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By Conrad Oort Chai

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1, as follows. 1, we see that Zp ⊗Z End(A) acts faithfully on A[p∞ ]. Hence, Qp ⊗Q End0 (A) acts faithfully on A[p∞ ] in the isogeny category of p-divisible groups over K. , finite) and A is an isotypic CM 32 1. 2) then Lp := Qp ⊗Q L acts faithfully and linearly on the vector space D(A[p∞ ])[1/p] of rank 2g over the absolutely unramified p-adic field W (K)[1/p]. 3) with lifting problems from positive characteristic to characteristic 0. (Note that when char(K) = 0, H1dR (A/K) provides essentially the same information as the CM type arising from the L-action on Lie(A) = H0 (A, Ω1A/K )∨ , in view of the Hodge filtration on H1dR (A/K); cf.

The following remarkable result relates Weil q-numbers to isogeny classes of simple abelian varieties over a finite field of size q. 1. Theorem (Honda–Tate). Let κ be a finite field of size q. The assignment A → πA defines a bijection from the set of isogeny classes of simple abelian varieties over κ to the set of Gal(Q/Q)-conjugacy classes of Weil q-numbers. 1. 1). We are not aware of a proof which avoids using abelian varieties in characteristic 0. We refer the reader to [33], [77], and [60] for a discussion of the proof of the Honda–Tate theorem.

By hypothesis the endomorphism [pN ]Γ = [p]N Γ of Γ induces an injective endomorphism of the coordinate ring, so f = 0. Now consider p-divisible groups over a field K of characteristic p > 0. Assuming that K is perfect, for any p-divisible group G = (Gn )n 1 over K with height h 1 we let D(G) denote the DK -module lim D(Gn ). By the same style of arguments used to work out the Z -module ←− structure of Tate modules of abelian varieties in characteristic = (resting on knowledge of the orders of the -power torsion subgroups), we use W -length to replace counting to infer that D(G) is a free W -module of rank h and D(G)/pn D(G) → D(Gn ) is an isomorphism for all n 1.

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