Explanipedia

A Formalization of Divided Powers in Lean Open

Antoine Chambert-Loir, María Inés de Frutos-Fernández · 2025

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!}\}_…

Balade newtonienne entre analyse et arithmétique Open

Antoine Chambert-Loir · 2024

Inventés par Kurt Hensel à la toute fin du xix e siècle sur le modèle des séries en une indéterminée, les nombres -adiques sont devenus non seulement un outil indispensable de l’arithmétique contemporaine, mais un sujet d’étude en soi. Dan…

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

Antoine Chambert-Loir · 2023

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 tex…

Potentiel et rationalité Open

Antoine Chambert-Loir, Camille Noûs · 2023

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…

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

Antoine Chambert-Loir · 2023

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 prov…

Burnside rings and volume forms with logarithmic poles Open

Antoine Chambert-Loir, Maxim Kontsevich, Yuri Tschinkel · 2023

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.…

Les conjectures de Weil : origines, approches, généralisations Open

Antoine Chambert-Loir · 2022

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, gene…

La conjecture de Mordell: origines, approches, généralisations Open

Antoine Chambert-Loir · 2021

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. Th…

Chapter VII: Arakelov Geometry, Heights, Equidistribution, and the Bogomolov Conjecture Open

Antoine Chambert-Loir · 2020

Le théorème de réduction stable de Deligne et Mumford Open

Antoine Chambert-Loir · 2019

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 com…

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

Antoine Chambert-Loir · 2018

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 enumerat…

A nonarchimedean Ax–Lindemanntheorem Open

Antoine Chambert-Loir, François Loeser · 2017

Motivated by the André–Oort conjecture, Pila has proved an analogue of the Ax–Lindemann theorem for the uniformization of classical modular curves. In this paper, we establish a similar theorem in nonarchimedean geometry. Precisely, we giv…

A. Chambert-Loir - Equidistribution theorems in Arakelov geometry and Bogomolov conjecture (part4) Open

Antoine Chambert-Loir, Jérémy Magnien · 2017

Let X be an algebraic curve of genus g⩾2 embedded in its Jacobian variety J. The Manin-Mumford conjecture (proved by Raynaud) asserts that X contains only finitely many points of finite order. When X is defined over a number field, Bogomol…

Motivic height zeta functions Open

Antoine Chambert-Loir, François Loeser · 2016

We consider a motivic analogue of the height zeta function for integral points of equivariant partial compactifications of affine spaces. We establish its rationality and determine its largest pole.

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