Note sur les corps 2-rationnels

Bornes sur le nombre de points rationnels des courbes -- en quete d'uniformite

The aim of this paper is to show how a conjectural lower bound on the canonical height function in the spirit of Lang and Silverman leads to an explicit uniform bound on the number of rational points on curves of genus $g\geq 2$ over a number field.

D{é}monstration d'une conjecture de Kruyswijk et Meijer sur le plus petit d{é}nominateur des nombres rationnels d'un intervalle

The average value of the smallest denominator of a rational number belonging to the interval $](j-1)/N,j/N]$, where~$j=1,\dots, N$, is proved to be asymptotically equivalent to~$16π^{-2}\sqrt{N}$, when $N$ tends to infinity. The result had been conjectured in 1977 by Kruyswijk and Meijer.

Points rationnels sur les quotients d'Atkin-Lehner de courbes de Shimura de discriminant $pq$

Let $p$ and $q$ be two distinct prime numbers, and $X^{pq}/w_q$ be the quotient of the Shimura curve of discriminant $pq$ by the Atkin-Lehner involution $w_q$. We describe a way to verify in wide generality a criterion of Parent and Yafaev to prove that if $p$ and $q$ satisfy some explicite congruence conditions, known as the conditions of the non ramified case of Ogg, and if $p$ is large enough compared to $q$, then the quotient $X^{pq}/w_q$ has no rational point, except possibly special points.

Familles de polynomes liees aux courbes modulaires X_1(l) unicursales et points rationnels non-triviaux de courbes elliptiques quotient

Soit l un entier et E_{c,l} la famille de Kubert des courbes elliptiques definies sur Q munies d'un point rationnel A d'ordre l. On note F_{c,l} la courbe elliptique quotient de E_{c,l} par le groupe engendre' par A, et f_l l'isogenie de E_{c,l} sur F_{c,l}. Pour l = 3, 4, 5 et 6, nous construisons explicitement, pour des parametrisations convenables de c, des elements non-triviaux de F_{c,l}(Q) / (f_l(E_{c,l}(Q))), autrement dit, des points explicites de F_{c,l}(Q) qui ne sont l'image par f_l d'aucun element de E_{c,l}(Q). Ces points sont en general d'ordre infini. Nous donnons des applicatio…

A program to test the p-rationality of any number field

Let K be a number field. We prove that its ray class group modulo p 2 (resp. 8) if p > 2 (resp. p = 2) characterizes its p-rationality. Then we give two short, very fast PARI Programs (§§3.1, 3.2) testing if K (defined by an irreducible monic polynomial) is p-rational or not. For quadratic fields we verify some densities related to Cohen-Lenstra-Martinet ones and analyse Greenberg's conjecture on the existence of p-rational fields with Galois groups (Z/2Z) t needed for the construction of some Galois representations with open image. We give examples for p = 3, t = 5 and t = 6 (§§5.1, 5.2) and …

Surfaces de del Pezzo de degré 4 sur un corps $C_1$

Sur toute surface de del Pezzo de degré 4 sur un corps $C_1$ de caractéristique zéro, tous les points rationnels sont R-équivalents. Plus généralement, ceci vaut sur tout corps parfait infini de caractéristique différente de 2. ---- On a del Pezzo surface of degree 4 over a characteristic zero $C_1$-field, all rational points are R-equivalent.This more generally holds over any infinite perfect $C_{1}$-field of characteristic different from 2.

Approximation faible pour les 0-cycles sur un produit de variétés rationnellement connexes

Consider weak approximation for 0-cycles on a smooth proper variety defined over a number field, it is conjectured to be controlled by its Brauer group. Let $X$ be a Châtelet surface or a smooth compactification of a homogeneous space of a connected linear algebraic group with connected stabilizer. Let $Y$ be a rationally connected variety. We prove that weak approximation for 0-cycles on the product $X\times Y$ is controlled by its Brauer group if it is the case for $Y$ after every finite extension of the base field. We do not suppose the existence of 0-cycles of degree $1$ neither on $X$ nor…

Produit eulérien motivique et courbes rationnelles sur les variétés toriques

We study the asymptotical behaviour of the moduli space of morphisms of given anticanonical degree from a rational curve to a split toric variety, when the degree goes to infinity. We obtain in this case a geometric analogue of Manin's conjecture about rational points of bounded height on varieties defined over a global field. The study is led through a generating series whose coefficients lie in a Grothendieck ring of motives, the motivic height zeta function. In order to establish convergence properties of this function, we use a notion of eulerian motivic product. It relies on a constructio…

Quelques questions d'approximation faible pour les tores algébriques

Let K be a global field, T a K-torus and S a finite set of places of K. Let K_v be the completion at a place v. Denote by T(O_v) the maximal compact subgroup of the group T(K_v) of K_v-points of T. We show that the diagonal map from T(K) to the product for v in S of the T(K_v)/T(O_v) need not be onto. As a corollary, for suitable v, the group T(O_v) does not cover all R-equivalence classes in T(K_v). While studying the height zeta functions of an algebraic torus over a function field in one variable over a finite field, D. Bourqui showed that Peyre's constant from the number field case should …

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 $\…

Remarques sur un article r'ecent de B. Poonen

B. Poonen recently produced smooth threefolds over a number field which do not have a rational point but have no Brauer-Manin obstruction even after descent to a finite 'etale cover. In this note I show that the varieties he produces have zero-cycles of degree 1. ----- B. Poonen a r'ecemment exhib'e des exemples de vari'et'es projectives et lisses de dimension 3 sur un corps de nombres qui n'ont pas de point rationnel et pour lesquelles il n'y a pas d'obstruction de Brauer-Manin apr`es rev^etement fini 'etale. Dans cette note, je montre que les vari'et'es qu'il construit poss`edent des z'ero-c…

Points de hauteur bornée sur les hypersurfaces de l'espace triprojectif

We establish an asymptotic formula for the number of points of bounded height on a singular hypersurface of the triprojective space. We will see that the final result is in accordance with Batyrev-Manin conjecture. The method used is a version of the circle method developped by Heath Brown and used by Robbiani for counting points of bounded height on an hypersurface of biprojective space. ----- Nous établissons ici une formule asymptotique pour le nombre de points de hauteur bornée sur une hypersurface singulière de l'espace triprojectif. Nous verrons que le résultat obtenu est en accord avec …

On the distribution of points of bounded height on equivariant compactifications of vector groups

We prove asymptotic formulas for the number of rational points of bounded height on smooth equivariant compactifications of the affine space. (Nous établissons un développement asymptotique du nombre de points rationnels de hauteur bornée sur les compactifications équivariantes lisses de l'espace affine.)

Résultat géométrique sur les représentations de groupes réductifs sur un corps ultramétrique

This article adapts the results of arXiv:1305.6557 to the ultrametric setting. This is a re-edition of the Chapter V of author thesis.

Zéro-cycles sur les fibrations au-dessus d'une courbe de genre quelconque

Let X be a smooth and proper variety over a number field k. Conjectures on the image of the Chow group of zero-cycles of X in the product of the corresponding groups over all completions of k were put forward by Colliot-Thélène, Kato and Saito. We prove these conjectures for the total space of fibrations, over curves with finite Tate-Shafarevich group, into rationally connected varieties which satisfy weak approximation, under an abelianness assumption on the singular fibers. ---- Soit X une variété propre et lisse sur un corps de nombres k. Des conjectures sur l'image du groupe de Chow des zé…

Les quartiers de la lune de Troie ou le bouclier d'Achille, essai sur le corps de classe

This is a text in french about classical class field theory, readable with a minimum of prerequisites

关于Borovoi-Kunyavskiı齐性空间

Sur les espaces homogènes de Borovoi-Kunyavski\uı

本文证明由Borovoi-Kunyavskiı构造的SL_n的某些齐性空间（几何稳定子幂零类2）满足Hasse原理和弱逼近，从而验证Colliot-Thélène关于Brauer-Manin障碍的猜想。

Sur la nullité du mu-invariant d'Iwasawa des corps totalement réels

The article is withdrawn. There is a gap in the proof of the vanishing of Iwasawa mu-invariant for the cyclotomic Z_p extension of a CM field, abelian over a totally real number field.

Sur la torsion de la distribution ordinaire universelle attachée à un corps de nombres

We study the torsion subgroup of the universal ordinary distribution related to a general number field. We describe a way to control this subgroup. We apply this method to the special case of an imaginary quadratic field, and we give examples of such fields where these torsion subgroups are non-trivial.

Sur le sous-groupe des éléments de hauteur infinie du K2 d'un corps de nombres

By using the logarithmic approach of the classical kernels for the K2 of number fields, we compute the 2-rank of the wild kernel WK2(F) and the 2-rank of the subgroup of infinite heigh elements in K2(F) in terms of positive class groups for any number field F.

Le problème inverse de Galois sur les corps des fractions tordus à indéterminée centrale

In this article, we show that the Inverse Galois Problem over a skew field $H$ of finite dimension over its center $k$ is equivalent to a variant of the Inverse Galois Problem over $k$ involving a polynomial constraint. As an application, we show that if $k$ contains an ample field, then the Inverse Galois Problem has a positive answer over the skew field $H(t)$ of rational fractions with central indeterminate.

Distributions de séries d'Eisenstein presque holomorphes sur un corps totalement réel

At first a type of Eisenstein series is defined as distributions giving nearly-holomorphic automorphic forms on a totally real field, with different expressions (integral, summation) ; then these are shown to satisfied the expected properties, and the Fourier coefficients are explicitly computed. Finally, an explicit computation shows that in the simplest case, the abstract object considered simplify down to the most classical version of Eisenstein series.

Types modulo $\ell$ pour les formes intérieures de $GL_n$ sur un corps local non archimédien

Let F be a non-Archimedean locally compact field of residue characteristic p, let D be a finite dimensional central division F-algebra and let l be a prime number different from p. We develop a theory of l-modular types for the group GL(m,D), m\>1, in preparation of the study of the l-modular smooth representations of this group.

Décompte dans une conjecture de Lang sur les corps de fonctions : cas des courbes

Let $X$ be a genus $d$ curve with $d\geq 2$ defined over a global function field $K$ of characteristic $p>0$ with $p>2d+1$. Suppose $X$ non-isotrivial. Let $Γ$ be a sub-group of $J(K_s)$, where $J$ is the jacobian of $X$ and $K_s$ is a separable closure of $K$, verifying $Γ/p Γ$ finite. Then one shows that $X\cap Γ$ has finite cardinal and one provides an explicit upper bound. This generalizes a result of Buium and Voloch.

Remarques sur le premier cas du théorème de Fermat sur les corps de nombres

The first case of Fermat's Last Theorem for a prime exponent $p$ can sometimes be proved using the existence of local obstructions. In 1823, Sophie Germain has obtained an important result in this direction by establishing that, if $2p+1$ is a prime number, the first case of Fermat's Last Theorem is true for $p$. In this paper, we investigate such obstructions over number fields. We obtain analogous results on Sophie Germain type criteria, for imaginary quadratic fields. Furthermore, extending a well known statement over ${\bf Q}$, we give an easily testable condition which allows occasionally…

Critère d'irréductibilité pour les courbes elliptiques semi-stables sur un corps de nombres

For a fixed number field and an elliptic curve defined and semi-stable over this number field, we consider the set of prime numbers p such that the Galois representation attached to the p-torsion points of the elliptic curve is reducible. When the number field satisfies a certain necessary condition, we give an explicit bound, depending only on the number field and not on the semi-stable elliptic curve, for these primes. This generalizes previous results of Kraus.

Sur la pro-p-extension localement cyclotomique maximale d'un corps de nombres

Let p be a prime number and F be a number field. We consider the Galois group G over the cyclotomic Z_p extension of F of the maximal unramified, p-decomposed, pro-p-extension of the cyclotomic Z_p extension of F. The question whether G is free pro-p was already asked by many authors. In this article, we highlight a link between the freeness of G and the Galois descent for some localisation kernels. Then we give explicit criterions to show that G is not a free pro-p-group.

Pentes des fibrés vectoriels adéliques sur un corps global

At the end of the twentieth century, J.-B. Bost developped a slope theory of hermitian vector bundles over number fields. A new method of diophantine approximation, the so-called slope method, has emerged from his research. Our article proposes a generalisation to adelic vector bundles over global fields. The norms at the archimedean places are no longer supposed to be hermitian. The link with adelic successive minima is also mentioned. To get these results, we use several concepts from the geometry of finite dimensional Banach spaces.

Minoration du rang des courbes elliptiques sur les corps de classes de Hilbert

Soit $E/\BmQ$ une courbe elliptique. Soit $D<0$ un discriminant fondamental suffisamment grand. Si $E(\bar{\BmQ})$ contient des points de Heegner de discriminant $D$, ces points engendrent un sous-groupe dont le rang est supérieur à $\pabs{D}^{0.0009}$. Ce résultat est en accord avec la conjecture de Birch et Swinnerton-Dyer. --- Let $E/\BmQ$ be an elliptic curve. Let $D<0$ be a sufficiently large fundamental discriminant. If $E(\bar{\BmQ})$ contains Heegner points of discriminant $D$, these points generate a subgroup of rank greater than $\pabs{D}^{0.0009}$. This result is in agreement with t…

Une base explicite de symboles modulaires sur les corps de fonctions

Modular symbols for the congruence subgroup $Γ_0(\mathfrak{n})$ of $GL_{2}(\mathbf{F}_q[T])$ have been defined by Teitelbaum. They have a presentation given by a finite number of generators and relations, in a formalism similar to Manin's for classical modular symbols. We completely solve the relations and get an explicit basis of generators when $\mathfrak{n}$ is a prime ideal of odd degree. As an application, we give a non-vanishing statement for $L$-functions of certain automorphic cusp forms for $\mathbf{F}_q(T)$. The main statement also provides a key-step for a result towards the uniform…

非阿基米德等分布与树的群作用

Equidistribution non-archimédienne et actions de groupes sur les arbres

本研究利用离散测地流的遍历性质证明树商空间中共公垂线的等分布定理，进而推导出有限域上函数域元素及二次无理数在完备局部域中的非阿基米德等分布结果。

Problèmes de plongement finis sur les corps non commutatifs

We extend finite embedding problems over fields, a central notion in inverse Galois theory, to the situation of a skew field $H$ of finite dimension over its center $h$. First, we show that solving a finite embedding problem over $H$ is equivalent to finding a solution to some finite embedding problem over $h$ fulfilling a polynomial constraint. Next, we show that every constant finite split embedding problem over the skew field of fractions $H(t)$ with central indeterminate $t$ has a solution, if $h$ is an ample field. This is a non-commutative analogue of a deep result of Pop. More generally…

Le théorème de Fermat sur certains corps de nombres totalement réels

Let $K$ be a totally real number field. For all prime number $p\geq 5$, let us denote by $F_p$ the Fermat curve of equation $x^p+y^p+z^p=0$. Under the assumption that $2$ is totally ramified in $K$, we establish some results about the set $F_p(K)$ of points of $F_p$ rational over $K$. We obtain a criterion so that the asymptotic Fermat's Last Theorem is true over $K$, criterion related to the set of Hilbert modular cusp newforms over $K$, of parallel weight $2$ and of level the prime ideal above $2$. It is often simply testable numerically, particularly if the narrow class number of $K$ is $1$…

Sur la capitulation des 2-classes d'idéaux du corps Q(\sqrt{2p_1p_2}, i)

Let $p_1$ and $p_2$ be two primes such that $p_1\equiv p_2\equiv1 \pmod4$ and at least two of the three elements $\{(\frac{2}{p_1}), (\frac{2}{p_2}), (\frac{p_1}{p_2})\}$ are equal to -1. Put $i=\sqrt{-1}$, $d=2p_1p_2$ and $k =Q(\sqrt{d}, i)$. Let $k_2^{(1)}$ be the Hilbert 2-class field of $k$ and $k^{(*)}=Q(\sqrt{p_1},\sqrt{p_2},\sqrt 2, i)$ be its genus field. Let $C_{k,2}$ denote the 2-part of the class group of $k$. The unramified abelian extensions of $k$ are $K_1=k(\sqrt{p_1})$, $K_2=k(\sqrt{p_2})$, $K_3=k(\sqrt{2})$ and $k^{(*)}$. Our goal is to study the capitulation problem of the 2-…

Summary of Delsarte's "Nombre de Solutions des Equations Polynomiales sur un Corps Fini"

An English summary is given of Jean Delsarte's article "Nombre de solutions des equations polynomiales sur un corps fini."

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 l…

Dualité sur un corps local de caractéristique positive à corps résiduel algébriquement clos

Let K be a complete discretely valued field with residue field k of characteristic p>0. There is a duality theory for cohomology with coefficients in commutative finite K-group schemes in the following cases : char(K)=0 and k finite (Tate), char(K)=p and k finite (Shatz), char(K)=0 and k algebraically closed (Bégueri). In this paper, we settle the case where char(K)=p and k is algebraically closed.

Sur le corps des modules de certaines variétés

To every covering of curves, we associate several varieties having the same field of moduli and same fields of definition. We deduce examples of curves having Q (the field of rationals) as field of moduli, that admit models over any completion of Q but no model over Q.

Sur le groupe de Chow de codimension deux des variétés sur les corps finis

Using the construction of Colliot-Thélène and Ojanguren, we exhibit an example of a smooth projective geometrically rational variety X defined over a finite field F_p with an algebraic closure \bar F_p and the absolute Galois group G, such that the group H^3_nr(X, Z/2) is nonzero and the map CH^2(X)\to CH^2(\bar X)^G is not surjective.

Zéro-cycles sur les espaces homogènes et problème de Galois inverse

Let X be a smooth compactification of a homogeneous space of a linear algebraic group G over a number field k. We establish the conjecture of Colliot-Thélène, Sansuc, Kato and Saito on the image of the Chow group of zero-cycles of X in the product of the same groups over all the completions of k. When G is semisimple and simply connected and the geometric stabiliser is finite and supersolvable, we show that rational points of X are dense in the Brauer-Manin set. For finite supersolvable groups, in particular for finite nilpotent groups, this yields a new proof of Shafarevich's theorem on the i…

Les variétés sur le corps à un élément

We propose a definition of varieties over the field with one element. These have extensions of scalars to the ring of integers which are varieties in the usual sense. We show that toric varieties can be defined over the field with one element. We also discuss zeta functions for such objects. We give a motivic interpretation of the image of the J-homomorphism defined by Adams. ~ ~ ~ ~

Sur la p-dimension des corps

Let A be an excellent integral henselian local noetherian ring, k its residue field of characteristic p>0 and K its fraction field. Using an algebraization technique introduced by the first named author, and the one-dimension case already proved by Kazuya KATO, we prove the following formula: cd_p(K) = dim(A) + p-rank(k), if k is separably closed and K of characteristic zero. A similar statement is valid without those assumptions on k and K.

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 …

Stabilisation de la formule des traces tordue V: intégrales orbitales et endoscopie sur le corps réel

It is one of a series of articles whose goal is to stabilize the twisted trace formula. In two previous papers, we have considered the stabilization of weighted orbital integrals over a non-archimedean local field. Here we consider the similar question over the real field. The principal result is the same: all expected properties are deduced from the stabilization of weighted orbital integrals relative to strongly regular elements.

Dualité 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 théorèmes de dualité classiques (Tate, Poitou, Nakayama) pour la cohomologie galoisienne à valeurs dans des tores et des modules abéliens finis. Nous établissons de tels théorèmes pour $K$ le corps des fonctions d'une courbe projective et lisse sur le corps $F={\bf C}((t))$ des séries formelles en une variable sur le corps des complexes. Cela permet de contrôler le défaut du principe de Hasse et l'approximation faible pour les espaces homogènes sous un tore. Il y a ici des diff…

Sur le rang des Jacobiennes sur un corps de fonctions

On the rank of Jacobians over function fields.} Let $f:\mathcal{X}\to C$ be a projective surface fibered over a curve and defined over a number field $k$. We give an interpretation of the rank of the Mordell-Weil group over $k(C)$ of the jacobian of the generic fibre (modulo the constant part) in terms of average of the traces of Frobenius on the fibers of $f$. The results also give a reinterpretation of the Tate conjecture for the surface $\mathcal{X}$ and generalizes results of Nagao, Rosen-Silverman and Wazir.

Stabilisation de la formule des traces tordue II: intégrales orbitales et endoscopie sur un corps local non-archimédien; définitions et énoncés des résultats

It is the second article of a series whose final goal is the stabilization of the twisted trace formula. Over a non archimedean field, we define (after Arthur) wheighted orbital integrals and their stable and endoscopic versions. We state the main theorems that will be proved in the following articles. We begin to study the local properties of our integrals, for instance expansion in Shalika germs.

Equivalence numérique et équivalence cohomologique pour les variétés abéliennes sur les corps finis

In characteristic zero, it was proven a long time ago by D. Lieberman that cohomological and numerical equivalence coincide for cycles on abelian varieties. In this paper we show this to be true also in a somewhat technical sense for abelian varieties defined over a finite field. The technicality is that the result is valid for l-adic cohomology for primes l of non-zero density.

Stabilisation de la formule des traces tordue III: intégrales orbitales et endoscopie sur un corps local non-archimédien; réductions et preuves

This is one of a series of articles, in collaboration with C. Moeglin, whose goal is to stabilize the twisted trace formula. In the previous article, we have stated several assertions about the stabilization of weighted orbital integrals over a non-archimedean field. Those integrals depend on a conjugacy class in our twisted space. Here we deduce all these assertions from a unique one, where the conjugacy class is assumed to be strongly regular.

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 …

关于有限域上某些三维乘积的整值Tate猜想

Autour de la conjecture de Tate entière pour certains produits de dimension $3$ sur un corps fini

本文研究有限域上满足b_2=ρ的曲面与曲线的三维乘积上的1-圈整值Tate猜想强形式，推广并给出了此前与Colliot-Thélène合作结果的若干无条件证明。

复代数曲线函数域上的强逼近纯度

Pureté de l'approximation forte sur le corps des fonctions d'une courbe algébrique complexe

该研究证明复代数曲线函数域上，半单代数群齐性空间及低次仿射光滑完全交的余维2闭子集补集满足去掉非空有限位集的强逼近性质。

Stabilisation de la formule des traces tordue VIII: l'application epsilon_tilde{M} sur un corps de base local non-archimédien

It is a step in the proof of the stabilization of the twisted trace formula. We generalize to the twisted case the proposition 3.1 of the third Arthur's paper on the stabilization. That is, consider the difference between an omega-equivariant weighted orbital integral (relative to a Levi subspace of a twisted space) and its endoscopic avatar. Then this difference is the ordinary omega-orbital integral of some function on the Levi subspace. Here, the base-field is non-archimedean.

Vers un algorithme pour la réduction stable des revêtements p-cycliques de la droite projective sur un corps p-adique

In his Ph. D. thesis, C. Lehr offers an algorithm which gives the stable model for p-cyclic covers of the projective line over a p-adic field under the conditions that the branch locus whose cardinal is m+1 has the so called equidistant geometry and m<p. In this note we give an algorithm also in the equidistant geometry case but without condition on m. In particular we are able to study the reduction at 2 of hyperelliptic curves with equidistant branch locus.