Reconstructing $p$-divisible groups from their truncations of small level

Abstract

Abstract. Let k be an algebraically closed field of characteristic p > 0. Let D be a p-divisible group over k. Let n D be the smallest non-negative integer for which the following statement holds: if C is a p-divisible group over k of the same codimension and dimension as D and suchTo the Dieudonné module of D we associate a non-negative integer`D which is a computable upper bound ofif the set I has at least two elements we also show that n D Ä maxf1; n D i ; n D i C n D j 1 j i; j 2 I; j ¤ i g. We show that we have n D Ä 1 if and only if`D Ä 1; this recovers the classification of minimal p-divisible groups obtained by Oort. If D is quasi-special, we prove the Traverso truncation conjecture for D. If D is F -cyclic, we explicitly compute n D . Many results are proved in the general context of latticed F -isocrystals with a (certain) group over k.

Level 𝑚 stratifications of versal deformations of 𝑝-divisible groups

“…Subsection 2.3 studies T 1 . Theorem 2.4 recalls properties of stabilizer subgroup schemes of T m we obtained in [Va4,Thm. 5.3].…”

“…26 to 40]. A significant change in their presentation was made independently of [Tr2] and [Tr3], in [Va3] for the case m = 1 and in [Va4,Sect. 5] for all m. The change allows a very easy description of the orbit spaces that leads to short, elementary, and foundational (computations and) proofs of all parts of the Basic Theorem; the change is explained in Remark 2.4.1.…”

“…In this Section we recall the group action T m over k we introduced in [Va4,Sect. 5]; its set of orbits parametrizes isomorphism classes of truncated Barsotti-Tate groups of level m over k that have codimension c and dimension d. Subsection 2.1 introduces certain group schemes that play a key role in the definition (see Section 2.2) of T m .…”

Computing isomorphism numbers of F -crystals using the level torsions

“…Moreover, the isomorphism number of a latticed F -isocrystal is invariant under duality. See [14,Fact 4.2.1] for a proof in the Dieudonné module case, which is easily adapted to the latticed F -isocrystal case. Proof.…”

“…The main result of [14] provides a computable upper bound for the isomorphism numbers; see Theorem 2.3.…”