Finite Flat Group Schemes
A group scheme G over a noetherian scheme S is finite and flat if and only if OG is a locally free OS-module of finite rank. The rank is a locally constant function on S, and is called the order of G. The order of a constant group scheme is equal to the order of the corresponding group, and in general, order behaves well with respect to base change and finite flat restriction of scalars.
Among the finite flat group schemes, the constants (cf. example above) form a special class, and over an algebraically closed field of characteristic zero, the category of finite groups is equivalent to the category of constant finite group schemes. Over bases with positive characteristic or more arithmetic structure, additional isomorphism types exist. For example, if 2 is invertible over the base, all group schemes of order 2 are constant, but over the 2-adic integers, μ2 is non-constant, because the special fiber isn't smooth. There exist sequences of highly ramified 2-adic rings over which the number of isomorphism types of group schemes of order 2 grows arbitrarily large. More detailed analysis of commutative finite flat group schemes over p-adic rings can be found in Raynaud's work on prolongations.
Commutative finite flat group schemes often occur in nature as subgroup schemes of abelian and semi-abelian varieties, and in positive or mixed characteristic, they can capture a lot of information about the ambient variety. For example, the p-torsion of an elliptic curve in characteristic zero is locally isomorphic to the constant elementary abelian group scheme of order p2, but over Fp, it is a finite flat group scheme of order p2 that has either p connected components (if the curve is ordinary) or one connected component (if the curve is supersingular). If we consider a family of elliptic curves, the p-torsion forms a finite flat group scheme over the parametrizing space, and the supersingular locus is where the fibers are connected. This merging of connected components can be studied in fine detail by passing from a modular scheme to a rigid analytic space, where supersingular points are replaced by discs of positive radius.
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