Moduli of $p$-divisible groups

Abstract

We prove several results about moduli spaces of p-divisible groups such as Rapoport-Zink spaces. Our main goal is to prove that Rapoport-Zink spaces at infinite level carry a natural structure as a perfectoid space, and to give a description purely in terms of padic Hodge theory of these spaces. This allows us to formulate and prove duality isomorphisms between basic Rapoport-Zink spaces at infinite level in general. Moreover, we identify the image of the period morphism, reproving results of Faltings, [Fal10]. For this, we give a general classification of p-divisible groups over O C , where C is an algebraically closed complete extension of Q p , in the spirit of Riemann's classification of complex abelian varieties. Another key ingredient is a full faithfulness result for the Dieudonné module functor for p-divisible groups over semiperfect rings (i.e. rings on which the Frobenius is surjective).

On torsion in the cohomology of locally symmetric varieties

“…Note that a result very similar in spirit was proved in joint work with Jared Weinstein, [56], for Rapoport-Zink spaces. Unfortunately, for a number of reasons, it is not possible to use that result to obtain a result for Shimura varieties (although the process in the opposite direction does work).…”

“…Conversely, all Q p -rational totally isotropic subspaces W ⊂ C 2g are in one GSp 2g (Z p )-orbit. By the classification result for p-divisible groups over O C , [56,Theorem B], it follows that if the Hodge-Tate filtration is Q p -rational, then…”

“…Although there may be a more direct way to prove this, we deduce it from the Hodge-Tate period map. Recall that the Hodge-Tate filtration of an abelian variety A over a complete and algebraically closed extension C of Q p is a short exact sequence In fact, observe that by [56,Theorem B], the C-valued points of Fℓ are in bijection with principally polarized p-divisible groups G over O C , with a trivialization of their Tate module. Thus, π HT is, at least on C-valued points of the locus of good reduction, the map sending an abelian variety over O C to its associated p-divisible group.…”

“…On the supersingular locus, one has the following description of π HT , using the isomorphism M LT,∞ ∼ = M Dr,∞ between Lubin-Tate and Drinfeld tower, cf. [29], [32], [56]:…”

Exterior Powers of Lubin-Tate Groups

“…For example, over base fields, these exterior powers with their universal property are used to construct explicitly the Lubin-Tate tower at infinite level in [23] (equal characteristic) and the Rapoport-Zink spaces at infinite level in [21] (mixed characteristic).…”