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Uniform bounds on the image of the arboreal Galois representations attached to non-CM elliptic curves. (English) Zbl 1464.11055

Summary: Let \(\ell\) be a prime number, let \(F\) be a number field, and let \(E/F\) be a non-CM elliptic curve with a point \(\alpha\in E(F)\) of infinite order. Attached to the pair \((E,\alpha)\) is the \(\ell\)-adic arboreal Galois representation \(\omega_{E,\alpha,\ell^{\infty}}:\text{Gal}(\overline{F}/F)\to \mathbb{Z}_{\ell }^2\rtimes\text{GL}_2(\mathbb{Z}_{\ell})\) describing the action of \(\text{Gal}(\overline{F}/F)\) on points \(\beta_n\) so that \(\ell^n \beta_n=\alpha\). We give an explicit bound on the index of the image of \(\omega_{E,\alpha,\ell^{\infty}}\) depending on how \(\ell\)-divisible the point \(\alpha\) is, and the image of the ordinary \(\ell\)-adic Galois representation. The image of \(\omega_{E,\alpha,\ell^{\infty}}\) is connected with the density of primes \(\mathfrak{p}\) for which \(\alpha\in E(\mathbb{F}_{\mathfrak{p}})\) has order coprime to \(\ell\).

MSC:

11F80 Galois representations
11G05 Elliptic curves over global fields
12G05 Galois cohomology
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