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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?

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$\newcommand{\Spec}{\mathrm{Spec}}$ 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.

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  • $\begingroup$ 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? $\endgroup$ – libofmath Apr 1 at 5:46
  • $\begingroup$ @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. $\endgroup$ – Alex Youcis Apr 1 at 5:48

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