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

allelliptic 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.

4more comments