Diamantine Picard functors of rigid spaces

Abstract

For a connected smooth proper rigid space X over a perfectoid field extension of Qp, we show that the Picard functor of X ♦ defined on perfectoid test objects is the diamondification of the rigid analytic Picard functor. In particular, it is represented by a rigid group variety if and only if the rigid analytic Picard functor is.As an application, we determine which line bundles are trivialized by pro-finiteétale covers, and prove unconditionally that the associated "topological torsion Picard functor" is represented by a divisible analytic group. We use this to generalize and geometrize a construction of Deninger-Werner in the p-adic Simpson correspondence: There is an isomorphism of rigid analytic group varieties between the moduli space of continuous characters of π1(X, x) and that of pro-finite-étale Higgs line bundles on X.This article is part II of a series about line bundles on rigid spaces as diamonds.1. The natural map (Pic X,ét ) ♦ → Pic ♦ X,ét is an isomorphism of sheaves on Perf K,ét .2. If moreover K is algebraically closed, then the v-Picard functor Pic ♦ X,v fits into a split exact sequence of abelian sheaves on Perf K,ét 0 → Pic ♦ X,ét → Pic ♦ X,v → H 0 (X, Ω 1 )(−1) ⊗ K G a → 0.

The p-adic Corlette–Simpson correspondence for abeloids

“…Over the base field K = C p , we have Pic tt (A) = Pic 0 (A) by [18,Lemma B.5]. Therefore, in this case, the pro-finite-étale Higgs bundles are precisely the ones with numerically flat reduction, in line with [41,Theorem 1.2].…”

“…One case in which a p-adic Corlette-Simpson correspondence for representations is known is the case of rank one [20]: In this case, characters of the étale fundamental group correspond to pro-finite-étale Higgs line bundles, and one can also explicitly describe pro-finite-étale line bundles as topological torsion points in the Picard variety [18]. However, a "full" p-adic Corlette-Simpson correspondence from a specific subcategory of Higgs bundles on a proper smooth variety X to the category of finite-dimensional continuous K -linear representations of the étale fundamental group of X has so far not been established yet, not even in non-trivial special cases of X .…”

“…Remark 4. 18 The analog of this story for the analytic topology is more subtle: One can show using [6, Proposition 8.5] that…”

“…continuous K -linear representations of π 1 (X , x)In order to understand the p-adic Corlette-Simpson correspondence in rank one, it thus remains to determine when a Higgs line bundle is pro-finite-étale. This is done as follows:Theorem 2.16[18, Theorem 5.1] Assume that the rigid Picard functor of X is represented by a rigid group variety G. Then a line bundle L on X is pro-finite-étale if and only if its associated point L ∈ G(K ) is topological torsion, i.e. there is N ∈ N such that…”

Pro-étale uniformisation of abelian varieties

“…]. For a rigid or perfectoid group G we have introduced in [13, §2] the "topological p-torsion subgroup" G ⊆ G, a sub-v-sheaf (often an adic subgroup) whose K-points are given by…”