Diophantine Geometry: towards the ultimate Bogomolov conjecture

Informacje na temat projektu

DiophGeo

Identyfikator umowy o grant: 101027237

DOI

10.3030/101027237

Realizacja projektu zakończyła się 31 Stycznia 2023

Data podpisania przez KE 5 Marca 2021

Data rozpoczęcia 1 Kwietnia 2021

Data zakończenia 30 Września 2023

Finansowanie w ramach

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Koszt całkowity

€ 207 312,00

Wkład UE

€ 207 312,00

Koordynowany przez

KOBENHAVNS UNIVERSITET
Denmark

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

Okres sprawozdawczy: 2021-04-01 do 2023-03-31

Podsumowanie kontekstu i ogólnych celów projektu

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.

Prace wykonane od początku projektu do końca okresu sprawozdawczego oraz najważniejsze dotychczasowe rezultaty

(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.

Innowacyjność oraz oczekiwany potencjalny wpływ (w tym dotychczasowe znaczenie społeczno-gospodarcze i szersze implikacje społeczne projektu)

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