On a conjecture of Rapoport and Zink (Q376219)

Let \(\mathbb{F}_{p}\) be the finite field with \(p\) elements, and let \(\mathbb{F}_{p}^{\text{alg}}\) denote its algebraic closure. Let \(K_{0} =W\left( \mathbb{F}_{p}\right) \left[ p^{-1}\right] ,\) and let \(\varphi\) be the Frobenius lift on \(K_{0}.\) For \(G\) a reductive linear algebraic group, fix a conjugacy class \(\left\{ \mu\right\} \) of cocharacters \(\mu :\mathbb{G}_{m}\rightarrow G\) defined over subfields of \(\mathbb{C}_{p}.\) Let \(E\) be the subfield of definition of \(\left\{ \mu\right\} .\) Two cocharacters are said to be equivalent if they induce the same weight filtration on Rep\(_{\mathbb{Q}_{p}}G\), the finite dimensional \(\mathbb{Q}_{p} \)-rational representations of \(G.\)Then there is a projective variety \(\mathcal{F}\) over \(E\) whose \(\mathbb{C}_{p}\)-valued points are in one-to-one correspondence with equivalence classes of cocharacters from \(\left\{ \mu\right\} .\) Let \(\mathcal{\breve{F}=F\otimes}_{E}\breve{E},\) where \(\breve{E}\) is the completion of the maximal unramified extension of \(E\)., denote the rigid analytic space associated to \(\mathcal{\breve{F}}\) by \(\mathcal{\breve{F}}^{\text{rig}}.\) For \(b\in G\left( K_{0}\right) ,\) the \(p\)-adic period space associated with \(\left( G,b,\left\{ \mu\right\} \right) ,\) denoted \(\left( \mathcal{\breve{F}}_{b}^{wa}\right) ^{\text{rig}},\) is the set of \(\mu\in\mathcal{\breve{F}}^{\text{rig}}\) such that \(\left( b,\mu\right) \) is weakly admissible (that is, the filtered isocrystal associated to a faithful representation of \(G\) is weakly admissible. The title of this work refers to a conjecture by \textit{M. Rapoport} and \textit{Th. Zink} [Period spaces for \(p\)-divisible groups. Annals of Mathematics Studies. 141. Princeton, NJ: Princeton Univ. Press. (1996; Zbl 0873.14039)], which addresses the existence of an étale bijective morphism \(\mathcal{F}^{a}\rightarrow\mathcal{F}^{wa}\) of rigid analytic spaces, and a universal local system of universal \(\mathbb{Q}_{p}\)-spaces on \(\left( \mathcal{\breve{F}}_{b}^{wa}\right) ^{\text{rig}}\). Here, the conjecture is refined somewhat, positing (among other things) the existence of a unique largest arcwise connected dense open \(\breve{E}\)-analytic subspace \(\mathcal{\breve{F}}_{b}^{a}\subset\) \(\mathcal{\breve{F}}_{b}^{wa}\) which agrees on all finite extensions of \(\breve{E}.\) For Hodge weights \(n-1\) and \(n\) the authors construct an instrinsic Berkovich open subspace \(\mathcal{F}^{0}\) of \(\mathcal{F}^{wa}\) and the universal local system on \(\mathcal{F}^{0}\). There are some exceptional cases where these are equal. \ For period spaces possessing PEL\ period morphisms, \(\mathcal{F}^{0}\) equals the image of the period morphism. More generally, it is conjectured that \(\mathcal{F}^{0}\) is connected and the maximal possible.\ Assuming the existence of PEL\ period morphisms, the author constructs a tower of covering spaces over this image and proves that it agrees the tower conjectured by Rapoport-Zink.