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Period spaces for Hodge structures in equal characteristic. (English) Zbl 1304.11050
This paper lays the foundation of the analogue of crystalline Galois representations of a \(p\)-adic field in the equal positive characteristic, as well as the analogue of a Rapoport-Zink period space. The author proves the analogue of “weakly admissible implies admissible” and the Rapoport-Zink conjecture on the existence of the “universal local system” on the admissible locus. For readers with background in usual \(p\)-adic Hodge theory, we highly recommend first going through the dictionary of the analogy listed in the author’s article [J. Number Theory 129, No. 7, 1734–1757 (2009; Zbl 1186.11071)].
In the usual \(p\)-adic Hodge theory, crystalline representations \(V\) of the absolute Galois group of a \(p\)-adic field \(K\) are defined to be those representations “likely” to come from the \(p\)-adic étale cohomology of some proper and smooth variety over \(K\) with good reduction. Fontaine associated a filtered \(\varphi\)-module to such a representation \(V\), satisfying a numerical condition called weakly admissible. An important achievement is to prove a converse theorem, namely, every weakly admissible filtered \(\varphi\)-module comes from a (unique) crystalline representation. While there are many proofs to this theorem, this paper focuses on making the analogy with one proof, given by L. Berger [in: Représentation \(p\)-adiques de groupes \(p\)-adiques. I. Représentations galoisiennes et \((\varphi, \Gamma)\)-modules. Paris: Société Mathématique de France, 13–38 (2008; Zbl 1168.11019)] which builds on the slope filtration theorem of K. S. Kedlaya [Doc. Math., J. DMV 10, 447–525 (2005; Zbl 1081.14028); erratum ibid. 12, 361–362 (2007)].
In the equal positive characteristic analogue, the base field (analogue of \(K\)) is denoted by \(L\), often a finite extension of \(\mathbb F_q(( \zeta ))\). Let \(\mathcal O_L\) and \(\ell\) denote its valuation ring and the residue field, respectively. Taking Witt vectors in the mixed characteristic corresponds to producing from the field \(\ell\) the formal power series \(\ell[[ z]]\), where \(z\) plays the role of \(p\). The Frobenius lift \(\sigma\) is to simply raise the coefficients of the power series in \(\ell[[ z ]]\) to their \(q\)th powers. We must point out that while \(p\) enters the usual \(p\)-adic Hodge theory in a twofold way: as the uniformizing parameter of the Witt vectors, and as an element of the base field, the two roles are separated in the equal characteristic analogue, as \(z\) and \(\zeta\) above.
The notion of usual isocrystal is replaced by \(z\)-isocrystal, namely a finite dimensional \(\ell(( z))\)-vector space \(D\) and an isomorphism \[ F_D: \sigma^*D: = D \otimes_{\ell((z)), \sigma} \ell((z)) \to D. \] Fix a section of \(\mathcal O_L \to \ell\), which then induces an inclusion \(\ell((z)) \to L[[ z-\zeta]]\). A Hodge-Pink structure over \(L\) on the \(z\)-isocrystal \((D, F_D)\) is an \(L [[ z-\zeta]]\)-lattice inside \(\sigma^*D \otimes_{\ell((z))} L((z-\zeta))\). One could obtain a filtration on \(D_L: = \sigma^*D \otimes_{\ell((z)), z\mapsto \zeta} L\) from the Hodge-Pink structure, but it is necessary to consider the Hodge-Pink structure to get a nice category, due to the separation of the roles of \(z\) and \(\zeta\). For a \(z\)-isocrystal with Hodge-Pink structure, one can define the analogous Newton and Hodge slopes, and hence the notation of weakly admissible.
The crystalline Galois representations are replaced by the local shtukas, namely, finite free \(\mathcal O_L[[ z]]\)-modules \(M\) together with an isomorphism \[ F_M: \sigma_*M[\tfrac 1{z-\zeta}] \rightarrow{\cong}M[\tfrac 1{z-\zeta}]. \] From such a local shtukas, one can on the one hand, recover a Galois representation \(\rho_M: \mathrm{Gal}(\bar L / L) \to \mathrm{GL}_n(\mathbb F_q((z))) \) from the action on the analogous Tate-modules, and on the other hand, associate a \(z\)-isocrystal with Hodge-Pink structure as constructed by A. Genestier and V. Lafforgue [Ann. Sci. Éc. Norm. Supér. (4) 44, No. 2, 263–360 (2011; Zbl 1277.14036)] (slightly generalized in this paper to arbitrary base field \(L\)).
The main theorem of the paper is to prove that every weakly admissible \(z\)-isocrystal with Hodge-Pink structure over \(L\) is one that is associated to a local shtuka, concluded in Section 2. In fact, the paper proves it for a general nonarchimedean base field \(L\) satisfying certain mild hypothesis that allows the rigidification of local shtukas up to isogeny. The essential ingredient of the proof of the main theorem is to establish an analogue of the “second generation” of the slope filtration theorem by Kedlaya (loc. cit.); this occupies the first section of the paper, making use of one technical core, namely to establish a Dieudonné-Manin classification over the analogue of Robba ring, carried out in an earlier work by the author and R. Pink in [Compos. Math. 140, No. 3, 689–716 (2004; Zbl 1074.14028)].
The third and last section of the paper constructs the period space for Hodge-Pink structures; this space is an analogue of the Rapoport-Zink space [M. Rapoport and Th. Zink, Period spaces for \(p\)-divisible groups. Princeton, NJ: Princeton Univ. Press (1996; Zbl 0873.14039)] of Fontaine’s filtered isocrystals. The author proves that the locus \(\mathcal H^a\) where the universal filtered \(z\)-isocrystal comes from local shtukas is a Berkovich open subspace of the weakly admissible locus \(\mathcal H^{wa}\) inside the period space. Moreover, there is an étale covering \(X\) of \(\mathcal H^a\) and a local shtuka on \(X\) giving rise to the universal \(z\)-isocrystal with Hodge-Pink structure on \(\mathcal H^a\).
This paper is very well written.

MSC:
11G09 Drinfel’d modules; higher-dimensional motives, etc.
13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
14G20 Local ground fields in algebraic geometry
14G22 Rigid analytic geometry
14D07 Variation of Hodge structures (algebro-geometric aspects)
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