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This paper gives an elementary proof of a theorem by de Felipe and Teissier in the paper “Valuations and henselization” (arxiv.org/abs/1903.10793v1), to appear in Math. Annalen. The theorem compares two henselizations of a local domain dominated by a valuation domain. Our proofs are written in the constructive Bishop style.

International audience; In my paper at Inventiones 2009, we determine the cohomology of Lubin-Tate spaces globally using the comparison theorem of Berkovich by computing the fibers at supersingular points of the perverse sheaf of vanishing cycle Ψ of some Shimura variety of Kottwitz-Harris-Taylor type. The most difficult argument deals with the control of maps of the spectral sequences computing the sheaf cohomology of both Harris-Taylor perverse sheaves and those of Ψ. In this paper, we bypass these difficulties using the classical theory of representations of the mirabolic group and a simple geometric argument.; Dans [2], on détermine les groupes de cohomologie des espaces de Lubin-Tate par voie globale en calculant les fibres des faisceaux de cohomologie du faisceau pervers des cyclesévanescentscyclesévanescents Ψ d'une variété de Shimura de type Kottwitz-Harris-Taylor. L'ingrédient le plus complexe consistè a contrôler lesfì eches de deux suites spectrales calculant l'une les faisceaux de cohomologie des faisceaux pervers d'Harris-Taylor, et l'autre ceux de Ψ. Dans cet article, nous contournons ces difficultés en utilisant la théorie classique des représentations du groupe mirabolique ainsi qu'un argument géométrique simple. Abstract (Mirabolic group, ramified Newton stratification and cohomology of Lubin-Tate spaces) In [2], we determine the cohomology of Lubin-Tate spaces globally using the comparison theorem of Berkovich by computing the fibers at supersingular points of the perverse sheaf of vanishing cycle Ψ of some Shimura variety of Kottwitz-Harris-Taylor type. The most difficult argument deals with the control of maps of the spectral sequences computing the sheaf coho-mology of both Harris-Taylor perverse sheaves and those of Ψ. In this paper, we bypass these difficulties using the classical theory of representations of the mirabolic group and a simple geometric argument.

Si la notion de validité fait l’objet de multiples recherches, notamment dans le champ de la psychométrie, elle n’a pas encore été beaucoup étudiée dans le cadre de la didactique des mathématiques. Nous proposons d’expliciter la façon dont les outils de ce champ de recherche, en particulier l’analyse a priori des tâches, peuvent être exploités pour apporter des preuves de validité complémentaires à celles de la psychométrie. Une première partie vise à proposer une méthodologie d’analyse et de conception du contenu d’une évaluation externe dans laquelle l’approche didactique de la validité est pensée en complémentarité de celle psychométrique. Nous illustrons, dans une deuxième partie, la mise en oeuvre de cette méthodologie sur l’étude de deux domaines mathématiques (l’arithmétique en fin d’école et l’algèbre en fin de collège) dans des évaluations menées nationalement en France à ces deux ordres d’enseignement., Even if many researches deal with validity, especially in psychometrics, this notion hasn’t been significantly discussed in the field of didactics of mathematics. This paper examines how didactics tools, especially a priori analysis of tasks, can be used to bring validity evidence in addition to those provided by psychometrics. A first part of the paper aims to expose a methodology, using a combination of didactic and psychometric approaches, to analyze and design external assessment tests. In a second part, we implement this methodology to study two national large scale assessments, in two mathematical domains at two levels of school (arithmetic at the end of primary school and algebra at the end of grade 9), in France., Embora o conceito de validade seja objeto de muitas investigações, particularmente no campo da psicometria, não foi ainda muito estudado no quadro da didática das matemáticas. Ora, o nosso propósito é explicar a maneira como as ferramentas deste campo de investigação, em particular a análise prévia de tarefas, podem ser exploradas para fornecer provas de validade complementares às da psicometria. A primeira parte visa propor uma metodologia de análise e conceção do conteúdo de uma avaliação externa na qual a abordagem didática da validade é pensada em complementaridade à da psicometria. Numa segunda parte, mostramos a implementação desta metodologia no estudo de dois domínios matemáticos (aritmética no final do 2.º ciclo e de álgebra no final do 3.º ciclo do ensino básico) nas avaliações realizadas a nível nacional em França nestes dois níveis de ensino.

Based on the framework of meta-didactic transposition analysis (Arzarello et al., 2014) and specifically the concepts of brokering and boundary object, this paper studies how knowledge to teach algebra is exchanged during a training program involving nine mathematics teachers and two researchers specialised in algebra teaching and learning. Organized in three half-day sessions, this program is based on a problem pointed out in the research literature as particularly rich to develop algebraic thinking. In addition, the materials used in training are come directly from the classes of the teachers participating in the program. In this sense, the program values knowledge that makes sense in both research and teaching practice. The analysis of interactions between researchers and teachers highlights three types of collaborative activities between the two groups and thus questions the potential of such a mechanism to foster integration of research results by teachers. En s’appuyant sur le cadre d’analyse de la transposition méta-didactique (Arzarello et al., 2014) et plus précisément sur les concepts de courtage en connaissances et d’objet frontière, cet article étudie la manière dont les connaissances pour enseigner l’algèbre sont échangées dans le cadre d’un programme de formation réunissant neuf enseignants de mathématiques et deux chercheurs spécialisés en didactique de l’algèbre. Organisé en trois séances d’une demi-journée, ce programme s’articule autour d’un problème pointé dans la littérature de recherche comme particulièrement riche pour développer la pensée algébrique des élèves. Les documents exploités en formation sont en outre directement issus des classes des enseignants participant au programme. En ce sens, le programme valorise des connaissances ayant du sens tant dans la recherche que dans la pratique enseignante. L’analyse des interactions entre les chercheurs et les enseignants met en évidence trois types d’activités de collaboration entre les deux groupes d’intervenants et questionnent le potentiel d’un tel dispositif pour favoriser une appropriation, par les enseignants, de résultats de recherches.

We study the geometry and the cohomology of the tamely ramified cover of Drinfeld’s p-adic symmetric space over a p-adic field K. For this tame level, we prove, in a purely local way, most of a conjecture of Harris on the form of the $$\ell $$ -adic cohomologies, as well as the torsion freeness of the integral cohomology. In this paper, we also compute the $$\ell $$ -adic cohomology of Coxeter Deligne–Lusztig variety associated to $$\mathrm{GL}_d,$$ and some results of independent interest on the coefficient systems over the Bruhat–Tits building associated to $$\mathrm{GL}_d(K)$$ have been established.

Euclid’s lemma can be derived from the algebraic gcd property, but it is not at all apparent that Euclid himself does this. We would be quite surprised if he didn’t use this property because he points it out early on and because we expect him to make use of the Euclidean algorithm in some significant way. In this paper, we explore the question of just how the algebraic gcd property enters into Euclid’s proof, if indeed it does. Central to Euclid’s development is the idea of four numbers being proportional: a is to b as c is to d. Euclid gives two different definitions of proportionality, one in Book VII for numbers (“Pythagorean proportionality”) and one in Book V for general magnitudes (“Eudoxean proportionality”). We will discover that it is essential to keep in mind the difference between these two definitions and that many authorities, possibly including Euclid himself, have fallen into the trap of believing that Eudoxean proportionality for numbers is easily seen to be the same as Pythagorean proportionality. Finally, we will suggest a way to make Euclid’s proof good after 2300 years.