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Let $E \to \mathbb{G}_m/k$ an elliptic curve over $ \mathbb{G}_m$ ($k$ field of char $p>0$) and $E[\ell]$ for $(\ell,p)=1$ the $\ell$-torsion group.

Let $f:T \to \mathbb{G}_m$ an finite etale covering.

I'm looking for reference dicussing criteria $f$ beeing trivializing cover for $\ell$- torsion (preferably ) encoded in data from etale fundamental group, ie when the $\ell$- torsion $E'[\ell]$ of elliptic curve $E'$ obtained as pullback along $f:T \to \mathbb{G}_m$ is isomorphic as group scheme over $T$ to constant group scheme $((\mathbb{Z}/\ell)^2)_T$?

More specifically motivated by Daniel Litt's answer (see the "algebraic version") I'm primary interested in such criteria for finite etale covers of the specific shape $[n]:\mathbb{G}_m \to \mathbb{G}_m $ ($(n,p)=1$).

For example, there seems to exist a criterion in case of $p \neq 2,3$ (compare with Noam Elkies' example) telling that if $E[\ell] \to \mathbb{G}_m$ has tame ramification only in $0, \infty$ (recall that the latter means if I understood it correctly that the induced map $\overline{E[\ell]} \to \mathbb{P}^1$ between smooth proper models of these curves is ramified only in $0, \infty$ in usual sense; note this information on ramification can be also encoded as inertia groups of fundamental group without "passing"explicitly to smooth proper models),
then the $\ell$-torsion trivializes under pullback along $[n]$ in above sense.
That looks pretty surprising at first glance since seemingly there is non dependence on $n$, except beeing prime to characteristic (=$[n]$ etale)

But I nowhere found a proof/ discussion of this result, or similar results treating this question on trivializating coverings of $\ell$-torsion based on data extractible from etale fundamental group.

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    $\begingroup$ If $\overline{E[\ell]}\to\mathbf P^1$ is tamely ramified at $\infty$, then the action of $\pi_1(\mathbf G_m)$ factors over the tame quotient $\pi_1^t(\mathbf G_m)$, which by SGA 1, Exp. XII, Cor. 2.12 is isomorphic to $\hat{\mathbf Z}^{(p')}$, the maximal pro-prime-to-$p$ quotient of $\hat{\mathbf Z}$. Explicitly, the tame fundamental group is the limit of the Galois groups of the covers $[n] \colon \mathbf G_m \to \mathbf G_m$ for $\gcd(n,p)=1$. So factoring over $\pi_1^t(\mathbf G_m)$ means exactly that it trivialises after pulling back along $[n]$ for some $n \in\mathbf N$ prime to $p$. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jan 1 at 21:40
  • $\begingroup$ @R.vanDobbendeBruyn I think you mean exposé XIII $\endgroup$
    – Will Chen
    Commented Jan 2 at 1:38
  • $\begingroup$ @WillChen whoops, you're right! $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jan 2 at 17:11
  • $\begingroup$ @R.vanDobbendeBruyn: just to clarify the final step in the argument: to show that the pullback cover along some $[n]$ trivialize, we have to show that the pulled back monodromy action $[n]^*\pi_1(\mathbb{G}_m)$ is trivial, but by your agument above factoring trought $\pi_1^t(\mathbf G_m)$ implies factoring trought some Galois group for some $[n] \colon \mathbf G_m \to \mathbf G_m$ which is isomorphic to $\pi_1^t(\mathbf G_m)/ [n]^*\pi_1^t(\mathbf G_m)$, and so the pulled back action along this $[n]$ is killed implying triviality of pullback cover $\endgroup$
    – user267839
    Commented Jan 2 at 18:31
  • $\begingroup$ I don't know what $[n]^*\pi_1(\mathbf G_m)$ means. At any rate, finite étale covers of a curve $C$ (resp. finite étale covers that are tame at $\bar C\setminus C$) correspond to finite sets with a continuous action of $\pi_1(C)$ (resp. $\pi_1^t(C)$). A profinite group $G$ acting continuously on a finite set means that the action factors through $G/H$ for some open (= closed of finite index) subgroup $H\subseteq G$. These $H$ are of the form $\pi_1(C')$ (resp. $\pi_1^t(C')$) for some (tame) finite étale cover $C'\to C$. For $\mathbf G_m$, they are given by $[n]\colon\mathbf G_m\to\mathbf G_m$. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jan 3 at 0:26

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