Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-02-04T01:42:12.495Z Has data issue: false hasContentIssue false
  • English
  • Français

Faltings extension and Hodge-Tate filtration for abelian varieties over p-adic local fields with imperfect residue fields

Part of: (Co)homology theory Abelian varieties and schemes Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  11 June 2020

Tongmu He
Tongmu He*
Affiliation:
Institut des Hautes Études Scientifiques, 35 route de Chartres, 91440Bures-sur-Yvette, France
*
e-mail: hetm15@ihes.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Let K be a complete discrete valuation field of characteristic $0$ , with not necessarily perfect residue field of characteristic $p>0$ . We define a Faltings extension of $\mathcal {O}_K$ over $\mathbb {Z}_p$ , and we construct a Hodge-Tate filtration for abelian varieties over K by generalizing Fontaine’s construction [Fon82] where he treated the perfect residue field case.

Keywords

Hodge-TateFaltings extensionabelian varietyp-adicimperfect residue field

MSC classification

Primary: 14F30: $p$-adic cohomology, crystalline cohomology
Secondary: 14G20: Local ground fields 14K15: Arithmetic ground fields
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 2 , June 2021 , pp. 247 - 263
Check for updates
Copyright
© Canadian Mathematical Society 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abbes, A. and Gros, M., Les suites spectrales de Hodge-Tate. Preprint, 2020. arxiv:2003.04714 Google Scholar
Abbes, A., Gros, M., and Tsuji, T., The $p$ -adic Simpson correspondence. Ann. Math. Stud., 193, Princeton University Press, Princeton, NJ, 2016. http://dx.doi.org/10.1515/9781400881239 Google Scholar
Caraiani, A. and Scholze, P., On the generic part of the cohomology of compact unitary Shimura varieties. Ann. of Math. (2) 186(2017), 649–766. http://dx.doi.org/10.4007/annals.2017.186.3.1 CrossRefGoogle Scholar
Deligne, P., Raynaud, M., Rim, D. S., and Grothendieck, A., Groupes de monodromie en géométrie algébrique. I. In: Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I), Lecture Notes in Mathematics, 288, Springer-Verlag, Berlin-New York, 1972.Google Scholar
Faltings, G., $p$ -adic Hodge theory. J. Amer. Math. Soc. 1(1988), 255–299. http://dx.doi.org/10.2307/1990970 Google Scholar
Faltings, G., Almost étale extensions. Cohomologies $p$ -adiques et applications arithmétiques, II. Astérisque 279(2002), 185–270.Google Scholar
Fontaine, J.-M., Formes différentielles et modules de Tate des variétés abéliennes sur les corps locaux. Invent. Math. 65(1981/82), 379–409. http://dx.doi.org/10.1007/BF01396625 CrossRefGoogle Scholar
Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I. Inst. Hautes Études Sci. Publ. Math. 20(1964), 259.Google Scholar
Hyodo, O., On the Hodge-Tate decomposition in the imperfect residue field case. J. Reine Angew. Math. 365(1986), 97–113. http://dx.doi.org/10.1515/crll.1986.365.97 Google Scholar
Jannsen, U., Continuous étale cohomology. Math. Ann. 280(1988), 207–245.CrossRefGoogle Scholar
Scholze, P., Perfectoid spaces: a survey. In: Current developments in mathematics, 2012, Int. Press, Somerville, MA, 2013, pp. 193–227.Google Scholar
Tate, J. T., $p$ -divisible groups. Proc. Conf. Local Fields (Driebergen, 1966), Springer, Berlin, 1967, pp. 158–183.CrossRefGoogle Scholar
The Stacks Project Authors, The stacks project. 2020. https://stacks.math.columbia.edu.Google Scholar
Tsuji, T., $p$ -adic étale cohomology and crystalline cohomology in the semi-stable reduction case. Invent. Math. 137(1999), 233–411. http://dx.doi.org/10.1007/s002220050330 CrossRefGoogle Scholar
Tsuji, T., Semi-stable conjecture of Fontaine-Jannsen: a survey. Cohomologies $p$ -adiques et applications arithmétiques, II. Astérisque 279(2002), 323–370.Google Scholar