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.


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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[1] A. Grothendieck, Élements de Géométrie Algébrique, New York: Springer-Verlag, vol. 4, 8, 11, 17, 20, 24, 28, 32.
[2] A. Grothendieck, Revêtements Étales et Groupe Fondamental, New York: Springer-Verlag, 1971, vol. 224.
[3] G. W. Anderson, ”\(t\)-motives,” Duke Math. J., vol. 53, iss. 2, pp. 457-502, 1986. · Zbl 0679.14001
[4] G. W. Anderson, ”On Tate modules of formal \(t\)-modules,” Internat. Math. Res. Notices, iss. 2, pp. 41-52, 1993. · Zbl 0777.11020
[5] Y. André, ”Slope filtrations,” Confluentes Math., vol. 1, iss. 1, pp. 1-85, 2009. · Zbl 1213.14039
[6] F. Andreatta, ”Generalized ring of norms and generalized \((\phi,\Gamma)\)-modules,” Ann. Sci. École Norm. Sup., vol. 39, iss. 4, pp. 599-647, 2006. · Zbl 1123.13007
[7] F. Andreatta and O. Brinon, ”Surconvergence des représentations \(p\)-adiques: le cas relatif,” in Représentations \(p\)-Adiques de Groupes \(p\)-Adiques. I. Représentations Galoisiennes et \((\phi,\Gamma)\)-Modules, , 2008, vol. 319, pp. 39-116. · Zbl 1168.11018
[8] F. Andreatta and A. Iovita, ”Global applications of relative \((\phi,\Gamma)\)-modules. I,” in Représentations \(p\)-Adiques de Groupes \(p\)-Adiques. I. Représentations Galoisiennes et \((\phi,\Gamma)\)-Modules, , 2008, vol. 319, pp. 339-420. · Zbl 1163.11051
[9] J. Ax, ”Zeros of polynomials over local fields-The Galois action,” J. Algebra, vol. 15, pp. 417-428, 1970. · Zbl 0216.04703
[10] L. Berger, ”Représentations \(p\)-adiques et équations différentielles,” Invent. Math., vol. 148, iss. 2, pp. 219-284, 2002. · Zbl 1113.14016
[11] L. Berger, ”Équations différentielles \(p\)-adiques et \((\phi,N)\)-modules filtrés,” in Représentations \(p\)-Adiques de Groupes \(p\)-Adiques. I. Représentations galoisiennes et \((\phi,\Gamma)\)-modules, , 2008, vol. 319, pp. 13-38. · Zbl 1168.11019
[12] L. Berger and P. Colmez, ”Familles de représentations de de Rham et monodromie \(p\)-adique,” in Représentations \(p\)-Adiques de Groupes \(p\)-Adiques. I. Représentations Galoisiennes et \((\phi,\Gamma)\)-Modules, , 2008, vol. 319, pp. 303-337. · Zbl 1168.11020
[13] V. G. Berkovich, Spectral Theory and Analytic Geometry over Non-Archimedean Fields, Providence, RI: Amer. Math. Soc., 1990, vol. 33. · Zbl 0715.14013
[14] V. G. Berkovich, ”Étale cohomology for non-Archimedean analytic spaces,” Inst. Hautes Études Sci. Publ. Math., vol. 78, pp. 5-161 (1994), 1993. · Zbl 0804.32019
[15] G. Böckle and U. Hartl, ”Uniformizable families of \(t\)-motives,” Trans. Amer. Math. Soc., vol. 359, iss. 8, pp. 3933-3972, 2007. · Zbl 1140.11030
[16] M. Bornhofen and U. Hartl, ”Pure Anderson motives and abelian \(\tau\)-sheaves,” Math. Z., 2011. · Zbl 1227.11077
[17] S. Bosch, Lectures on formal and rigid geometry. · Zbl 1314.14002
[18] S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean Analysis, New York: Springer-Verlag, 1984, vol. 261. · Zbl 0539.14017
[19] S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron Models, New York: Springer-Verlag, 1990, vol. 21. · Zbl 0705.14001
[20] S. Bosch and W. Lütkebohmert, ”Formal and rigid geometry. I. Rigid spaces,” Math. Ann., vol. 295, iss. 2, pp. 291-317, 1993. · Zbl 0808.14017
[21] S. Bosch and W. Lütkebohmert, ”Formal and rigid geometry. II. Flattening techniques,” Math. Ann., vol. 296, pp. 403-429, 1993. · Zbl 0808.14018
[22] N. Bourbaki, Éléments de Mathématique. Algèbre. Chapitres 1 à 3, Paris: Hermann, 1970. · Zbl 0211.02401
[23] N. Bourbaki, Éléments de Mathématique. Topologie Générale. Chapitres 1 à 4, Paris: Hermann, 1971. · Zbl 1107.54001
[24] C. Breuil, Schemas en groupes et corps des normes, 1998. · Zbl 0961.14010
[25] C. Breuil, ”Groupes \(p\)-divisibles, groupes finis et modules filtrés,” Ann. of Math., vol. 152, iss. 2, pp. 489-549, 2000. · Zbl 1042.14018
[26] O. Brinon, Représentations \(p\)-Adiques Cristallines et de de Rham dans le Cas Relatif, , 2008, vol. 112. · Zbl 1170.14016
[27] P. Colmez, ”Espaces de Banach de dimension finie,” J. Inst. Math. Jussieu, vol. 1, iss. 3, pp. 331-439, 2002. · Zbl 1044.11102
[28] P. Colmez, ”Espaces vectoriels de dimension finie et représentations de de Rham,” in Représentations \(p\)-Adiques de Groupes \(p\)-Adiques. I. Représentations Galoisiennes et \((\phi,\Gamma)\)-Modules, , 2008, vol. 319, pp. 117-186. · Zbl 1168.11021
[29] P. Colmez and J. Fontaine, ”Construction des représentations \(p\)-adiques semi-stables,” Invent. Math., vol. 140, iss. 1, pp. 1-43, 2000. · Zbl 1010.14004
[30] P. Deligne and L. Illusie, ”Relèvements modulo \(p^2\) et décomposition du complexe de de Rham,” Invent. Math., vol. 89, iss. 2, pp. 247-270, 1987. · Zbl 0632.14017
[31] P. Deligne and J. S. Milne, ”Tannakian categories,” in Hodge Cycles, Motives, and Shimura Varieties, New York: Springer-Verlag, 1982, vol. 900, pp. 101-228. · Zbl 0477.14004
[32] V. G. Drinfeld, ”Elliptic modules,” Math. USSR-Sb., vol. 23, pp. 561-592, 1976. · Zbl 0321.14014
[33] V. G. Drinfeld, ”Moduli varieties of \(F\)-sheaves,” Funktsional. Anal. i Prilozhen., vol. 21, iss. 2, pp. 23-41, 1987.
[34] D. Eisenbud, Commutative Algebra with a View toward Algebraic Geometry, New York: Springer-Verlag, 1995, vol. 150. · Zbl 0819.13001
[35] G. Faltings, ”Crystalline cohomology and \(p\)-adic Galois-representations,” in Algebraic Analysis, Geometry, and Number Theory, Baltimore, MD, 1989, pp. 25-80. · Zbl 0805.14008
[36] G. Faltings, ”Almost étale extensions,” in Cohomologies \(p\)-adiques et Applications Arithmétiques, II, Paris: Math. Soc. France, 2002, vol. 279, pp. 185-270. · Zbl 1027.14011
[37] G. Faltings, ”Coverings of \(p\)-adic period domains,” J. Reine Angew. Math., vol. 643, pp. 111-139, 2010. · Zbl 1208.14039
[38] J. Fontaine, ”Modules galoisiens, modules filtrés et anneaux de Barsotti-Tate,” in Journées de Géométrie Algébrique de Rennes, Vol. III, Paris, 1979, pp. 3-80. · Zbl 0429.14016
[39] J. Fontaine, ”Sur certains types de représentations \(p\)-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate,” Ann. of Math., vol. 115, iss. 3, pp. 529-577, 1982. · Zbl 0544.14016
[40] A. Genestier and V. Lafforgue, Théorie de Fontaine en égales charactéristiques, preprint, 2008. · Zbl 1277.14036
[41] A. Genestier and V. Lafforgue, Structures de Hodge-Pink pour les \(\phi/\mathfrakS\)-modules de Breuil et Kisin, 2010. · Zbl 1328.11112
[42] L. Gerritzen, ”Zerlegungen der Picard-Gruppe nichtarchimedischer holomorpher Räume,” Compositio Math., vol. 35, iss. 1, pp. 23-38, 1977. · Zbl 0353.14009
[43] A. Grothendieck, Groupes de Barsotti-Tate et Cristaux de Dieudonné, Les Presses de l’Université de Montréal, Montreal, Que., 1974, vol. 45. · Zbl 0331.14021
[44] U. Hartl, ”Uniformizing the stacks of abelian sheaves,” in Number Fields and Function Fields-Two Parallel Worlds, Boston, MA: Birkhäuser, 2005, vol. 239, pp. 167-222. · Zbl 1137.11322
[45] U. Hartl, ”On period spaces for \(p\)-divisible groups,” C. R. Math. Acad. Sci. Paris, vol. 346, iss. 21-22, pp. 1123-1128, 2008. · Zbl 1161.14032
[46] U. Hartl, ”A dictionary between Fontaine-theory and its analogue in equal characteristic,” J. Number Theory, vol. 129, iss. 7, pp. 1734-1757, 2009. · Zbl 1186.11071
[47] U. Hartl, On a conjecture of Rapoport and Zink. · Zbl 1285.14027
[48] U. Hartl and R. Pink, ”Vector bundles with a Frobenius structure on the punctured unit disc,” Compos. Math., vol. 140, iss. 3, pp. 689-716, 2004. · Zbl 1074.14028
[49] U. Hartl and E. Viehmann, The Newton stratification on deformations of local \(G\)-shtukas. · Zbl 1225.14036
[50] M. J. Hopkins and B. H. Gross, ”The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory,” Bull. Amer. Math. Soc., vol. 30, iss. 1, pp. 76-86, 1994. · Zbl 0857.55003
[51] M. J. Hopkins and B. H. Gross, ”Equivariant vector bundles on the Lubin-Tate moduli space,” in Topology and Representation Theory, Providence, RI, 1994, pp. 23-88. · Zbl 0807.14037
[52] L. Illusie, ”Cohomologie de de Rham et cohomologie étale \(p\)-adique (d’après G. Faltings, J.-M. Fontaine et al.),” in Séminaire Bourbaki, Vol. 1989/90, , 1990, vol. 189-190, p. exp. no. 726, 325-374. · Zbl 0736.14005
[53] A. J. de Jong, ”Étale fundamental groups of non-Archimedean analytic spaces,” Compositio Math., vol. 97, iss. 1-2, pp. 89-118, 1995. · Zbl 0864.14012
[54] A. J. de Jong, ”Homomorphisms of Barsotti-Tate groups and crystals in positive characteristic,” Invent. Math., vol. 134, iss. 2, pp. 301-333, 1998. · Zbl 0929.14029
[55] J. de Jong and M. van der Put, ”Étale cohomology of rigid analytic spaces,” Doc. Math., vol. 1, p. no. 01, 1-56, 1996. · Zbl 0922.14012
[56] N. M. Katz, ”\(p\)-adic properties of modular schemes and modular forms,” in Modular Functions of One Variable, III, New York, 1973, pp. 69-190. · Zbl 0271.10033
[57] N. M. Katz, ”Slope filtration of \(F\)-crystals,” in Journées de Géométrie Algébrique de Rennes, Vol. I, Paris, 1979, pp. 113-163. · Zbl 0426.14007
[58] K. S. Kedlaya, ”A \(p\)-adic local monodromy theorem,” Ann. of Math., vol. 160, iss. 1, pp. 93-184, 2004. · Zbl 1088.14005
[59] K. S. Kedlaya, ”Slope filtrations revisited,” Doc. Math., vol. 10, pp. 447-525, 2005. · Zbl 1081.14028
[60] R. Kiehl, ”Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie,” Invent. Math., vol. 2, pp. 256-273, 1967. · Zbl 0202.20201
[61] W. Kim, Galois deformation theory for norm fields and its arithmetic applications.
[62] M. Kisin, ”Crystalline representations and \(F\)-crystals,” in Algebraic Geometry and Number Theory, Boston, MA: Birkhäuser, 2006, vol. 253, pp. 459-496. · Zbl 1184.11052
[63] M. Kisin, ”Potentially semi-stable deformation rings,” J. Amer. Math. Soc., vol. 21, iss. 2, pp. 513-546, 2008. · Zbl 1205.11060
[64] L. Lafforgue, ”Chtoucas de Drinfeld et conjecture de Ramanujan-Petersson,” Astérisque, vol. 243, p. ii, 1997. · Zbl 0899.11026
[65] S. Lang, ”Algebraic groups over finite fields,” Amer. J. Math., vol. 78, pp. 555-563, 1956. · Zbl 0073.37901
[66] S. G. Langton, ”Valuative criteria for families of vector bundles on algebraic varieties,” Ann. of Math., vol. 101, pp. 88-110, 1975. · Zbl 0307.14007
[67] G. Laumon, Cohomology of Drinfeld Modular Varieties. Part I, Cambridge: Cambridge Univ. Press, 1996, vol. 41. · Zbl 0837.14018
[68] M. Lazard, ”Les zéros des fonctions analytiques d’une variable sur un corps valué complet,” Inst. Hautes Études Sci. Publ. Math., vol. 14, pp. 47-75, 1962. · Zbl 0119.03701
[69] T. Liu, ”Torsion \(p\)-adic Galois representations and a conjecture of Fontaine,” Ann. Sci. École Norm. Sup., vol. 40, iss. 4, pp. 633-674, 2007. · Zbl 1163.11043
[70] W. Lütkebohmert, ”Vektorraumbündel über nichtarchimedischen holomorphen Räumen,” Math. Z., vol. 152, iss. 2, pp. 127-143, 1977. · Zbl 0333.32024
[71] J. I. Manin, ”Theory of commutative formal groups over fields of finite characteristic,” Uspehi Mat. Nauk, vol. 18, iss. 6, pp. 3-90, 1963. · Zbl 0128.15603
[72] R. Pink, Hodge structures over function fields. · Zbl 0981.20036
[73] I. Y. Potemine, ”Drinfeld-Anderson motives and multicomponent KP hierarchy,” in Recent Progress in Algebra, Providence, RI, 1999, pp. 213-227. · Zbl 0926.11041
[74] D. Quillen, ”Higher algebraic \(K\)-theory. I,” in Algebraic \(K\)-theory, I: Higher \(K\)-Theories, New York, 1973, pp. 85-147. · Zbl 0292.18004
[75] M. Rapoport, ”Non-Archimedean period domains,” in Proceedings of the International Congress of Mathematicians, Vol. 1, 2, Basel, 1995, pp. 423-434. · Zbl 0874.11046
[76] M. Rapoport and M. Richartz, ”On the classification and specialization of \(F\)-isocrystals with additional structure,” Compositio Math., vol. 103, iss. 2, pp. 153-181, 1996. · Zbl 0874.14008
[77] M. Rapoport and T. Zink, Period Spaces for \(p\)-Divisible Groups, Princeton, NJ: Princeton Univ. Press, 1996, vol. 141. · Zbl 0873.14039
[78] M. Raynaud, ”Géométrie analytique rigide d’après Tate, Kiehl,\(\,\ldots\, \),” in Table Ronde d’Analyse Non Archimédienne, Paris, 1974, pp. 319-327. · Zbl 0299.14003
[79] J. J. Rotman, An Introduction to Homological Algebra, New York: Academic Press [Harcourt Brace Jovanovich Publishers], 1979, vol. 85. · Zbl 0441.18018
[80] P. Schneider, ”Points of rigid analytic varieties,” J. Reine Angew. Math., vol. 434, pp. 127-157, 1993. · Zbl 0774.14021
[81] J. Tate, ”\(p\)-divisible groups,” in Proc. Conf. Local Fields, New York, 1967, pp. 158-183. · Zbl 0157.27601
[82] J. Tate, ”Rigid analytic spaces,” Invent. Math., vol. 12, pp. 257-289, 1971. · Zbl 0212.25601
[83] M. Temkin, ”On local properties of non-Archimedean analytic spaces,” Math. Ann., vol. 318, iss. 3, pp. 585-607, 2000. · Zbl 0972.32019
[84] T. Tsuji, On purity for \(p\)-adic representations. · Zbl 1239.14014
[85] T. Zink, Cartiertheorie Kommutativer Formaler Gruppen, Leipzig: BSB B. G. Teubner Verlagsgesellschaft, 1984, vol. 68. · Zbl 0578.14039
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