Universal homotheties for elliptic curves

Question

Let $K$ be a number field and $E_1, \cdots, E_n$ elliptic curves over $K$. Let $\ell$ be a prime. Then there exists an element $\sigma \in \text{Gal}(\overline{K}/K)$ such that $\sigma$ acts on $T_\ell(E_i)$ via a non-root-of-unity scalar for all $i$. (Here $T_\ell(E_i)$ denotes the $\ell$-adic Tate module of $E_i$.)

This is a result of Bogomolov, who shows that for any Abelian variety $A/K$, there exists an element of $\text{Gal}(\overline{K}/K)$ which acts on $T_\ell(A)$ via a non-root-of-unity scalar; applying this result to $\prod_i E_i$ gives the claim.

I would like to know if an analogous statement is true for infinite sets of elliptic curves. In particular:

Fix a prime $\ell$. Does there exist an element $\sigma\in \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ such that for all elliptic curves $E/\mathbb{Q}$, $\sigma$ acts on $T_\ell(E)$ via a non-root-of-unity scalar?

I am also interested in the following stronger variant:

Fix a prime $\ell$. Does there exist an element $\sigma\in \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ such that for all number fields $K\subset \overline{\mathbb{Q}}$ and all elliptic curves $E/K$, there exists an integer $N$ such that $\sigma^N$ acts via a non-root-of-unity scalar on $T_\ell(E)$?

Note that the above makes sense because for $N$ sufficiently divisible, $\sigma^N\in \text{Gal}(\overline{\mathbb{Q}}/K)$.

Of course this question is related to uniform boundedness conjectures, but my hope is that it can be answered independently from them.

Answer 1

Fix the prime $\ell$. I hope: There exists a constant $C$ such that for any elliptic curve $E/\mathbb{Q}$ the image of the Galois representation $\rho\colon \operatorname{Gal} ( \bar{\mathbb{Q}}/\mathbb{Q}) \to \operatorname{GL}_2(\mathbb{Z}_{\ell})$ contains $\bigl\{x^C \bigm\vert x\in \mathbb{Z}_{\ell}^{\times}\bigr\}$.

Carola Eckstein in her thesis http://cds.cern.ch/record/897530/files/cer-002567978.pdf answers this partially. If $E$ has complex multiplication then $C=54$ will do. If $\ell>163$ then $C=12$ is ok. Agnès David https://arxiv.org/abs/1007.4725 makes further progress and shows that for $\ell>23$ and different from $37$, $43$, $67$ and $163$ then $C=2$ works. I don't know more about the question for smaller $\ell$, but as ulrich hints in his comments above, one should be able to prove the existence of $C$.

Suppose there is such a $C$. Given an elliptic curve $E/\mathbb{Q}$, let $K_E$ be the field fixed by the image of $\operatorname{im}(\rho)\to \operatorname{PGL}_2(\mathbb{Z}_\ell)$ (as in ulrich's comment). Set $K_\infty = \mathbb{Q}(\mu_{\ell^{\infty}})$. The image of the homotheties under $\det:\operatorname{im}(\rho) \to \operatorname{Gal}(K_\infty/\mathbb{Q})\cong \mathbb{Z}_{\ell}^{\times}$ will contain all $2C$-th powers. Therefore there is a finite extension $F/\mathbb{Q}$ inside $K_\infty$ such that $K_E\cap K_\infty\subset F$ for all elliptic curves $E$.

In particular the compositum $\mathcal{K}$ of any (or all) $K_E$ intersects $K_\infty$ in $F$ and hence any element of $\operatorname{Gal}(\bar{\mathbb{Q}}/\mathcal{K})$ that maps to an element $\sigma$ of infinite order in $\operatorname{Gal}(K_\infty/F)$ will work.

The reference above will also show that this works for many $\ell$ over any fixed number field instead of $\mathbb{Q}$.

I have no answer for the second question, yet.