Let $\pi: \mathcal A \to S$ be an abelian scheme, i.e. a proper, smooth group scheme over a scheme $S$ whose geometric fibers are connected and of dimension $g$. Is $\pi$ necessarily flat? What is the proof/reference?
Motivation: say we are given an Artinian local ring $R$ with the residue field $k$. The deformation theory is interested in seeking for flat liftings of varieties $X/k$ to $R$. However, when lifting abelian varieties (e.g. Serre-Tate theory) one doesn't mention flatness.
This question bothers me, since it is something I should already know by now...