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?