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By Conrad Oort Chai

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We know that D(Z)[1/p] is free of rank 1 over K0 ⊗Qp F , so D(f ) acts as multiplication by some c ∈ K0 ⊗Qp F . Since D(f ) also commutes with the action of F that is semilinear over the absolute Frobenius σ of K0 , we have (σ ⊗ 1)(c) = c. This forces c ∈ F , as desired. We finish our summary of the theory of p-divisible groups by recording (for later reference) two important results. The first is a deep result of Tate. 6. Theorem (Tate). Let R be a complete discrete valuation ring with perfect residue field of characteristic p > 0 and fraction field M of characteristic 0.

A p-divisible group of height h 0 over a scheme S is a directed system G = (Gn )n 1 of commutative S-groups Gn such that: Gn is killed by pn , each Gn → S is finite and locally free, G1 → S has constant degree ph , and Gn is identified with Gn+1 [pn ] for all n 1. 4. 3. Example. Let R be a complete local noetherian ring with residue characteristic p > 0. For any finite flat commutative R-group scheme H, the connected component H 0 of the identity section is an open and closed R-subgroup scheme and the quotient H ´et := H/H 0 is finite ´etale; these properties can be seen via the special fiber.

38 1. 1 can be formulated with a general ground field K of characteristic 0, but the nature of the descent becomes a bit more subtle. Namely, if A is an abelian variety over a field K of characteristic 0 and if A admits sufficiently many complex multiplications, then there is a finite extension K /K such that AK descends (along with its entire endomorphism algebra) to an abelian variety over a number field contained in K . In this formulation it is crucial to introduce the finite extension K /K, even if we just wish to descend the abelian variety (and not any specific endomorphisms).

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