Etale group schemes and p-divisible groups

“Etale groups schemes are entirely determined by their geometric points.”
The basic idea for deformation theory is to understand when an object \(X/S\) for \(S\) a scheme lifts to an infinitesimal neighborhood of \(S\).
- Example: When does an elliptic curve \(E/{\mathbf{Z}}_p\) lift to an elliptic scheme \(\mathcal E / {\mathbf{Z}}_p\)?
“Deformation theory for ’blah’s is the study of smoothness for the moduli space of ’blah’s.”
“The deformation theory of an abelian scheme coincides with the deformation theory of its \(p\)-divisible group.”
\(p{\hbox{-}}\)divisible groups are systems (of finite, flat group schemes) in which each piece is the torsion of the following piece.