: \Gamma \to \mathbf{GL}_d(\mathbb{F}_p)$ lifts to a representation $\rho_2: \Gamma \to \mathbf{GL}_d(\mathbb{Z}/p^2)$. [In case i), it is understood these are continuous.] The actual statement is much stronger: for all $r \geq 1$, under "suitable" assumptions, triangular representations $\rho_r: \Gamma \to \mathbf{B}_d(\mathbb{Z}/p^r)$ lift to $\rho_{r+1}: \Gamma \to \mathbf{B}_d(\mathbb{Z}/p^{r+1})$, in the strongest possible step-by-step sense. Here "suitable" is made precise by the concept of $\textit{Kummer flag}$. An essential aspect of this work is to identify the common properties of groups i) and ii) that suffice to ensure the existence of such lifts.

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arXiv2022

Étale homotopy groups of algebraic groups and homogeneous spaces

We show the vanishing of the second homotopy group of the étale homotopy type of a smooth connected algebraic group over a separably closed field, completed away from the characteristic. This is an algebraic analogue of a classical theorem of Elie Cartan. Based on this result, we establish an explicit formula for the similarly completed second homotopy group of a homogeneous space.

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arXiv2020

Duality for complexes of tori over a global field of positive characteristic

If K is a number field, arithmetic duality theorems for tori and complexes of tori over K are crucial to understand local-global principles for linear algebraic groups over K. When K is a global field of positive characteristic, we prove similar arithmetic duality theorems, including a Poitou-Tate exact sequence for Galois hypercohomology of complexes of tori. One of the main ingredients is Artin-Mazur-Milne duality theorem for fppf cohomology of finite flat commutative group schemes.

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arXiv2019

Artin-Mazur-Milne duality for fppf cohomology

We provide a complete proof of a duality theorem for the fppf cohomology of either a curve over a finite field or a ring of integers of a number field, which extends the classical Artin-Verdier Theorem in étale cohomology. We also prove some finiteness and vanishing statements.

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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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arXiv2018

Splitting families in Galois cohomology

Let $k$ be a field, with absolute Galois group $Γ$. Let $A/k$ be a finite étale group scheme of multiplicative type, i.e. a discrete $Γ$-module. Let $n \geq 2$ be an integer, and let $x \in H^n(k,A)$ be a cohomology class. We show that there exists a countable set $I$, and a familiy $(X_i)_{i \in I}$ of (smooth, geometrically integral) $k$-varieties, such that the following holds. For any field extension $l/k$, the restriction of $x$ vanishes in $H^n(l,A)$ if and only if (at least) one of the $X_i$'s has an $l$-point. We moreover show that the $X_i$'s can be made into an ind-variety. In the case $n=2$, we note that one variety is enough.

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arXiv2018

Comparing descent obstruction and Brauer-Manin obstruction for open varieties

We provide a relation between Brauer-Manin obstruction and descent obstruction for torsors over open varieties under a connected linear algebraic group or a group of multiplicative type is given. Such a relation is further refined for torsors under a torus. As an appliaction, we prove that the semi-simple part of a connected linear algebraic group will satisfy strong approximation with Brauer-Manin obstruction if G iteself satisfies strong approximation with Brauer-Manin obstruction.The equivalence between descent obstruction and etale Brauer-Manin obstruction for smooth projective varieties is extended to smooth quasi-projective varieties, which provides the perspective to study integral points.

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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

Le groupe fondamental d'un espace homogène d'un groupe algébrique linéaire

Soit X un espace homogène d'un groupe algébrique linéaire connexe G sur C. Soit x un C-point de X. On désigne par H le stabilisateur de x dans G. On montre qu'on peut définir algébriquement le groupe fondamental topologique π_1(X(C),x), si ce groupe fondamental topologique est abélien. Si Pic(G)=0 et H est connexe ou abélien, on calcule π_1(X(C),x) en termes des groupes de caractères de G et H. En outre, si G et X sont définis sur un corps algébriquement clos de caractéristique p quelconque, on calcule la partie première à p du groupe fondamental étale de X en termes des groupes de caractères de G et H (si Pic(G)=0 et H est connexe). Let X be a homogeneous space of a connected linear algebraic group G defined over C. Let x be a C-point of X. We denote by H the stabilizer of x in G. We show that if the topological fundamental group π_1(X(C),x) is abelian, then it can be defined algebraically. If Pic(G)=0 and H is connected or abelian, we compute π_1(X(C),x) in terms of the character groups of G and H. Furthermore, when G and X are defined over an algebraically closed field of arbitrary characteristic p, we compute the prime-to-p étale fundamental group of X in terms of the character groups of G and H (if Pic(G)=0 and H is connected).

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arXiv2013

Hasse principle and weak approximation for multinorm equations

In this note, we are interested in local-global principles for multinorm equations of the form $\prod_{i=1}^n N_{L_i /k}(z_i) = a$ where $k$ is a global field, $L_i/k$ are finite separable field extensions and $a \in k^*$. In particular, we prove a result relating weak approximation for this equation to weak approximation for some classical norm equation $N_{F/k}(w) = a$ where $F := \bigcap_{i=1}^n L_i$. It provides a proof of a "weak approximation" analogue of a recent conjecture by Pollio and Rapinchuk about multinorm principle. We also provide a counterexample to the original conjecture concerning Hasse principle.

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arXiv2012

Complexes de groupes de type multiplicatif et groupe de Brauer non ramifié des espaces homogènes

Let k be a field, G a smooth connected linear algebraic group and X a homogeneous space of G over k, such that the geometric stabilizers are extensions of a smooth group of multiplicative type by a smooth connected characterfree group. If k has characteristic zero and if X^c is a smooth compactification of X over k, we obtain a formula for the algebraic Brauer group of X^c. Several variants are obtained in positive characteristic p, including the finite field case and the global field case, where the formulae describe the prime-to-p part of the algebraic unramified Brauer group of X, without assuming the existence of a smooth compactification of X. Moreover, assuming that stabilizers are connected, then our formulae hold for the prime-to-p part of the whole unramified Brauer group.

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arXiv2010

Une formule pour le groupe de Brauer algébrique d'un torseur

For a homogeneous space X of a connected algebraic group G (with connected stabilizers) over a field k of characteristic zero, we construct a canonical complex of Galois modules of length 3 and a canonical isomorphism between an hypercohomology group of this complex and an explicit subgroup of the Brauer group of X. This result is obtained as a consequence of a formula describing the "algebraic Brauer group of a torsor", and it generalizes recent results by Borovoi and van Hamel, by considering non-linear groups G and by taking into account some transcendental elements in the Brauer group of X.

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arXiv2010

Manin obstruction to strong approximation for homogeneous spaces

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-Thélène and Xu for homogeneous spaces of simply connected semisimple groups, and our proof uses those results.

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arXiv2009

Théorèmes de dualité pour les complexes de tores

We consider a complex of tori of length 2 defined over a number field k. We establish here some local and global duality theorems for the (étale or Galois) hypercohomology of such a complex. We prove the existence of a Poitou-Tate exact sequence for such a complex, which generalizes the Poitou-Tate exact sequences for finite Galois modules and tori. In particular, we obtain a Poitou-Tate exact sequence for k-groups of multiplicative type. The general results proven here lie at the root of recent results about the defect of strong approximation in connected linear algebraic groups and about some arithmetic duality theorems for the (non-abelian) Galois cohomology of such groups.

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