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H2020Individual fellowship2018–2020

LowDegModCurve · Low Degree Points on Modular Curves

Horizon 2020 — Marie Skłodowska-Curie Actions

Duration
2018-09-03 → 2020-09-02
EU contribution
€195,455
Participants
1
Scheme
MSCA-IF

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Results in brief

Low Degree Points on Modular Curves

"Number theory, and in particular diophantine equations (equations with integers), is a theme of mathematics almost as old as civilisation itself, as seen on Babylonian clay tablets from 1800 B.C. More recently, the study of so-called Galois representations of elliptic curves is at the heart of modern arithmetic geometry, and intimately related to the celebrated proof of Fermat's Last Theorem by Wiles. Our focus is on the possible images of a Galois representation: to each possibility we can associate a modular curve, which is a moduli space of elliptic curves with representation having that image. The central conjecture regarding these images, called Serre's uniformity conjecture, states that these modular curves have no nontrivial rational points when their genus is large enough, and can be split in three main cases. Two of those cases have been solved in the fundamental works of Mazur, Kamienny, Merel, Bilu, Parent and Rebolledo (Mazur's being crucial to the proof of Wiles), but there was a persistent obstruction to the last one, ""non-split Cartan"". In this project, we study low degree points on interesting modular curves (or even rational points where this type of obstruction holds), by developing and extending powerful methods including an overdetermined version of Chabauty's method in the symmetric power setting, and a recent breakthrough called ""quadratic Chabauty method"" for the non-split Cartan modular curves. Any advance towards the resolution of Serre's uniformity conjecture is expected to have applications in multiple fields ranging from cryptography to arithmetic geometry. Moreover, the tools developed on the way should shed a new light on the general problem of explicitly determining rational points on curves of large genus."

Data: CORDIS, © European Union

Project objective

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.

Original text from CORDIS.

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Data: CORDIS, © European Union