# Equivalent definitions of elliptic curves over a scheme

I would like to compare the two definitions of elliptic curves over an arbitrary scheme.

Scholze: A morphism $$p: E \to S$$ of schemes with a section $$e: S \to E$$ such that $$p$$ is proper, flat, and all geometric fibers are elliptic curves (with zero section given by $$e$$).

(from The Langlands-Kottwitz approach for the modular curve)

Katz, Mazur: proper, smooth curve $$f:E \to S$$ with geometrically connected fibers all of genus one, given with a section $$0: S \to E$$.

(from Arithmetic Moduli of Elliptic Curves)

• Scholze => Katz, Mazur: Do we use the fact that a morphism that is flat, locally of finite presentation, and all fibers smooth, is smooth morphism? If so, how do we show that $$p$$ is locally of finite presentation?

• Katz, Mazur => Scholze: Fiber of $$f:E_s \to Spec(\kappa(s))$$ is a proper, smooth, geometrically connected curve of genus one with a section given by $$0$$.

1. This becomes projective because $$E_s$$ is an abelian variety, and proper abelian variety over a field $$k$$ is projective?

2. Why do we impose the condition geometrically connected? Isn't the fiber still an elliptic curve over a field with geometrically connected condition?

$$\newcommand{\Spec}{\mathrm{Spec}}$$ Scholze---> Katz-Mazur: I really wouldn't stress too much about this, to be honest. Probably Scholze should say that $$p$$ is locally of finite presentation and/or $$S$$ is locally Noetherian. Since the moduli spaces of such objects constructed is locally Noetherian, you really have no harm restricting to such a thing. Then, proper implies finite type and since S is locally Noetherian this implies that $$p$$ is locally of finite presentation. And then, yes, we use [Tag01V8][1] If it makes you feel any better, his ultimate goal with this paper, and subsequent ones (which, incidentally, my thesis is a generalization of one of these papers) is to work in the same realm as the work of Harris-Taylor. In Harris-Taylor's seminal book/paper where they prove local Langlands for $$\mathrm{GL}_n(F)$$ they explicitly restrict only the schemes which are locally Noetherian (as does Kottwitz, if I recall correctly, in his original paper "On the points of some Shimura varieties over finite fields).
Katz-Mazur ---> Scholze: A smooth proper connected curve over a field is automatically projective. We may assume we're over $$\overline{k}$$. Let $$X$$ be a smooth proper conneced curve. Let $$U$$ be an affine open subscheme. Then, by taking a projectivization of $$U$$ (i.e. locally closed immerse $$U$$ into some $$\mathbb{P}^n$$ and take closure) and normalizations you can find an $$X'$$ which is smooth and projective containing $$U$$. Then, you get a birational map $$X\dashrightarrow X'$$. One can then use the valuative criterion to deduce this is an isomorphism.
An elliptic curve is connected. Note then that if $$X/k$$ is finite type, connected, and $$X(k)\ne \varnothing$$ then $$X$$ is automatically geometrically connected. Since any idempotents in $$\mathcal{O}(X_{\overline{k}})$$ must show up at some finite extension, it suffices to show that $$X_L$$ is connected for every finite extension $$L/k$$. Note that since $$\Spec(L)\to \Spec(k)$$ is flat and finite then same is true for $$X_L\to X$$, and thus $$X_L\to X$$ is clopen. Thus, if $$C$$ is a connected component of $$X_L$$ it's clopen (since $$X_L$$ is Noetherian) and thus its image under $$X_L\to X$$ is clopen, and thus all of $$X$$. Suppose that there exists another connected component $$C'$$ of $$X_L$$. Then, by what we just said the image of $$C$$ and $$C'$$ both contain any $$x\in X(k)$$. Note though that if $$\pi:X_L\to X$$ is our projection, then $$\pi^{-1}(x)$$ can be identified set theoretically as $$\Spec(L\otimes_k k)=\Spec(L)$$ and co consists of one point. This means that $$C$$ and $$C'$$, since they both hit $$x$$, have an intersection point. This is a contradiction. So an elliptic curve, being connected and having $$E(k)\ne \varnothing$$, is automatically geometrically connected.
• Thank you for the prompt reply. For the part you said to "not stress too much about", do you mean the following? Since the representing scheme $\mathcal{M}_n$ of moduli space of elliptic curves with "naive" level-$N$-structure is a curve over $\mathrm{Spec}~\mathbb{Z}[1/N]$, the moduli space of elliptic curves (with level-$N$-structure) coincides whether you use Scholze's definition and Katz-Mazur's definition? – libofmath Apr 1 at 5:46
• @libofmath Yes, that $\mathcal{M}_n$ is locally Noetherian (not just $\mathrm{Spec}(\mathbb{Z}[1/N]$) so that the representing space for moduli of enhanced elliptic curves lives in the category of locally Noetherian schemes. – Alex Youcis Apr 1 at 5:48