Etale group schemes and p-divisible groups

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  • “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.
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