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Diophantine Problems, o-Minimality, and Heights

Applicant Habegger Philipp
Number 165525
Funding scheme Project funding (Div. I-III)
Research institution Departement Mathematik und Informatik Universität Basel
Institution of higher education University of Basel - BS
Main discipline Mathematics
Start/End 01.04.2016 - 30.09.2019
Approved amount 379'404.00
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Keywords (14)

families of abelian varieties; diophantine geometry; CM points; Mordell-Lang Conjecture; integral points; model theory; heights; Zilber-Pink Conjecture; special points; o-minimal structures; unlikely intersections; isogenies; Faltings height; Colmez Conjecture

Lay Summary (German)

Lead
Diophantische Gleichungen lassen sich ausschliesslich mit den Grundrechenarten ausdrucken. Aufgrund ihrer einfachen Bauweise wurden sie bereits in der Antike untersucht. Die grundlegende Schwierigkeit liegt darin, dass ganzzahlige oder rationale Lösungen gesucht werden. Fermats Letzter Satz und die Mordell Vermutung sind prominente Beispiele und wurden von Wiles resp. Faltings gelöst. Im Zentrum dieses Projekts stehen Vermutungen über atypische Schnitte, die auf Arbeiten von Bombieri-Masser-Zannier, Pink und Zilber zurückgehen. Sie entsprechen einer grossen Klasse von diophantischen Gleichungen zusammen mit Verallgemeinerungen.
Lay summary
In der diophantischen Geometrie werden diophantische Gleichungen mittels Methoden der algebraischen Geometrie behandelt. Vermutung über atypische Schnitte beschäftigen sich mit klassischen Objekte der algebraischen Geometrie wie abelsche Varietäten, Modulkurven und Shimura Varietäten, die mit einer analytischen Überlagerungsabbildungen ausgestattet werden können. In den letzten Jahren hat sich hier die o-minimale Geometrie aus der mathematischen Logik als mächtiges Hilfsmittel etabliert. Sie stellt den richtigen Rahmen, um den analytischen Aspekt zu behandeln sowie den wichtigen Satz von Pila-Wilkie und seine Weiterentwicklungen. Die Brücke zwischen Arithmetik und Geometrie schlagen Höhenfunktionen, welche in diesem Projekt in verschiedenen Formen zum Einsatz kommen. Wir werden insgesamt vier Teil Projekte behandeln. Teile I) und II) dienen der übertragen der Mordell Vermutung  auf Familien abelscher Varietäten. In Teil III) werden Verallgemeinerungen der Vermutung atypischer Schnitte untersucht. Im vierten Teil beschäftigen wir uns mit Eigenschaften der Faltingshöhe einer abelschen Varietät.
Direct link to Lay Summary Last update: 30.03.2016

Responsible applicant and co-applicants

Employees

Publications

Publication
CM Relations Im Fibered Powers Of Elliptic Families
Barroero Fabrizio (2019), CM Relations Im Fibered Powers Of Elliptic Families, in Journal of the Institute of Mathematics of Jussieu, 18(5), 941-956.
Unlikely intersections in families of abelian varieties and the polynomial Pell equation
Barroero F., Capuano L. (2019), Unlikely intersections in families of abelian varieties and the polynomial Pell equation, in Proceedings of the London Mathematical Society, 120(2), 192-219.
Heights in families of abelian varieties and the Geometric Bogomolov Conjecture
Gao Ziyang, Habegger Philipp (2019), Heights in families of abelian varieties and the Geometric Bogomolov Conjecture, in Annals of Mathematics, 189(2), 527-527.
Unlikely intersections in products of families of elliptic curves and the multiplicative group
Barroero Fabrizio, Capuano Laura (2017), Unlikely intersections in products of families of elliptic curves and the multiplicative group, in The Quarterly Journal of Mathematics, 68(4), 1117-1138.
Generalized Vojta-Rémond Inequality
DillGabriel, Generalized Vojta-Rémond Inequality, in International Journal of Number Theory.
Unlikely Intersections between Isogeny Orbits and Curves
DillGabriel, Unlikely Intersections between Isogeny Orbits and Curves, in Journal of the European Mathematical Society.

Collaboration

Group / person Country
Types of collaboration
Ziyang Gao, CNRS, Paris, France France (Europe)
- Publication
Gaël Rémond, CNRS, Institut Fourier, Grenoble France (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Exchange of personnel

Scientific events

Active participation

Title Type of contribution Title of article or contribution Date Place Persons involved
Seminar of Geometry, Università Roma Tre Individual talk Unlikely intersections with isogeny orbits 14.03.2019 Rom, Italy Dill Gabriel;
Real Algebraic Geometry and Model Theory (RAGMT) Talk given at a conference Unlikely intersections between isogeny orbits and curves 08.10.2018 Konstanz, Germany Dill Gabriel;
Workshop on effectivity and ineffectivity for unlikely intersections Talk given at a conference Unlikely intersections in semi abelian varieties 24.07.2018 Manchester, Great Britain and Northern Ireland Barroero Fabrizio;
Around Functional Transcendence Talk given at a conference Variation of the Néron-Tate Height in a Family of Abelian Varieties 26.06.2018 Oxford, Great Britain and Northern Ireland Habegger Philipp;
CIRM Conference "Diophantine Geometry" Talk given at a conference Unlikely Intersections in families of abelian varieties and applications 21.05.2018 Luminy, France Barroero Fabrizio;
CIRM Conference "Diophantine Geometry" Talk given at a conference Unlikely intersections between isogeny orbits and curves 21.05.2018 Luminy, France Dill Gabriel;
Simons Symposium on Periods and L-values of Motives Talk given at a conference Variation of the Néron-Tate Height in a Family of Abelian Varieties and the Bogomolov Conjecture for Function Fields 29.04.2018 Elmau, Germany Habegger Philipp;
SFB-Seminar, Universität Regensburg Individual talk On the Bogomolov Conjecture over Function Fields 27.04.2018 Univ. Regensburg, Germany Habegger Philipp;
The fourth mini symposium of the Roman Number Theory Association Talk given at a conference Unlikely Intersections in families of abelian varieties 18.04.2018 Rom, Italy Barroero Fabrizio;
Seminari di Geometria del Politecnico di Torino Individual talk Unlikely intersections in families of abelian varieties 28.03.2018 Turin, Italy Barroero Fabrizio;
Séminaire de théorie des nombres, Institut Fourier, Grenoble Individual talk Intersections atypiques entre les orbites d’isogénie et les courbes 08.02.2018 Grenoble, France Dill Gabriel;
Joint Number Theory / Logic seminar, University of Oxford Individual talk Counting lattice points and O-minimal structures 06.02.2018 Oxford, Great Britain and Northern Ireland Barroero Fabrizio;
Séminaire de théorie des nombres, Institut Fourier, Grenoble Individual talk Unlikely intersections in families of abelian varieties 25.01.2018 Grenoble, France Barroero Fabrizio;
Algebra & Number Theory seminar, University of Freiburg Individual talk Unlikely intersections between isogeny orbits and curves 19.01.2018 Freiburg, Germany Dill Gabriel;
Workshop on Arithmetic and Complex Dynamics Talk given at a conference Unlikely Intersections on families of abelian varieties (Part I) 12.11.2017 Oaxaca, Mexico Barroero Fabrizio;
Specialization Problems in Diophantine Geometry Talk given at a conference On the Bogomolov Conjecture over Function Fields 09.07.2017 Cetraro, Italy Habegger Philipp;
Workshop on O-minimality and Diophantine Applications, Fields Institute Talk given at a conference Unlikely intersections in families of abelian varieties and some polynomial Diophantine equations 19.06.2017 Toronto, Canada Barroero Fabrizio;
Oberwolfach Workshop "O-Minimality and its Applications to Number Theory and Analysis" Talk given at a conference nlikely intersections in products of families of elliptic curves 30.04.2017 Oberwolfach, Germany Barroero Fabrizio;


Self-organised

Title Date Place
Topics in Rational and Integral Points 02.09.2019 Basel, Switzerland

Associated projects

Number Title Start Funding scheme
184623 Diophantine Equations: Special Points, Integrality, and Beyond 01.10.2019 Project funding (Div. I-III)
184623 Diophantine Equations: Special Points, Integrality, and Beyond 01.10.2019 Project funding (Div. I-III)

Abstract

The study of diophantine equations is one of the oldest parts of number theory. Its ultimate goal is to describe integer or rational solutions of polynomial equations. For polynomials in two variables, major results are: Siegel's Theorem on integral points on curves and Faltings' resolution of the Mordell Conjecture. They imply, under natural hypotheses, that a polynomial equation in two variables has only finitely many integral and rational solutions, respectively. In a greater number of variables, finiteness statements are known by work of Vojta, Faltings, and others in the context of the Mordell-Lang Conjecture when working inside an abelian variety. Roughly a decade ago, Zilber and Pink independently stated conjectures on unlikely intersections. Here points of arithmetic interest, e.g. those in the Mordell-Lang Conjecture, appear as intersections that are improbable for geometric reasons. Pink's version contains the André-Oort Conjecture on the distribution of CM or special points on certain moduli spaces. A strategy developed by Zannier has been one driving force towards these general conjectures. Central to this approach is a counting result by Pila and Wilkie in an o-minimal structure, a concept that originated in and has connections to model theory in mathematical logic. Part of this proposal aims to study new cases of the Zilber and Pink Conjecture by combining Vojta’s approach to the Mordell Conjecture with the counting strategy. We also plan to investigate unlikely intersections in non-reductive groups and shed new light on connections between Siegel's Theorem and CM points.
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