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4 COMMUTATIVE FORMAL GROUPS â EXERCISES (64) Show that the contravariant DieudonneÌ functor intertwines FG (resp. VG ) with the operator F (resp. V) on M (G). (65) Let M be the contravariant DieudonneÌ module of a p-divisible group G over a perfect field k of char p. Show that there canonical k-linear isomorphisms (LieG)â â¼ = M/FM (p) (which respects V) â â¼ (p) and LieG = M /VM (which respects F). (66) Using DieudonneÌ theory, show that there exists one-dimensional height two p-divisible groups over Fq which are not isogenous to the p-divisible groups of any (supersingular) elliptic curve over Fq . (67) Let (M, Ï) be an isocrystal over W (Fq ) where q = pr . Let {x1 , Â· Â· Â· , xn } be the multi-set of eigenvalues of Ïr (which is a linear operator on M ). Show that the slopes of (M, Ï) are given by the multi-set {Î»1 , Â· Â· Â· , Î»n } where Î»i = 1r valp (xi ). (68) Show that the isocrystal MÎ» â Isocry(k) is simple ( k algebraically closed, Î» â Q). (69) Consider isocrystals over k algebraically closed. How is the tensor product MÎ» â MÂµ decomposed into simple objects? (70) Let M a finitely generated W (k)-module and Ï : M â M be a Ï-linear map. Show that M is a direct sum of W (k)-submodules M1 â M2 , both stable under Ï and Ï acts bijectively on M1 and topologically nilpotently on M2 . Such a decomposition is unique. (71) Let k be a perfect field of char p. Classify p-divisible groups over k with height two and dimension one.

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