Diophantine Geometry: towards the ultimate Bogomolov conjecture

Informations projet

DiophGeo

N° de convention de subvention: 101027237

DOI

10.3030/101027237

Projet clôturé le 31 Janvier 2023

Date de signature de la CE 5 Mars 2021

Date de début 1 Avril 2021

Date de fin 30 Septembre 2023

Financé au titre de

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Coût total

€ 207 312,00

Contribution de l’UE

€ 207 312,00

Coordonné par

KOBENHAVNS UNIVERSITET
Denmark

Periodic Reporting for period 1 - DiophGeo (Diophantine Geometry: towards the ultimate Bogomolov conjecture)

Période du rapport: 2021-04-01 au 2023-03-31

Résumé du contexte et des objectifs généraux du projet

Diophantine equations are equations in which solely addition, subtraction, multiplication, whole numbers, and a finite number of unknowns appear. Since antiquity, mathematicians have tried to determine the solutions of such equations in whole numbers or integers. The results of the project (proofs of the uniform Bogomolov and Mordell-Lang conjecture) have provided us with much better "geometric" tools for this quest, though many open problems remain in this long-running cultural endeavor of mankind spanning several centuries.

Travail effectué depuis le début du projet jusqu’à la fin de la période considérée dans le rapport et principaux résultats atteints jusqu’à présent

(1) The main results of the project have been strong uniform results of Manin-Mumford, Bogomolov, and Mordell-Lang type. As yet, the results have been submitted and are available as preprints.
(2) The extension to a relative Bogomolov conjecture has been explored, but only proven for fibered products of families of elliptic curves. The results are published in Journal für Reine und Angewandte Mathematik.
(3) In addition, the Pink-Zilber conjecture on unlikely intersections for curves in semiabelian varieties has been proven and the proof published in a research article in Selecta Mathematica. As a continuation, new ideas for its proof beyond the case of curves have been obtained.
(4) Finally, the Investigator taught 2 student seminars, 1 lecture, supervised a master student, and co-supervised a PhD student at KU, improving his track record in teaching.

Progrès au-delà de l’état des connaissances et impact potentiel prévu (y compris l’impact socio-économique et les conséquences sociétales plus larges du projet jusqu’à présent)

The work on uniformity in diophantine geometry conducted during this project has led to long-expected, but previously unproven results that will shape future research in diophantine geometry and related areas (e.g. arithmetic dynamics). Though this will shape future research in the field, no wider societal implications are expected.

a (happy) family of abelian varieties