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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 1 - LowDegModCurve (Low Degree Points on Modular Curves)

Teaser

Summary

\"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.\"

Work performed

- Work Package 1.2: progress has been made in this package, and the researcher had several discussions with Jennifer Park (Ohio State University), an expert on this topic, about strengthening the currently existing methods. The researcher has also collaborated with Filip Najman (Zagreb) on a paper (accepted for publication in Mathematics Research Letters) where they determine all possible torsion subgroups of elliptic curves over quadratic fields.

- Works Packages 2.1, and 2.2 and 2.3: mentioned together, as they have been dealt with simultaneously. These packages have been fully completed with the production of a submitted paper with Netan Dogra (University of Oxford). This completion also required algorithms to finish the proof, available online.

Final results

\"The project has been terminated after 12 months due to the researcher obtaining a permanent position as a \"\"MaÃ®tre de ConfÃ©rences\"\" in Grenoble, France. Nevertheless, the progress has been the following: the researcher has proven with Netan Dogra that all the nonsplit Cartan modular curves (and another type of modular curve) are amenable to a refined version of the quadratic Chabauty method (which proves the finiteness of their rational points and allows in principle to determine them algorithmically, as opposed to Faltings\'s theorem or even Vojta\'s approach). This goes beyond what was expected of the full Research Package 2, as improved conditions have been found for quadratic Chabauty, and there is now hope for its general applicability without full knowledge of the Jacobian of the curve. Furthermore, the results do not rely anymore on the Birch and Swinnerton-Dyer conjecture, which seemed unavoidable at the time of writing the research project.\"