Diophantine Geometry: towards the ultimate Bogomolov conjecture

Informazioni relative al progetto

DiophGeo

ID dell’accordo di sovvenzione: 101027237

DOI

10.3030/101027237

Progetto concluso il 31 Gennaio 2023

Data della firma CE 5 Marzo 2021

Data di avvio 1 Aprile 2021

Data di completamento 30 Settembre 2023

Finanziato da

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Costo totale

€ 207 312,00

Contributo UE

€ 207 312,00

Coordinato da

KOBENHAVNS UNIVERSITET
Denmark

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

Periodo di rendicontazione: 2021-04-01 al 2023-03-31

Sintesi del contesto e degli obiettivi generali del progetto

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.

Lavoro eseguito dall’inizio del progetto fino alla fine del periodo coperto dalla relazione e principali risultati finora ottenuti

(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.

Progressi oltre lo stato dell’arte e potenziale impatto previsto (incluso l’impatto socioeconomico e le implicazioni sociali più ampie del progetto fino ad ora)

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