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

Davide Lombardo and Sebastiano Tronto

Vol. 320 (2022), No. 1, 133–175
Abstract

We give explicit uniform bounds for several quantities relevant to the study of Galois representations attached to elliptic curves E. 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().

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
Authors
Davide Lombardo
Dipartimento di Matematica
Università di Pisa
Pisa
Italy
Sebastiano Tronto
Department of Mathematics
University of Luxembourg
Esch-sur-Alzette
Luxembourg