The authors give explicit uniform bounds for several quantities relevant to the study of Galois representations attached to elliptic curves \(E\) over \(\mathbb Q\). Such types of bounds were studied before by the present authors and many others (including Greenberg, Lawson and Wuthrich, Mazur, Serre, Zywina).
For any prime \(l\), let \(G_{l^{\infty}}\) denote the image of the \(l\)-adic Galois representation attached to \(E\), and by \(G_{\infty}\) the image of the adelic representation. They prove:

(i)

a uniform upper bound for the index \([\mathbb Z_{l^{\times}} : \mathbb Z_{l^{\times}} \cap G_{l^{\infty}}]\) (Theorem 3.16);

(ii)

a uniform upper bound on the exponent of the cohomology group \(H^1(G_{\infty},T)\), for any Galois submodule \(T\) of \(E_{tors}\) (Theorem 4.9);

(iii)

a uniform lower bound for the closed \(\mathbb Z_l\)-subalgebra \(\mathbb Z_l[G_{l^{\infty}}]\) of \(\text{Mat}_{2\times 2}(\mathbb Z_l)\) generated by \(G_{l^{\infty}}\) (Theorem 5.8);

(iv)

a uniform lower bound on the degrees of the relative “Kummer extensions” (Theorem 6.5). This part was their original motivation for the present work; a similar result had already shown in their paper [D. Lombardo and S. Tronto, Int. Math. Res. Not. 2022, No. 22, 17662–17712 (2022; Zbl 1514.11036)].

In the last section, the authors give examples showing that most of their results are sharp or close to being sharp. They also point out that most of their results can be extended to number fields having at least one real place, at least if one is ready to believe the generalised Riemann hypothesis.