Searcharxiv⌕ Search

arXiv subjects

Antoine Chambert-Loir

Publications and source records attributed to Antoine Chambert-Loir.

At least 19 recordsLinked to original sources
arXiv2026

Potentiel et rationalité

Nous étendons aux courbes de genre arbitraire le théorème de rationalité de Cantor, lui-même une extension de théorèmes de Borel, Pólya, Dwork, Bertrandias et Robinson. La démonstration s'effectue en deux étapes. La première est un critère d'algébricité, démontré par une méthode d'approximation diophantienne. La seconde repose sur le théorème de l'indice de Hodge en théorie d'Arakelov. -- We extend to algebraic curves of arbitrary genus the rationality theorem of Cantor, itself an extension of theorems of Borel, Pólya, Dwork, Bertrandias and Robinson. The proof runs in two steps. The first step is an algebraicity criterion, which is proved using a method of diophantine approximation. The second step relies on the Hodge index theorem in Arakelov geometry.

math.NT↗
arXiv2026

Burnside rings and volume forms with logarithmic poles

We develop a theory of Burnside rings in the context of birational equivalences of algebraic varieties equipped with logarithmic volume forms. We introduce a residue homomorphism and construct an additive invariant of birational morphisms. We also define a specialization homomorphism. -- Nous proposons une théorie d'anneaux de Burnside dans le contexte de la géométrie birationnelle des variétés algébriques munies d'une forme volume à pôles logarithmiques. Nous introduisons un homomorphisme « résidu », construisons un invariant additif des morphismes birationnels. Nous définissons aussi un homomorphisme de spécialisation.

math.AG↗
arXiv2025

Formalizing Polynomial Laws and the Universal Divided Power Algebra

The goal of this paper is to present an ongoing formalization, in the framework provided by the Lean/Mathlib mathematical library, of the construction by Roby (1965) of the universal divided power algebra. This is an analogue, in the theory of divided powers, of the classical algebra of polynomials. It is a crucial tool in the development of crystalline cohomology; it is also used in $p$-adic Hodge theory to define the crystalline period ring. As an algebra, this universal divided power algebra has a fairly simple definition that shows that it is a graded algebra. The main difficulty in Roby's theorem lies in constructing a divided power structure on its augmentation ideal. To that aim, Roby identified the graded pieces with another universal structure: homogeneous polynomial laws.We formalize the first steps of the theory of polynomial laws and show how future work will allow to complete the formalization of the above-mentioned divided power structure. We report on various difficulties that appeared in this formalization: taking care of universes, extending to semirings some aspects of the Mathlib library, and coping with several instances of "invisible mathematics".

cs.LO↗
arXiv2025

Formes différentielles réelles et courants sur les espaces de Berkovich

We define a theory of real $(p,q)$-forms and currents on Berkovich spaces which is parallel to the theory of differential forms on complex spaces. It is based on Lagerberg's theory of superforms in tropical geometry and on the consideration of tropicalization maps and skeleta on domains of non archimedean analytic spaces in the sense of Berkovich. We construct canonical calibrations of skeleta of analytic spaces, which give rise to integrals of $(n,n)$-forms, and a variant of Stokes formula. The theory of currents furnishes analogues of the Poincaré-Lelong formula, as well as the formulas of Bochner-Martinelli and Levine. We define a notion of plurisubharmonic functions and develop an analogue of Bedford-Taylor's theory of products of closed positive currents. Smooth metrized line bundles have a Chern form; the integrals of products of these Chern forms is compatible with numerical intersection theory. The case of psh metrics gives rise to Chern currents. In the case of formal metrics, we compute these product currents in terms of intersection numbers of the special fiber. In a final chapter, we detail how the uniformization of abelian varieties allows to study the canonical metrics on their line bundles. The theory allows to reinterpret tropical intersection theory and is presented in the general context of so-called "tropical spaces" which we introduce in a first part of the book.

math.AG↗
arXiv2025

A Formalization of Divided Powers in Lean

Given an ideal $I$ in a commutative ring $A$, a divided power structure on $I$ is a collection of maps $\{γ_n \colon I \to A\}_{n \in \mathbb{N}}$, subject to axioms that imply that it behaves like the family $\{x \mapsto \frac{x^n}{n!}\}_{n \in \mathbb{N}}$, but which can be defined even when division by factorials is not possible in $A$. Divided power structures have important applications in diverse areas of mathematics, including algebraic topology, number theory and algebraic geometry. In this article we describe a formalization in Lean 4 of the basic theory of divided power structures, including divided power morphisms and sub-divided power ideals, and we provide several fundamental constructions, in particular quotients and sums. This constitutes the first formalization of this theory in any theorem prover. As a prerequisite of general interest, we expand the formalized theory of multivariate power series rings, endowing them with a topology and defining evaluation and substitution of power series.

cs.LO↗
arXiv2025

La logique continue des corps globalement valués

The continuous logic of globally valued fields -- A globally valued field is a field endowed with a family of absolute values that satisfy a product formula. Number fields and function fields in one variable give classical and fundamental examples; Nevanlinna theory also gives rise to such structures on the field of meromorphic functions on $\mathbf C$. These globally valued fields can be studied in the context of continuous logic (for which the predicates are real valued), and such a study has been undertaken some 10 years ago by Ben Yaacov and Hrushovski, thus providing a model-theoretic framework for the diophantine theory of heights. One of the first fundamental results in the tehory states the the field of algebraic numbers, with its essentially unique structure of a globally valued field, is existentially closed: every system involving polynomial equalities and inequalities, as well as strict inequalities in heights, possesses a solution in algebraic numbers as soon as it possesses some solution in a globally valued extension. The proof, due to Szachniewicz, is inspired by the proof proposed by Ben Yaacov and Hrushovski in the case of function fields: the latter used in a crucial way the description by Boucksom, Demailly, P\u aun and Peternell of the cone of mobile curves in a complex projective variety, the case of number fields relies on recent results in Arakelov geometry.

math.LO↗
arXiv2023

Balade newtonienne entre analyse et arithmétique (Newtonian promenade between analysis and arithmetic)

Invented by Kurt Hensel at the very end of 19th century on the model of power series in one indeterminate, the $p$-adic numbers have not only become an indispensable tool of contemporary arithmetic, but a research topic per se. In this text, stemming out two talks at the 2023 X-UPS lectures, I shall explain their construction, how they fit in a vaster framework, between analysis and arithmetic, where two constructions bearing the name of Isaac Newton play a central role : Newton's method and Newton's polygon. Inventés par Kurt Hensel à la toute fin du 19e siècle sur le modèle des séries formelles en une indéterminée, les nombres $p$-adiques sont devenus non seulement un outil indispensable de l'arithmétique contemporaine, mais un sujet d'étude en soi. Dans ce texte, issu de deux exposés aux journées X-UPS 2023, j'expliquerai leur construction, comment ils s'insèrent dans un cadre plus vaste, entre analyse et arithmétique, où deux constructions portant le nom d'Isaac Newton jouent un rôle central: la méthode de Newton et la notion de polygone de Newton.

math.HO↗
arXiv2023

Un experimento de demostración formal de un teorema de nivel intermedio en álgebra (Formalizing the proof of an intermediate-level algebra theorem -- An experiment)

Proof assistants are computer softwares that allow us to write mathematical proofs so as to assess their correctness. In November 2021, I started the project of checking the simplicity of the alternating groups within the Lean theorem prover and its mathlib library. This text aims at reviewing this experiment. -- (French) Les assistants de preuves sont des logiciels qui permettent de rédiger des démonstrations mathématiques et d'en garantir leur correction. En novembre 2021, j'ai débuté un projet de vérification de la simplicité des groupes alternés au sein de l'assistant de preuve Lean, et de sa librairie mathlib. Ce texte est un essai de compte rendu de cette expérience. -- Published version in Spanish

math.GR↗
arXiv2022

Les conjectures de Weil : origines, approches, généralisations

Je retracerai l'histoire des conjectures de Weil sur le nombre de solutions d'équations polynomiales dans un corps fini et quelques unes des approches qui ont été proposées pour les résoudre. The Weil conjectures: origins, approaches, generalizations. I recount the history of the conjectures by Weil on the number of solutions of polynomial equations in finite fields, and some of the approaches that have been proposed to solve them.

math.NT↗
arXiv2021

La conjecture de Mordell: origines, approches, généralisations

The Mordell conjecture: origins, approaches, generalizations -- The Mordell conjecture predicts that a diophantine equation defining a smooth projective curve of genus at least two has only finity many solutions in a given number field. The century that ran since its statement, in 1922, gave rise to several approaches, several proofs, and vast extensions most of which are still conjectural. This text is based on the oral presentation and aims at recalling this story. La conjecture de Mordell prédit qu'une équation diophantienne définissant une courbe projective lisse de genre au moins deux n'a qu'un nombre fini de solutions dans un corps de nombres donné. Le siècle qui s'est écoulé depuis son énoncé, en 1922, a vu plusieurs approches, plusieurs démonstrations, ainsi que de vastes extensions dont la plupart sont encore conjecturales. Ce texte, qui reprend l'exposé oral, s'efforce de retracer cette histoire.

math.NT↗
arXiv2020

Arakelov geometry, heights, equidistribution, and the Bogomolov conjecture

This is an introduction to the topics of the title, from the 2017 Grenoble Summer school on Arakelov geometry and arithmetic applications. We review Arithmetic intersection numbers, explain the definition of the height of a variety and its properties, both in the framework of classical Arakelov geometry and of Zhang's adelic formalism. We then discuss arithmetic ampleness and its application to the equidistribution theorem of Szpiro-Ullmo-Zhang-Yuan, both at complex and nonarchimedean places. We conclude with Ullmo-Zhang's proof of the Bogomolov conjecture over number fields.

math.NT↗
arXiv2019

Relations de Hodge--Riemann et combinatoire des matroïdes (d'après K. Adiprasito, J. Huh et E. Katz)

Finite matroids are combinatorial structures that express the concept of linear independence. In 1964, G.-C. Rota conjectured that the coefficients of the "characteristic polynomial" of a matroid $M$, polynomial whose coefficients enumerate its subsets of given rank, form a log-concave sequence. K. Adiprasito, J. Huh et E. Katz have proved this conjecture using methods which, although entirely combinatorial, are inspired by algebraic geometry. From the Bergman fan of the matroid $M$, they define a graded "Chow ring" $A(M)$ for which they prove analogs of the Poincaré duality, the Hard Lefschetz theorem, and the Hodge--Riemann relations. The sought for log-concavity inequalities are then analogous to the Khovanskii--Teissier inequalities.

math.AG↗
arXiv2019

Le théorème de réduction stable de Deligne et Mumford

The stable reduction theorem of Deligne and Mumford --- The moduli space of smooth projective curves of genus $g$ is a quasi-projective algebraic variety, but is not projective. To understand its geometry, it may be crucial to consider compactifications of this space. By allowing to parameterize as well curves with controlled singularities (the so called stable curves), Deligne and Mumford constructed a projective compactification. The properness of this compactification translates into the stable reduction theorem that they prove, its projectivity is a later theorem of Knudsen and Mumford. This text is based on the oral presentation and aims at introducing these objects.

math.AG↗
arXiv2017

A non-archimedean Ax-Lindemann theorem

We prove a statement of Ax-Lindemann type for the uniformization of products of Mumford curves whose associated fundamental groups are non-abelian Schottky subgroups of $\mathop{\rm PGL}(2,\bar{\mathbf Q_p})$ contained in $\mathop{\rm PGL}(2,\bar{\mathbf Q})$. In particular, we characterize bi-algebraic irreducible subvarieties of the uniformization.

math.AG↗
arXiv2015

Motivic height zeta functions

Let $C$ be a projective smooth connected curve over an algebraically closed field of characteristic zero, let $F$ be its field of functions, let $C_0$ be a dense open subset of $C$. Let $X$ be a projective flat morphism to $C$ whose generic fiber $X_F$ is a smooth equivariant compactification of $G$ such that $D=X_F\setminus G_F$ is a divisor with strict normal crossings, let $U$ be a surjective and flat model of $G$ over $C_0$. We consider a motivic height zeta function, a formal power series with coefficients in a suitable Grothendieck ring of varieties, which takes into account the spaces of sections $s$ of $X\to C$ of given degree with respect to (a model of) the log-anticanonical divisor $-K_{X_F}(D)$ such that $s(C_0)$ is contained in $U$. We prove that this power series is rational, that its "largest pole" is at $\mathbf L^{-1}$, the inverse of the class of the affine line in the Grothendieck ring, and compute the "order" of this pole as a sum of dimensions of various Clemens complexes at places of $ C\setminus C_0$. This is a geometric analogue of a result over number fields by the first author and Yuri Tschinkel (Duke Math. J., 2012). The proof relies on the Poisson summation formula in motivic integration, established by Ehud Hrushovski and David Kazhdan (Moscow Math. J, 2009).

math.AG↗
arXiv2012

Diophantine Geometry and Analytic Spaces

This text is the write-up of a talk at the Bellairs Workshop in Number Theory on Tropical and Non-Archimedean Geometry that took place at the Bellairs Research Institute, Barbados, in May 2011. The goal of this text is to present recent work by in Diophantine Geometry over function fields due to Gubler and Yamaki, where analytic geometry in the sense of Berkovich plays a significant place. I also give an introduction to basic concepts and notions on Diophantine Geometry, such as heights, the Manin-Mumford conjecture, the Bogomolov conjecture, and its proof by Ullmo and Zhang.

math.NT↗
arXiv2012

On the canonical degrees of curves in varieties of general type

A widely believed conjecture predicts that curves of bounded geometric genus lying on a variety of general type form a bounded family. One may even ask whether the canonical degree of a curve $C$ in a variety of general type is bounded from above by some expression $aχ(C)+b$, where $a$ and $b$ are positive constants, with the possible exceptions corresponding to curves lying in a strict closed subset (depending on $a$ and $b$). A theorem of Miyaoka proves this for smooth curves in minimal surfaces, with $a>3/2$. A conjecture of Vojta claims in essence that any constant $a>1$ is possible provided one restricts oneself to curves of bounded gonality. We show by explicit examples coming from the theory of Shimura varieties that in general, the constant $a$ has to be at least equal to the dimension of the ambient variety. We also prove the desired inequality in the case of compact Shimura varieties.

math.AG↗