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

TL;DR

This paper provides explicit uniform bounds for key Galois-theoretic quantities associated with elliptic curves over the rationals, enhancing understanding of their Galois representations and related extensions.

Contribution

It introduces new explicit bounds for Galois image subgroups, cohomology groups, and Kummer extensions for elliptic curves over ield.

Findings

Explicit bounds for the scalar subgroup in Galois images

Bounds for the first Galois cohomology group with torsion coefficients

Limits on Kummer extensions generated by rational points

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 $E(\mathbb Q)$.