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Local shtukas, Hodge-Pink structures and Galois representations. (English) Zbl 1440.14113
Böckle, Gebhard (ed.) et al., \(t\)-motives: Hodge structures, transcendence and other motivic aspects. EMS Series of Congress Reports. Zürich: European Mathematical Society (EMS). 183-259 (2020).
Summary: We review the analog of Fontaine’s theory of crystalline \(p\)-adic Galois representations and their classification by weakly admissible filtered isocrystals in the arithmetic of function fields over a finite field. There crystalline Galois representations are replaced by the Tate modules of so-called local shtukas. We prove that the Tate module functor is fully faithful. In addition to this étale realization of a local shtuka we discuss also the de Rham and the crystalline cohomology realizations and construct comparison isomorphisms between these realizations. We explain how local shtukas and these cohomology realizations arise from Drinfeld modules and Anderson’s \(t\)-motives. As an application we construct equi-characteristic crystalline deformation rings, establish their rigid-analytic smoothness and compute their dimension.
For the entire collection see [Zbl 1441.14003].

MSC:
14F30 \(p\)-adic cohomology, crystalline cohomology
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
11F70 Representation-theoretic methods; automorphic representations over local and global fields
14G20 Local ground fields in algebraic geometry
14G35 Modular and Shimura varieties
11G09 Drinfel’d modules; higher-dimensional motives, etc.
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[1] A. Grothendieck, Élements de Géométrie Algébrique. Publ. Math. IHES 4, 8, 11, 17, 20, 24, 28, 32, Bures-Sur-Yvette, 1960-1967; see also Grundlehren 166, Springer-Verlag, Berlin etc. 1971; also available at http://www.numdam.org/numdam-bin/recherche?au=Grothendieck. · Zbl 0118.36206
[2] V. Abrashkin, Galois modules arising from Faltings’s strict modules. Compos. Math. 142(4) (2006), 867-888; also available asarXiv:math/0403542. · Zbl 1102.14032
[3] G. Anderson,t-Motives. Duke Math. J. 53 (1986), 457-502.
[4] G. Anderson, On Tate Modules of Formalt-Modules. Internat. Math. Res. Not. 2 (1993), 41-52. · Zbl 0777.11020
[5] E. Arasteh Rad and U. Hartl, LocalP-shtukas and their relation to globalG-shtukas. Muenster J. Math. 7 (2014), 623-670; open access athttp://miami.uni-muenster.de. · Zbl 1348.14110
[6] J. Ax, Zero of polynomials over local fields – the Galois action. J. Algebra 15 (1970), 417-428. · Zbl 0216.04703
[7] L. Berger, Représentationsp-adiques et équations différentielles. Invent. Math. 148 (2002), 219-284; also available asarXiv:math.NT/0102179. · Zbl 1113.14016
[8] L. Berger, Limites de représentations cristallines. Compos. Math. 140(6) (2004), 1473- 1498; also available asarXiv:math.NT/0201262. · Zbl 1071.11067
[9] L. Berger, An introduction to the theory ofp-adic representations. In Geometric aspects of Dwork theory, Vol. I, II, de Gruyter, Berlin, 2004, 255-292; also available as arXiv:math.NT/0210184.
[10] P. Berthelot, L. Breen, and W. Messing, Théorie de Dieudonné cristalline II, LNM 930, Springer, Berlin, 1982. · Zbl 0516.14015
[11] P. Berthelot and A. Ogus,F-isocrystals and de Rham cohomology I. Invent. Math. 72(2) (1983), 159-199. · Zbl 0516.14017
[12] G. Böckle and U. Hartl: Uniformizable Families oft-motives. Trans. Amer. Math. Soc. 359(8) (2007), 3933-3972; also available asarXiv:math.NT/0411262. · Zbl 1140.11030
[13] M. Bornhofen and U. Hartl, Pure Anderson motives and abelian-sheaves. Math. Z. 268 (2011), 67-100; also available asarXiv:0709.2809. · Zbl 1227.11077
[14] S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean Analysis, Grundlehren 261, Springer, Berlin, etc., 1984. · Zbl 0539.14017
[15] S. Bosch and W. Lütkebohmert, Formal and Rigid Geometry I. Rigid Spaces. Math. Ann. 295 (1993), 291-317. · Zbl 0808.14017
[16] N. Bourbaki, Élements de Mathématique, Algèbre, Chapitres 1 à 3, Hermann, Paris, 1970. · Zbl 0211.02401
[17] C. Breuil, Groupesp-divisibles, groupes finis et modules filtrés. Ann. Math. 152 (2000), 489-549; also available asarXiv:math.NT/0009252. · Zbl 1042.14018
[18] O. Brinon and B. Conrad,p-adic Hodge theory, notes from the CMI summer school 2009; also available athttp://math.stanford.edu/conrad.
[19] D. Brownawell and M. Papanikolas, Linear independence of gamma values in positive characteristic. J. Reine Angew. Math. 549 (2002), 91-148; also available as arXiv:math.NT/0106054. · Zbl 1002.11051
[20] L. Carlitz, On certain functions connected with polynomials in a Galois field. Duke Math. J. 1(2) (1935), 137-168. · Zbl 0012.04904
[21] P. Colmez and J.-M. Fontaine, Construction des représentationsp-adiques semi-stables. Invent. Math. 140(1) (2000), 1-43. · Zbl 1010.14004
[22] A.J. de Jong, Crystalline Dieudonné module theory via formal and rigid geometry. Inst. Hautes Études Sci. Publ. Math. 82 (1995), 5-96; also available at http://www.math.columbia.edu/dejong/papers/. · Zbl 0864.14009
[23] V. G. Drinfeld, Moduli variety ofF-sheaves. Funct. Anal. Appl. 21(2) (1987), 107-122.
[24] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, GTM 150, Springer, Berlin, etc., 1995. · Zbl 0819.13001
[25] G. Faltings, Crystalline cohomology andp-adic Galois-representations, Algebraic analysis, geometry, and number theory (Baltimore 1988), pp. 25-80, Johns Hopkins Univ. Press, Baltimore, 1989. · Zbl 0805.14008
[26] G. Faltings, Group schemes with strictO-action. Mosc. Math. J. 2(2) (2002), 249-279. · Zbl 1013.11079
[27] L. Fargues, Quelques résultats et conjectures concernant la courbe. Astérisque 369 (2015), 325-374; also available athttp://www.math.jussieu.fr/fargues. · Zbl 1326.14109
[28] L. Fargues and J.-M. Fontaine, Courbes et fibrés vectoriels en théorie de Hodgep-adique (French), Curves and vector bundles inp-adic Hodge theory. With a preface by P. Colmez. Astérisque 406 (2018), xiii+382. · Zbl 07005651
[29] J.-M. Fontaine, Modules galoisiens, modules filtrés et anneaux de Barsotti-Tate. In Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. III, pp. 3-80, Astérisque 65, Soc. Math. France, Paris, 1979. · Zbl 0429.14016
[30] J.-M. Fontaine, Sur certains types de représentationsp-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate. Ann. Math. (2) 115(3) (1982), 529-577. · Zbl 0544.14016
[31] J.-M. Fontaine, Représentationsp-adiques des corps locaux I. In The Grothendieck Festschrift, Vol. II, pp. 249-309, Progr. Math. 87, Birkhäuser, Boston, MA, 1990. · Zbl 0743.11066
[32] J.-M. Fontaine, Le corps des périodesp-adiques, in Périodesp-adiques (Bures-surYvette, 1988). Astérisque 223 (1994), 59-111.
[33] F. Gardeyn, A Galois criterion for good reduction of-sheaves. J. Number Theory 97 (2002), 447-471. · Zbl 1053.11054
[34] T. Gee and M. Kisin, The Breuil-Mézard conjecture for potentially BarsottiTate representations, Forum of Math, Pi 2 (2014), e1, 56 pp;available at http://journals.cambridge.org/. · Zbl 1408.11033
[35] E.-U. Gekeler, On the de Rham isomorphism for Drinfel’d modules. J. Reine Angew. Math. 401 (1989), 188-208. · Zbl 0672.14011
[36] A. Genestier and V. Lafforgue,Théorie de Fontaine en égales charactéristiques. Ann. Sci. École Norm. Supér. 44(2) (2011), 263-360; also available at http://www.math.jussieu.fr/vlafforg/. · Zbl 1277.14036
[37] D. Goss, Drinfel’d modules: cohomology and special functions. In Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math. 55, Part 2, pp. 309-362, Amer. Math. Soc., Providence, RI, 1994. · Zbl 0827.11035
[38] A. Grothendieck, Groupes de Barsotti-Tate et cristaux de Dieudonné, Séminaire de Mathématiques Supérieures 45, Les Presses de l’Université de Montréal, Montreal, 1974.
[39] U. Hartl, Uniformizing the Stacks of Abelian Sheaves, in Number Fields and Function fields - Two Parallel Worlds, Papers from the 4th Conference held on Texel Island, April 2004, Progress in Math. 239, Birkhäuser, Basel, 2005, 167-222; also available asarXiv:math.NT/0409341. · Zbl 1137.11322
[40] U. Hartl, A Dictionary between Fontaine-Theory and its Analogue in Equal Characteristic. J. Number Th. 129 (2009), 1734-1757; also available asarXiv:math.NT/0607182. · Zbl 1186.11071
[41] U. Hartl, Period Spaces for Hodge Structures in Equal Characteristic. Ann. Math. 173(3) (2011), 1241-1358; also available asarXiv:math.NT/0511686. · Zbl 1304.11050
[42] U. Hartl, On a Conjecture of Rapoport and Zink. Invent. Math. 193 (2013), 627-696; also available asarXiv:math.NT/0605254. · Zbl 1285.14027
[43] U. Hartl, Isogenies of abelian AndersonA-modules andA-motives, Preprint 2015 available athttp://www.math.uni-muenster.de/u/urs.hartl.
[44] U. Hartl and E. Hellmann, The universal familly of semi-stablep-adic Galois representations, preprint 2013 onarXiv:math/1312.6371.
[45] U. Hartl and A.-K. Juschka, Pink’s theory of Hodge structures and the Hodge conjecture over function fields. Int-motives: Hodge structures, transcendence and other motivic aspects (G. Böckle, D. Goss, U. Hartl, and M. Papanikolas, eds.), EMS Congr. Rep., Eur. Math. Soc., Berlin, 2020; also available asarxiv:math/1607.01412.
[46] U. Hartl and R. Pink,Vector bundles with a Frobenius structure on the punctured unit disc. Comp. Math. 140(3) (2004), 689-716;also available at http://www.math.uni-muenster.de/u/urs.hartl/Publikat/. · Zbl 1074.14028
[47] U. Hartl and R. K. Singh, Local Shtukas and Divisible Local Anderson Modules, preprint 2015 onarXiv:1511.03697. · Zbl 07106608
[48] U. Hartl and E. Viehmann, The Newton stratification on deformations of localG-shtukas. J. Reine Angew. Math. (Crelle) 656 (2011), 87-129; also available asarXiv:0810.0821. · Zbl 1225.14036
[49] R. Hartshorne, Algebraic Geometry, GTM 52, Springer, Berlin, etc., 1977. · Zbl 0367.14001
[50] E. Hellmann, On arithmetic families of filtered’-modules and crystalline representations. J. Inst. Math. Jussieu 12(4) (2013), 677-726; also available asarXiv:1010.4577. · Zbl 1355.11065
[51] R. Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Math., E30, Vieweg & Sohn, Braunschweig, 1996. · Zbl 0868.14010
[52] W. Kim, Galois deformation theory for norm fields and its arithmetic applications,PhD-Thesis, University of Michigan, June 3, 2009;available at https://deepblue.lib.umich.edu/handle/2027.42/63878.
[53] W. Kim, Galois deformation theory for norm fields and flat deformation rings. J. Number Th. 131 (2011), 1258-1275; also available asarXiv:1005.3147. · Zbl 1228.11171
[54] M. Kisin, Crystalline representations andF-crystals, in Algebraic geometry and number theory, pp. 459-496, Progress in Math. 253, Birkhäuser, Boston, MA, 2006; also available atwww.math.harvard.edu/kisin. · Zbl 1184.11052
[55] M. Kisin, Potentially semi-stable deformation rings. J. Amer. Math. Soc. 21(2) (2008), 513-546; also available atwww.math.harvard.edu/kisin. · Zbl 1205.11060
[56] M. Kisin, Moduli of finite flat group schemes, and modularity. Ann. Math. (2) 170(3) (2009), 1085-1180; also available atwww.math.harvard.edu/kisin. · Zbl 1201.14034
[57] M. Kisin, The Fontaine-Mazur conjecture for GL2. J. Amer. Math. Soc. 22(3) (2009), 641-690; also available atwww.math.harvard.edu/kisin. · Zbl 1251.11045
[58] S. Lang, Algebraic groups over finite fields. Amer. J. Math. 78 (1956), 555-563. · Zbl 0073.37901
[59] M. Lazard, Les zéros des fonctions analytiques d’une variable sur un corps valué complet. Inst. Hautes Études Sci. Publ. Math. 14 (1962), 47-75; also available at http://www.numdam.org/item?id=PMIHES 1962 14 47 0. · Zbl 0119.03701
[60] H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge, 1986. · Zbl 0603.13001
[61] B.Mazur,DeformingGaloisrepresentations.InGaloisgroupsoverQ, pp. 385-437, Math. Sci. Res. Inst. Publ., New York, 1989; also available at http://www.math.harvard.edu/mazur/.
[62] Y. Mishiba, Onv-adic periods oft-motives. J. Number Theory 132(10) (2012), 2132- 2165; also available asarXiv:1105.6243. · Zbl 1266.11075
[63] J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften 323, Springer, Berlin, 2008. · Zbl 1136.11001
[64] W. Niziol, Crystalline conjecture viaK-theory. Ann. Sci. École Norm. Sup. (4) 31(5) (1998), 659-681; also available athttp://www.math.utah.edu/niziol/. · Zbl 0929.14009
[65] M. Papanikolas, Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms. Invent. Math. 171 (2008), 123-174; also available as arxiv:math.NT/0506078. · Zbl 1235.11074
[66] G. Pappas and M. Rapoport,ˆ-modules and coefficient spaces. Moscow Math. J. 9(3) (2009), 625-664, also available asarXiv:0811.1170. · Zbl 1194.14032
[67] M. Raynaud, Schémas en groupes de type.p; : : : ; p/. Bul. Soc. Math. France 102 (1974), 241-280; also available athttp://www.numdam.org/item?id=BSMF 1974 102 · Zbl 0325.14020
[68] M. Schlessinger, Functors of Artin rings. Trans. Amer. Math. Soc. 130 (1968), 208-222; available athttp://www.ams.org/journals/tran/1968-130-02/S0002-9947-1968-0217093-3/. · Zbl 0167.49503
[69] J.-P. Serre, Classes de corps cyclotomiques, d’après K. Iwasawa, Séminaire Bourbaki, vol. 5, Exp. No. 174 (1958-1960), 83-93, Soc. Math. France, Paris, 1995.
[70] J. Tate,p-Divisible Groups, Proc. Conf. Local Fields (Driebergen, 1966), 158-183, Springer, Berlin, 1967.
[71] T. Tsuji,p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case. Invent. Math. 137(2) (1999), 233-411. · Zbl 0945.14008
[72] J. Yu, On periods and quasi-periods of Drinfeld modules. Compositio Math. 74(3) (1990), 235-245. · Zbl 0703.11067
[73] T.
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