Abelian scheme

Remarks

For each point $s \in S$, the fibre $A_s$ is an abelian variety over the residue field $\kappa(s)$. An abelian scheme over $\operatorname{Spec} k$, for a field $k$, is the same as an abelian variety over $k$. The group law of an abelian scheme is commutative.

Example: let $S = \operatorname{Spec} \Z[1/2]$ and let $E \subseteq \mathbb{P}^2_S$ be the closure of $y^2 = x^3 - x$, with unit section the point at infinity. The discriminant of this Weierstrass equation is $64 = 2^6$, a unit on $S$. So every fibre is a smooth curve of genus $1$, and $E$ is an abelian scheme of relative dimension $1$.

Non-example: the same equation over $\operatorname{Spec} \Z$ fails clause 1. Modulo $2$, the curve $y^2 = x^3 + x$ is singular at the point $(1, 0)$, where both partial derivatives $2y$ and $3x^2 + 1$ vanish.