Some uniform bounds for elliptic curves over $\mathbb{Q}$

Davide Lombardo and Sebastiano Tronto

Vol. 320 (2022), No. 1, 133–175

DOI: 10.2140/pjm.2022.320.133

Abstract

We give explicit uniform bounds for several quantities relevant to the study of Galois representations attached to elliptic curves $E∕\mathbb{Q}$. We consider, in particular, the subgroup of scalars in the image of Galois, the first Galois cohomology group with values in the torsion of $E$, and the Kummer extensions generated by points of infinite order in $E(\mathbb{Q})$.

Keywords

Elliptic curves, Galois representations, Kummer theory, division points

Mathematical Subject Classification

Primary: 11F80, 11G05, 14K15

Milestones

Received: 7 July 2021

Revised: 11 May 2022

Accepted: 16 July 2022

Published: 16 October 2022