2019/12/01 by Ding, Zeyu
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1912.01424
Let k be an algebraically closed field of characteristic p>0. Let c,d∈ ℕ be such that h=c+d>0. Let H be a p-divisible group of codimension c and dimension d over k. For m∈ℕ^∗ let H[pm]=ker([pm]:H→ H). It is a finite commutative group scheme over k of p power order, called a Barsotti-Tate group of level m. We study a particular type of p-divisible groups Hπ, where π is a permutation on the set \1,2,…,h\. Let (M,φπ) be the Dieudonné module of Hπ. Each Hπ is uniquely determined by Hπ[p] and by the fact that there exists a maximal torus T of GLM whose Lie algebra is normalized by φπ in a natural way. Moreover, if H is a p-divisible group of codimension c and dimension d over k, then H[p]≅ Hπ[p] for some permutation π. We call these Hπ canonical lifts of Barsotti-Tate groups of level 1. We obtain new formulas of combinatorial nature for the dimension of \boldsymbolAut(Hπ[pm]) and for the number of connected components of \boldsymbolEnd(Hπ[pm]).