Objectif The study of Galois representations of elliptic curves is at the heart of modern arithmetic geometry, and intimately related to modularity theorems and the proof of Fermat's Last Theorem. Galois representations of elliptic curves are classified by their images. Associated to a possible image is a modular curve which is a moduli space of elliptic curves with representation having that image. The study of rational and low degree points on modular curves underlies the celebrated theorems of Mazur, Kamienny, Merel, Bilu, Parent and Rebolledo. A common theme in all these works is the existence of a rank zero quotient of the modular Jacobian, and the validity of a formal immersion criterion. In this project, motivated by Serre's uniformity conjecture, we study rational and low degree points on interesting modular curves where these conditions fail, developing and extending powerful methods including an overdetermined version of Chabauty in the symmetric power setting, and quadratic Chabauty for the non-split Cartan modular curves.The University of Warwick has a strong and active number theory group, making it a natural host for the project. The Supervisor, Professor Siksek, is a leading expert on curves, Galois representations and modularity, with considerable experience in supervising research including 11 postdocs and 12 completed PhD students. The Researcher, Dr Le Fourn, did his PhD at Bordeaux (completed November 2015) with Professor Pierre Parent, including a 3 months internship at McGill with Professor Henri Darmon. Since September 2014 he has held the position of Agrégé préparateur at the École Normale Supérieure de Lyon. He has made excellent breakthroughs both in the theory of Q-curves, and in the arithmetic of Siegel modular varieties. The envisioned research will make the Researcher influential in modular curves and adjacent subjects, and allow him to realize his ambition of becoming an independent researcher at a leading European institution. Champ scientifique natural sciencescomputer and information sciencescomputer securitycryptographynatural sciencesmathematicspure mathematicsarithmeticsnatural sciencesmathematicspure mathematicsgeometry Mots‑clés Modular Curves Galois Representations Chabauty Quadratic Chabauty Rational Points Serre's Uniformity Programme(s) H2020-EU.1.3. - EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions Main Programme H2020-EU.1.3.2. - Nurturing excellence by means of cross-border and cross-sector mobility Thème(s) MSCA-IF-2017 - Individual Fellowships Appel à propositions H2020-MSCA-IF-2017 Voir d’autres projets de cet appel Régime de financement MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF) Coordinateur THE UNIVERSITY OF WARWICK Contribution nette de l'UE € 195 454,80 Adresse Kirby Corner Road - University House CV4 8UW Coventry Royaume-Uni Voir sur la carte Région West Midlands (England) West Midlands Coventry Type d’activité Higher or Secondary Education Establishments Liens Contacter l’organisation Opens in new window Site web Opens in new window Participation aux programmes de R&I de l'UE Opens in new window Réseau de collaboration HORIZON Opens in new window Coût total € 195 454,80
