Th\'eor\`eme des p\'eriodes et degr\'es minimaux d'isog\'enies

TL;DR

This paper improves the period theorem for elliptic curves, providing explicit bounds and a new formulation involving all archimedean places, leading to enhanced bounds on elliptic isogenies and applications to Serre's uniformity problem.

Contribution

It offers a sharpened, explicit version of the period theorem and new bounds for elliptic isogenies, extending previous results and applications.

Findings

A new explicit period theorem for elliptic curves.

Improved bounds for elliptic isogenies over number fields.

Application to Serre's uniformity problem in the split Cartan case.

Abstract

We give a new, sharpened version of the period theorem of Masser and W\"ustholz, which is moreover totally explicit. We also present a new formulation involving all archimedean places. We then derive new bounds for elliptic isogenies, improving those of Pellarin. The small numerical constants obtained allow an application to Serre's uniformity problem in the split Cartan case, thanks to the work of Bilu, Parent and Rebolledo.