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Subtle invariants for p-divisible groups and Traverso’s conjectures

We report on the joint works [1] and [2] aimed towards the classification of p-divisible groups over an algebraically closed field k of positive characteristic p. Let W (k) be the p-typical Witt ring of k. Let σ be the Frobenius automorphism of W (k). Let D be a p-divisible group over k of positive height r and let (M,φ, θ) be its (contravariant) Dieudonné module. Thus M is a free W (k)-module of rank r, φ : M → M is a σ-linear endomorphism, and θ : M → M is a σ−1-linear endomorphism such that we have φ ◦ θ = θ ◦ φ = p1M . Let d = dimk(M/φ(M)) and c = dimk(M/θ(M)) be the dimension and the codimension (respectively) of D. We have c+ d = r. Let t := min{c, d}.