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

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?

• It already makes sense to ask whether a field extension $K \to L$ (where $K = k(t)$ is the function field of $\mathbf G_m$) is tamely ramified at $0$ and $\infty$, so by the same logic it also makes sense to ask this for a finite morphism like $E[\ell] \to \mathbf G_m$. In both cases, it means you take the unique smooth projective model (of the function field $L$ or of the curve $E[\ell]$), which is the same thing as the relative integral closure of $\operatorname{Spec} L \to \mathbf P^1$ or of $E[\ell] \to \mathbf P^1$. Commented Dec 29, 2023 at 16:19
• @R.vanDobbendeBruyn: so just to clarify if I understood the philosophy correctly: The quoted formulation above "the map $E[\ell]\to \mathbb{G}_m$ has tame ramification at $0, \infty$" should be regarded as a kind of abusing of notation when one tacitly wants to reason about - let me say in usual algeom framing - ramification behavior of the assoc map between uniquely(!; therefore it's "legal") associated smooth projective models, right? Essentially what one does the whole time in algebraic number theory Commented Dec 31, 2023 at 3:22
• Right; it is the same use of 'unramified' as in unramified primes in a number field $\mathbf Q \hookrightarrow K$. The valuation-theoretic notion, not the geometric one, if you will (although they are closely related). Commented Dec 31, 2023 at 3:25

## 1 Answer

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 $$\mathbb{A}^1$$).

• regarding part two: so far I understand the idea correctly the aim is that $Z$ is even $\mathbb{G}_m$ and the etale base change map of the specific form $[n]$. But what I still not understand is why in order to reach this (ie triviality of $\ell$ torsion ) we have to show that there are infinitely many $\ell$ satisfying this condition on tame ramification at $0, \infty$? Eg, why doesn't it suffice to find a specific big, but single $\ell$ one? Commented Dec 31, 2023 at 3:44
• @user267839 You’re right, you just need to do this for a big enough $\ell$ (for any given $p$). I think his exposition was probably just a bit loose in this regard. Commented Dec 31, 2023 at 4:20
• Do you know a reference for this result to which Daniel Litt is refering to there? Namely, that if we indeed succeed to find such (big enough) $\ell$ with the distinguishing property that $E[\ell]\to \mathbb{G}_m$ beeing **tamely ramified** in $0$ and $\infty$ (...in the sense I learned now that it translates to demanding that the assoc map between compactrifications $\overline{E[\ell]} \to \mathbb{P}^1$ is unramified at these points...) - then the pullback of such $E[\ell]$ along $[n]$ on the base is isomorphic to constant sheaf $((\mathbb{Z}/ \ell)^2)_{\mathbb{G}_m}$? Commented Dec 31, 2023 at 17:32
• First, tamely ramified is not the same as unramified. The result for which you seek a reference is a direct consequence, using the theory of the fundamental group, of the fact that $E[\ell]$ is a finite etale group scheme with fiber $(Z/\ell)^2$ and hence its monodromy group is a subgroup of $GL_2(\ell)$. That it is a finite etale group scheme follows from the fact that it is a finite subgroup scheme of an elliptic curve, which is smooth. Commented Dec 31, 2023 at 23:24
• yes yes, sorry for confusion. In my last comment I intended indeed to write : "... assoc map between compactrifications $\overline{E[\ell]} \to \mathbb{P}^1$ is unramified everywhere, except at these points $0,\infty$ , where we demand it to be tamely ramified. " Commented Jan 1 at 16:56