Construction refuting the existence of nonisotrivial elliptic curve over $\mathbb{G}_m$

Question

I have some troubles to understand the construction in detail presented here by Daniel Litt used to show that there cannot exist an elliptic curve over $\mathbb{G}_m/k$, $k$ of characteristic $p >3$.

The construction works as follows:

Suppose $E\to \mathbb{G}_m/k$ is an elliptic curve. If we can find some $n$ so that the pullback of $E$ along $\mathbb{G}_m\overset{[n]}\longrightarrow \mathbb{G}_m$ has trivial $\ell$-torsion, with $(\ell, p)=1$ and $\ell \gg 0$, we're done, because choosing a trivialization we get a map from $\mathbb{G}_m$ to a high genus modular curve, which must be constant as $\mathbb{G}_m$ is rational.

To do this, we must show that for infinitely many $\ell$, the map $E[\ell]\to \mathbb{G}_m$ has tame ramification at $0, \infty$. It suffices to find $\ell$ so that $GL_2(\mathbb{Z}/\ell\mathbb{Z})$ has order prime to $p$. But this order is $(\ell^2-1)(\ell^2-\ell)=\ell(\ell-1)^2(\ell+1).$ But by Dirichlet's theorem on primes in arithmetic progressions there exist an infinite number of primes satisfying this and we win.

Questions:
(1) Why, if we succeed in finding such a $n$ with the property that the pullback ell curve - let us call it $E'/ \mathbb{G}_m$ - of putative $E\to \mathbb{G}_m/k$ along this multiplication map $[n]: \mathbb{G}_m \to \mathbb{G}_m$ has trivial $\ell$- torsion for big enough $\ell$ satisfying $(\ell, p)=1$, then we get a map from $\mathbb{G}_m $ to certain higher genus modular curve after choosing a trivialization?

Firstly, what in this context concretely means "to choose a trivialization"? Trivializing what?
Secondly, the modular curves $X(d)$ parametrize so far I know pairs $(E, N_d)$ where $E$ is elliptic curve, and $N_d$ certain $d$-dependent datum assoc to $E$, eg a point at the curve of order $d$ or a a subgroup of this order, plus some cusps parametrizing certain degenerated curves.

So, why in this context the condition that we require the pullback ell curve $E'/ \mathbb{G}_m$ to have trivial $\ell$-torsion assures that $\mathbb{G}_m$ is mapped to certain modular curve if we choose sometrivialization? Could the reason for this be elaborated in a bit more details?

(2) I not understand the formulation in second part that in order to find such $n$ it has to be shown that there exist infinitely many $\ell$'s such that the restricted map to torsion points $E[\ell] \to \mathbb{G}_m$ has tame ramification at $0, \infty$.

But $0, \infty \not \in \mathbb{G}_m$, so I not understand the meaning of this condition. Or is here probably meant the extension $E[\ell] \to \mathbb{P}^1$, otherwise I not understand the meaning of this statement. Could somebody elaborate what is meant by this requirement on the ramification behavior in these two points.

Especially, over which base we consider $E$ resp. $E[\ell]$ at that point?

Answer 1

For an elliptic curve over a scheme $T$, a trivialization of $E[n]$ is a choice of an isomorphism between $E[n]$ and the constant group scheme $(\mathbb{Z}/n)^2$ over $T$ (such a trivialization is also called a level $n$ structure). To answer your first question, one should think of modular curves as moduli stacks. By the definition of the moduli stack $M(n)$ of elliptic curves with level $n$ structure, an elliptic curve over a scheme $T$ with level $n$ structure is the same as a map $T\rightarrow M(n)$. It's known that $M(n)$ is a disjoint union of (isomorphic) curves, and that their common genus grows (unboundedly) with $n$. Thus for large $n$, there cannot exist an elliptic curve over $\mathbb{G}_m$ with trivial $n$-torsion.

For your second question, the point is that $E[\ell]$ is always etale over $\mathbb{G}_m$, and hence $E[\ell]$ can always be trivialized after base changing via a finite etale map $Z\rightarrow \mathbb{G}_m$. This $Z$ is just a curve, whose compactification is a covering of $\mathbb{P}^1$ ramified above $0,\infty$. However, for his purposes he wants to show that $Z$ can be chosen to have genus 0. This is certainly true in characteristic 0 (any branched cover of $\mathbb{P}^1$ with 2 branch points has genus 0), but it is also true under the assumption of tame ramification. Here the tameness requirement is necessary to rule out the possibility of stuff like Artin-Schreier covers, which can give you high-genus etale covers of $\mathbb{G}_m$ (even $\mathbb{A}^1$).