Diophantine Geometry: towards the ultimate Bogomolov conjecture

Projektinformationen

DiophGeo

ID Finanzhilfevereinbarung: 101027237

DOI

10.3030/101027237

Das Projekt endete am 31 Januar 2023

EK-Unterschriftsdatum 5 März 2021

Startdatum 1 April 2021

Enddatum 30 September 2023

Finanziert unter

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Gesamtkosten

€ 207 312,00

EU-Beitrag

€ 207 312,00

Koordiniert durch

KOBENHAVNS UNIVERSITET
Denmark

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

Berichtszeitraum: 2021-04-01 bis 2023-03-31

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

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.

Arbeit, die ab Beginn des Projekts bis zum Ende des durch den Bericht erfassten Berichtszeitraums geleistet wurde, und die wichtigsten bis dahin erzielten Ergebnisse

(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.

Fortschritte, die über den aktuellen Stand der Technik hinausgehen und voraussichtliche potenzielle Auswirkungen (einschließlich der bis dato erzielten sozioökonomischen Auswirkungen und weiter gefassten gesellschaftlichen Auswirkungen des Projekts)

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