Diophantine Geometry: towards the ultimate Bogomolov conjecture

Información del proyecto

DiophGeo

Identificador del acuerdo de subvención: 101027237

DOI

10.3030/101027237

Proyecto finalizado el 31 Enero 2023

Fecha de la firma de la CE 5 Marzo 2021

Fecha de inicio 1 Abril 2021

Fecha de finalización 30 Septiembre 2023

Financiado con arreglo a

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Coste total

€ 207 312,00

Aportación de la UE

€ 207 312,00

Coordinado por

KOBENHAVNS UNIVERSITET
Denmark

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

Período documentado: 2021-04-01 hasta 2023-03-31

Resumen del contexto y de los objetivos generales del proyecto

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.

Trabajo realizado desde el comienzo del proyecto hasta el final del período abarcado por el informe y los principales resultados hasta la fecha

(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.

Avances que van más allá del estado de la técnica e impacto potencial esperado (incluida la repercusión socioeconómica y las implicaciones sociales más amplias del proyecto hasta la fecha)

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