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 $0$.
This becomes projective because $E_s$ is an abelian variety, and proper abelian variety over a field $k$ is projective?
Why do we impose the condition geometrically connected? Isn't the fiber still an elliptic curve over a field with geometrically connected condition?