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David Harari

Publications and source records attributed to David Harari.

17 recordsLinked to original sources
arXiv2026

Real approximation for homogeneous spaces with finite stabilizers

We prove some new cases of real appoximation for homogeneous spaces with finite stabilizers and describe the state of the art around this question, giving proofs that are well-known to experts but that, to our knowledge, cannot be found in the literature. Our main new result needs the latest advances in the topic of the Brauer--Manin obstruction for homogeneous spaces with supersolvable stabilizers. It states that any finite $k$-group that is split by a $2$-primary extension satisfies real approximation.

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arXiv2026

On the defect in the generalized Grunwald--Wang problem

The classical Grunwald--Wang theorem asserts that, unless we are in the so-called special case, local cyclic Galois extensions at finitely many completions of a number field can be approximated by a global cyclic extension. In the special case the obstruction is measured by a group of order 2. It has been known for a long time that the Grunwald--Wang theorem extends to a very general context of valued fields. Therefore it is natural to ask whether in the special case the obstruction is always measured by a finite group and if so, is the order of this group bounded independently of the number of places under consideration. We show that the answer to both questions is negative in general, already for rational function fields and discrete valuations coming from points of the affine line. This has some interesting links to the arithmetic of function fields over Q or Q_p.

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arXiv2021

On Tate--Shafarevich groups of one-dimensional families of commutative group schemes over number fields

Given a smooth geometrically connected curve $C$ over a field $k$ and a smooth commutative group scheme $G$ of finite type over the function field $K$ of $C$ we study the Tate--Shafarevich groups given by elements of $H^1(K,G)$ locally trivial at completions of $K$ associated with closed points of $C$. When $G$ comes from a $k$-group scheme and $k$ is a number field (or $k$ is a finitely generated field and $C$ has a $k$-point) we prove that the Tate--Shafarevich group is finite, generalizing a result of Sa\"idi and Tamagawa for abelian varieties. We also give examples of nontrivial Tate--Shafarevich groups in the case when $G$ is a torus and prove other related statements.

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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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arXiv2018

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 \'etale cohomology. We also prove some finiteness and vanishing statements.

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arXiv2018

L'espace ad\'elique d'un tore sur un corps de fonctions

Let $k$ be a field of characteristic 0 and let $K$ be the function field of a smooth projective geometrically integral $k$-curve $X$. Let $T$ be a $K$-torus. In this article, we aim at studying the space of adelic points $T(S,\mathbb{A}_K)$ of $T$ outside a finite set $S$ of closed points of $X$. We start by proving that the group $T(K)$ of rational points of $T$ is always discrete (hence closed) in $T(S,\mathbb{A}_K)$. We then describe the quotient $T(\emptyset,\mathbb{A}_K)/T(K)$ in each of the following three cases: $k$ is an algebraically closed field, $k$ is the field of Laurent series $\mathbb{C}((t))$, and $k$ is a $p$-adic field. Soient $k$ un corps de caract\'eristique 0 et $K$ le corps des fonctions d'une $k$-courbe projective lisse g\'eom\'etriquement int\`egre $X$. Soit $T$ un $K$-tore. Dans cet article, on cherche \`a \'etudier l'espace des points ad\'eliques $T(S,\mathbb{A}_K)$ de $T$ hors d'un ensemble fini $S$ de points ferm\'es de $X$. On commence par montrer que le groupe $T(K)$ des points rationnels de $T$ est toujours ferm\'e discret dans $T(S,\mathbb{A}_K)$. On d\'ecrit ensuite le quotient $T(\emptyset,\mathbb{A}_K)/T(K)$ dans chacun des trois cas suivants: $k$ corps alg\'ebriquement clos, $k=\mathbb{C}((t))$ et $k$ corps $p$-adique.

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arXiv2014

Dualit\'e et principe local-global pour les tores sur une courbe au-dessus de C((t))

Pour $K$ un corps global (corps de nombres ou corps de fonctions d'une variable sur un corps fini $F$), on dispose de th\'eor\`emes de dualit\'e classiques (Tate, Poitou, Nakayama) pour la cohomologie galoisienne \`a valeurs dans des tores et des modules ab\'eliens finis. Nous \'etablissons de tels th\'eor\`emes pour $K$ le corps des fonctions d'une courbe projective et lisse sur le corps $F={\bf C}((t))$ des s\'eries formelles en une variable sur le corps des complexes. Cela permet de contr\^oler le d\'efaut du principe de Hasse et l'approximation faible pour les espaces homog\`enes sous un tore. Il y a ici des diff\'erences avec le cas classique ($K$ corps global), et aussi avec le cas r\'ecemment \'etudi\'e o\`u $K$ est un corps de fonctions d'une variable sur un corps $p$-adique. Par exemple, la $K$-rationalit\'e d'un tore n'implique pas ici la validit\'e du principe de Hasse pour ses espaces principaux homog\`enes. ---------------- Over a global field $K$ (number field, or function field of a curve over a finite field $F$), arithmetic duality theorems for the Galois cohomology of tori and finite Galois modules have long been known. More recent work investigates the case where $K$ is the function fields of of a curve over a $p$-adic field. For $K$ the function field of a curve over the formal series field $F={\bf C}((t))$, we establish analogous duality theorems. We thus control the obstruction to the local-global principle and to weak approximation for homogeneous spaces of tori. There are differences with the afore described cases. For example the Hasse principle need not hold for principal homogeneous spaces of a $K$-rational torus.

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arXiv2014

Local-global questions for tori over $p$-adic function fields

We study local-global questions for Galois cohomology over the function field of a curve defined over a p-adic field (a field of cohomological dimension 3). We define Tate-Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of the base field coming from a closed point of the curve. In the case of a torus we establish a perfect duality between the first Tate-Shafarevich group of the torus and the second Tate-Shafarevich group of the dual torus. As an application, we show that the failure of the local-global principle for rational points on principal homogeneous spaces under tori is controlled by a certain subquotient of a third etale cohomology group. We also prove a generalization to principal homogeneous spaces of certain reductive group schemes in the case when the base curve has good reduction.

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arXiv2014

Weak approximation for tori over $p$-adic function fields

This is the companion piece to "Local-global questions for tori over p-adic function fields" by the first and third authors. We study local-global questions for Galois cohomology over the function field of a curve defined over a p-adic field, the main focus here being weak approximation of rational points. We construct a 9-term Poitou--Tate type exact sequence for tori over a field as above (and also a 12-term sequence for finite modules). Like in the number field case, part of the sequence can then be used to analyze the defect of weak approximation for a torus. We also show that the defect of weak approximation is controlled by a certain subgroup of the third unramified cohomology group of the torus.

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arXiv2013

Approximation forte en famille

Let $k$ be a number field and $X$ a smooth integral affine variety equipped with a morphism $f : X \to A^1_k$ to the affine line. Assume that all fibres of $f$ are split, for instance that they are geometrically integral. Assume that the generic fibre of $f$ is a homogeneous space of a simply connected, almost simple, semisimple group $G/k(t)$, and that the geometric stabilizers are connected reductive groups. Let $v$ be a place of $k$ such that the fibration $f$ acquires a rational section over the completion $k_v$ at $v$. Assume moreover that at almost all points $x \in A^1(k_v)$ the specialized group $G_x$ is isotropic over $k_v$. If the Brauer group of $X$ is reduced to the Brauer group of $k$, then strong approximation holds for $X$ away from the place $v$.

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arXiv2012

Complexes de groupes de type multiplicatif et groupe de Brauer non ramifi\'e des espaces homog\`enes

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

Descent theory for open varieties

We extend the descent theory of Colliot-Thélène and Sansuc to arbitrary smooth algebraic varieties by removing the condition that every invertible regular function is constant. This links the Brauer--Manin obstruction for integral points on arithmetic schemes to the obstructions defined by torsors under groups of multiplicative type.

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arXiv2010

Descent obstruction and fundamental exact sequence

A torsor under a k-group scheme G on a variety X over a number field k imposes a descent obstruction against the existence of rational points on X. We discuss the finite descent obstruction, that is for all such torsors under finite k-groups G, in view of a local-global interpolation property for sections of the fundamental group short exact sequence of X/k. There are applications to the Brauer-Manin obstruction, to the descent obstruction by torsors under linear groups, and to the birational version of Grothendieck's section conjecture over number fields. In particular, we obtain examples of families of curves over number fields, such that the birational section conjecture is true in a non-trivial way.

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arXiv2009

Galois sections for abelianized fundamental groups

Given a smooth projective curve $X$ of genus at least 2 over a number field $k$, Grothendieck's Section Conjecture predicts that the canonical projection from the étale fundamental group of $X$ onto the absolute Galois group of $k$ has a section if and only if the curve has a rational point. We show that there exist curves where the above map has a section over each completion of $k$ but not over $k$. In the appendix Victor Flynn gives explicit examples in genus 2. Our result is a consequence of a more general investigation of the existence of sections for the projection of the étale fundamental group `with abelianized geometric part' onto the Galois group. We give a criterion for the existence of sections in arbitrary dimension and over arbitrary perfect fields, and then study the case of curves over local and global fields more closely. We also point out the relation to the elementary obstruction of Colliot-Thélène and Sansuc.

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arXiv2007

Local-global principles for 1-motives

Building upon our arithmetic duality theorems for 1-motives, we prove that the Manin obstruction related to a finite subquotient $\Be (X)$ of the Brauer group is the only obstruction to the Hasse principle for rational points on torsors under semiabelian varieties over a number field, assuming the finiteness of the Tate-Shaferevich group of the abelian quotient. This theorem answers a question by Skorobogatov in the semiabelian case and is a key ingredient of recent work on the elementary obstruction for homogeneous spaces over number fields. We also establish a Cassels-Tate type dual exact sequence for 1-motives, and give an application to weak approximation.

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arXiv2004

Arithmetic Duality Theorems for 1-Motives

We prove several duality theorems for the Galois and etale cohomology of 1-motives defined over local and global fields and establish a 12-term Poitou-Tate type exact sequence. The results give a common generalisation and sharpening of well-known theorems by Tate on abelian varieties as well as results by Tate/Nakayama and Kottwitz on algebraic tori.

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