arXiv subjects

Diego Izquierdo

Publications and source records attributed to Diego Izquierdo.

15 recordsLinked to original sources

arXiv2025

Transfer principles for Galois cohomology and Serre's conjecture II

In this article, we prove several transfer principles for the cohomological dimension of fields. Given a fixed field $K$ with finite cohomological dimension $δ$, the two main ones allow to: - construct totally ramified extensions of $K$ with cohomological dimension $\leq δ- 1$ when $K$ is a complete discrete valuation field; - construct algebraic extensions of $K$ with cohomological dimension $\leq δ-1$ and satisfying a norm condition. We then apply these results to Serre's conjecture II and to some variants for fields of any cohomological dimension that are inspired by conjectures of Kato and Kuzumaki. In particular, we prove that Serre's conjecture II for characteristic $0$ fields implies Serre's conjecture II for positive characteristic fields.

math.NT↗

arXiv2025

Central simple algebras, Milnor $K$-theory and homogeneous spaces over complete discretely valued fields of dimension 2

Let $K$ be a complete discretely valued field with residue field $\bar K$ of dimension $1$ (not necessarily perfect). This occurs if and only if $K$ has dimension $2$. We prove the following statements on the arithmetic of such fields: - The "period equals index" property holds for central simple $K$-algebras. - For every prime $p$, every class in the Milnor $\mathrm{K}$-theory modulo $p$ is represented by a symbol. - Serre's Conjecture II holds for the field $K$. That is, for every semisimple and simply connected $K$-group $G$, the set $H^1(K,G)$ is trivial.

math.RA↗

arXiv2025

Product of Brauer--Manin obstruction for 0-cycles over number fields and function fields

It is conjectured that the Brauer--Manin obstruction is expected to control the existence of 0-cycles of degree 1 on smooth proper varieties over number fields. In this paper, we prove that the existence of Brauer--Manin obstruction to Hasse principle for 0-cycles of degree 1 on the product of smooth (non-necessarily proper) varieties is equivalent to the simultaneous existence of such an obstruction on each factor. We also prove an analogous statement for smooth varieties defined over function fields of $\mathbb{C}((t))$-curves.

math.AG↗

arXiv2024

Tessellations of an affine apartment by affine weight polytopes

Let $\A$ be a finite dimensional vector space and $Φ$ be a finite root system in $\A$. To this data is associated an affine poly-simplicial complex. Motivated by a forthcoming construction of connectified higher buildings, we study "affine weight polytopes" associated to these data. We prove that these polytopes tesselate $\A$. We also prove a kind of "mixed" tessellation, involving the affine weight polytopes and the poly-simplical structure on $\A$.

math.GR↗

arXiv2024

$Λ$-buildings associated to quasi-split groups over $Λ$-valued fields

Let $\mathbf{G}$ be a quasi-split reductive group and $\mathbb{K}$ be a Henselian field equipped with a valuation $ω:\mathbb{K}^{\times}\rightarrow Λ$, where $Λ$ is a non-zero totally ordered abelian group. In 1972, Bruhat and Tits constructed a building on which the group $\mathbf{G}(\mathbb{K})$ acts provided that $Λ$ is a subgroup of $\mathbb{R}$. In this paper, we deal with the general case where there are no assumptions on $Λ$ and we construct a set on which $\mathbf{G}(\mathbb{K})$ acts. We then prove that it is a $Λ$-building, in the sense of Bennett.

math.GR↗

arXiv2023

On Kato and Kuzumaki's properties for the Milnor $K_2$ of function fields of $p$-adic curves

Let $K$ be the function field of a curve $C$ over a $p$-adic field $k$. We prove that, for each $n, d \geq 1$ and for each hypersurface $Z$ in $\mathbb{P}^n_{K}$ of degree $d$ with $d^2 \leq n$, the second Milnor $K$-theory group of $K$ is spanned by the images of the norms coming from finite extensions $L$ of $K$ over which $Z$ has a rational point. When the curve $C$ has a point in the maximal unramified extension of $k$, we generalize this result to hypersurfaces $Z$ in $\mathbb{P}^n_{K}$ of degree $d$ with $d \leq n$.

math.AG↗

arXiv2023

On composition of torsors

Let $K$ be a field, let $X$ be a connected smooth $K$-scheme and let $G,H$ be two smooth connected $K$-group schemes. Given $Y \to X$ a $G$-torsor and $Z \to Y$ an $H$-torsor, we study whether one can find an extension $E$ of $G$ by $H$ so that the composite $Z \to X$ is an $E$-torsor. We give both positive and negative results, depending on the nature of the groups $G$ and $H$.

math.AG↗

arXiv2021

Local-global principles for homogeneous spaces over some two-dimensional geometric global fields

In this article, we study the obstructions to the local-global principle for homogeneous spaces with connected or abelian stabilizers over finite extensions of the field $\mathbb{C}((x,y))$ of Laurent series in two variables over the complex numbers and over function fields of curves over $\mathbb{C}((t))$. We give examples that prove that the usual Brauer-Manin obstruction is not enough to explain the failure of the local-global principle, and we then construct a variant of this obstruction using torsors under quasi-trivial tori which turns out to work. In the end of the article, we compare this new obstruction to the descent obstruction with respect to torsors under tori. For that purpose, we use a result on towers of torsors, that is of independent interest and therefore is proved in a separate appendix.

math.AG↗

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.

math.AG↗

arXiv2018

Vanishing theorems and Brauer-Hasse-Noether exact sequences for the cohomology of higher-dimensional fields

Let $k$ be a finite field, a $p$-adic field or a number field. Let $K$ be a finite extension of the Laurent series field in $m$ variables $k((x_1,...,x_m))$ or, more generally, a finite extension of the field of rational functions $k((x_1,...,x_m))(y_1,...,y_n)$. When $r$ is an integer, we consider the Galois module $\mathbb{Q}/\mathbb{Z}(r)$ over $K$ and we prove several vanishing theorems for its cohomology. In the particular case when $K$ is a finite extension of the Laurent series field in two variables $k((x_1,x_2))$, we also prove exact sequences that play the role of the Brauer-Hasse-Noether exact sequence for the field $K$ and that involve some of the cohomology groups of $\mathbb{Q}/\mathbb{Z}(r)$ which do not vanish.

math.AG↗

arXiv2018

L'espace adélique 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 caractéristique 0 et $K$ le corps des fonctions d'une $k$-courbe projective lisse géométriquement intègre $X$. Soit $T$ un $K$-tore. Dans cet article, on cherche à étudier l'espace des points adéliques $T(S,\mathbb{A}_K)$ de $T$ hors d'un ensemble fini $S$ de points fermés de $X$. On commence par montrer que le groupe $T(K)$ des points rationnels de $T$ est toujours fermé discret dans $T(S,\mathbb{A}_K)$. On décrit ensuite le quotient $T(\emptyset,\mathbb{A}_K)/T(K)$ dans chacun des trois cas suivants: $k$ corps algébriquement clos, $k=\mathbb{C}((t))$ et $k$ corps $p$-adique.

math.AG↗

arXiv2017

Autour d'une conjecture de Kato et Kuzumaki

In 1986, Kato and Kuzumaki stated several conjectures in order to give a diophantine characterization of cohomological dimension of fields. In this article, we first prove a local-global principle in this context for number fields. This allows us to give a new proof of one of Kato and Kuzumaki's conjectures for totally imaginary number fields (the first proof was given by Olivier Wittenberg). Our arguments can be generalized to get results for global fields of positive characteristic. We then establish all the conjectures for the fields $\mathbb{C}(x_1,...,x_n)$ and $\mathbb{C}(x_1,...,x_n)((t))$. We finally prove a partial result for the field of Laurent series in two variables $\mathbb{C}((x,y))$. En 1986, Kato et Kuzumaki ont formulé des conjectures cherchant à donner une caractérisation diophantienne de la dimension cohomologique des corps. Dans cet article, nous montrons d'abord un énoncé de type principe local-global dans ce contexte pour les corps de nombres. Cela nous permet de donner une nouvelle démonstration d'une des conjectures de Kato et Kuzumaki pour les corps de nombres totalement imaginaires (la première preuve étant due à Olivier Wittenberg). Nos arguments nous permettent aussi d'obtenir des résultats pour les corps globaux de caractéristique positive. Dans la suite de l'article, nous établissons toutes les conjectures de Kato et Kuzumaki pour les corps $\mathbb{C}(x_1,...,x_n)$ et $\mathbb{C}(x_1,...,x_n)((t))$. Nous montrons finalement un résultat partiel pour le corps de séries de Laurent à deux variables $\mathbb{C}((x,y))$.

math.AG↗

arXiv2016

Dualité et principe local-global sur des corps locaux de dimension 2

Let $k$ be an algebraically closed field, a finite field or a $p$-adic field. Let $K_0=k((x,y))$ be the field of Laurent series in two variables over $k$. We define Tate-Shafarevich groups of a commutative group scheme over $K_0$ via cohomology classes locally trivial at each completion of $K_0$ coming from a codimension 1 point of $\text{Spec}\; k[[x,y]]$. We establish duality theorems between Tate-Shafarevich groups for finite groups schemes and for tori. We apply these results to the study of the obstruction to the local-global principle for $K_0$-torsors under a connected linear algebraic group, answering in that way a question of Colliot-Thélène, Parimala and Suresh, and to the weak approximation for tori over $K_0$. Soit $k$ un corps algébriquement clos, un corps fini, ou encore un corps $p$-adique. Soit $K_0=k((x,y))$ le corps des séries de Laurent à deux variables sur $k$. On définit les groupes de Tate-Shafarevich d'un $K_0$-schéma en groupes commutatif en considérant les classes de cohomologie qui deviennent triviales sur chaque complété de $K_0$ provenant d'un point codimension 1 de $\text{Spec}\; k[[x,y]]$. On établit des théorèmes de dualité arithmétique entre des groupes de Tate-Shafarevich pour les modules finis et pour les tores. On applique ces résultats à l'étude du principe local-global pour les $K_0$-torseurs sous un groupe linéaire connexe, répondant ainsi à une question de Colliot-Thélène, Parimala et Suresh, ainsi qu'à l'approximation faible pour les tores sur $K_0$.

math.AG↗

arXiv2015

Variétés abéliennes sur les corps de fonctions de courbes sur des corps locaux supérieurs

Let $k$ be a higher-dimensional local field and $X$ be a smooth projective geometrically integral curve over $k$. Let $K$ be the function field of $X$. We define Tate-Shafarevich groups of an abelian variety via cohomology classes locally trivial at each completion of $K$ coming from a closed point of $X$. We prove local duality theorems for abelian varieties over $k$, as well as global duality theorems for Tate-Shafarevich groups of abelian varieties over $K$. Soient $k$ un corps local supérieur et $X$ une courbe projective lisse géométriquement intègre de corps de fonctions $K$. On définit les groupes de Tate-Shafarevich d'une variété abélienne en considérant les classes de cohomologie qui deviennent triviales sur chaque complété de $K$ provenant d'un point fermé de $X$. On établit des théorèmes de dualité locale pour les variétés abéliennes sur $k$, ainsi que des théorèmes de dualité globale pour les groupes de Tate-Shafarevich des variétés abéliennes sur $K$.

math.AG↗

arXiv2014

Théorèmes de dualité pour les corps de fonctions sur des corps locaux supérieurs et applications arithmétiques

Let $K$ be the function field of a smooth projective curve $X$ over a higher-dimensional local field $k$. We define Tate-Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of $K$ coming from a closed point of $X$. We establish duality theorems between Tate-Shafarevich groups for finite groups schemes, for tori, for groups of multiplicative type, and even for 2-term complexes of tori. We apply these results to the weak approximation for tori over $K$ and to the study of the obstruction to the local-global principle for $K$-torsors under a connected linear algebraic group. We also give examples and counter-examples to the local-global principle for central simple algebras over $K$. Soit $K$ le corps des fonctions d'une courbe projective lisse X sur un corps local supérieur $k$. On définit les groupes de Tate-Shafarevich d'un schéma en groupes commutatif en considérant les classes de cohomologie qui deviennent triviales sur chaque complété de $K$ provenant d'un point fermé de $X$. On établit des théorèmes de dualité arithmétique entre des groupes de Tate-Shafarevich pour les modules finis, pour les tores, pour les groupes de type multiplicatif, et même pour les complexes à deux termes de tores. On applique ces résultats à l'approximation faible pour les tores sur $K$ et à l'étude du principe local-global pour les $K$-torseurs sous un groupe linéaire connexe. On exhibe aussi des exemples et des contre-exemples au principe local-global pour les algèbres simples centrales sur $K$.

math.AG↗