Power-integral points on elliptic curves

Informations projet

PIP

N° de convention de subvention: 235210

Projet clôturé

Date de début 1 Juillet 2009

Date de fin 30 Juin 2011

Financé au titre de

• Specific programme "People" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013)

Coût total Aucune donnée

Contribution de l’UE € 151 863,51

Coordonné par

UNIVERSITEIT UTRECHT
Pays-Bas

Objectif

The study of Diophantine equations is one of the oldest branches of pure mathematics. The 20th century saw Siegel’s theorem about the finitness of integral points on elliptic curves, the negative solution of Hilbert’s 10th problem, Faltings' Theorem and the proof of Fermat’s Last Theorem. In the 21st century much work has already been done to resolve extensions of Hilbert’s 10th problem and to make Faltings' theorem effective. Moreover, solving generalized Fermat equations has, for example, led to all of the perfect powers in the Fibonacci sequence being found, an unsolved problem for over 50 years. In 2006 the applicant proved that for each positive integer larger than 2, there corresponds a finite set of rational points on an elliptic curve which contains the integral points. The points in these finite sets have an important number theoretic structure and are called power-integral points. However, the proof given by the applicant uses Faltings' theorem and so gives no way to find them. Remarkably, in many cases the power-integral points can be found by solving generalized Fermat equations and by finding the perfect powers in an elliptic divisibility sequence. An elliptic divisibility sequence is in many ways an analogue of the Fibonacci sequence and its properties are receiving a lot of attention due to links with extensions of Hilbert’s 10th problem and Cryptography. It is believed that the study of these sequences combined with advances in solving Diophantine equations will achieve the objectives of this proposal. These are: to find all of the power-integral points on families of elliptic curves, and to give a quantitative bound for the number of power-integral points on an arbitrary elliptic curve.

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• sciences naturelles › mathématiques › mathématiques pures

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Mots‑clés

• Diophantine
• Number
• curves
• elliptic
• equations
• theory

Programme(s)

• FP7-PEOPLE - Specific programme "People" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013)

Thème(s)

• FP7-PEOPLE-IEF-2008 - Marie Curie Action: "Intra-European Fellowships for Career Development"

Appel à propositions

FP7-PEOPLE-IEF-2008
Voir d’autres projets de cet appel

Régime de financement

MC-IEF -

Coordinateur

UNIVERSITEIT UTRECHT

• Contribution de l’UE: € 151 863,51
• Adresse: HEIDELBERGLAAN 8 ; 3584 CS Utrecht ; Pays-Bas
• Type d’activité: Établissements d’enseignement supérieur ou secondaire
• Coût total: Aucune donnée