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Mosc. Math. J., 2002, Volume 2, Number 2, Pages 249–279 (Mi mmj55)  

This article is cited in 23 scientific papers (total in 23 papers)

Group schemes with strict $\mathcal{O}$-action

G. Faltings

Max Planck Institute for Mathematics

Abstract: Let $\mathcal{O}$ denote the ring of integers in a $p$-adic local field. Recall that $\mathcal{O}$-modules are formal groups with an $\mathcal{O}$-action such that the induced action on the Lie algebra is via scalars. In the paper this notion is generalised to finite flat group schemes. It is shown that the usual properties carry over. For example, Cartier duality holds with the multiplicative group replaced by the Lubin–Tate group. We also show that liftings over $\mathcal{O}$-divided powers are controlled by Dieudonné modules or, better, by complexes. For these facts new proofs have to be invented, because the classical recipe of embedding into abelian varieties is not available.

Key words and phrases: Finite flat group schemes, Lubin–Tate groups, $\mathcal{O}$-modules.

DOI: https://doi.org/10.17323/1609-4514-2002-2-2-249-279

Full text: http://www.ams.org/.../abst2-2-2002.html
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MSC: 14L15
Received: February 18, 2002; in revised form May 28, 2002
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Citation: G. Faltings, “Group schemes with strict $\mathcal{O}$-action”, Mosc. Math. J., 2:2 (2002), 249–279

Citation in format AMSBIB
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\by G.~Faltings
\paper Group schemes with strict $\mathcal{O}$-action
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\yr 2002
\vol 2
\issue 2
\pages 249--279
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    This publication is cited in the following articles:
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    2. Strauch M., “Geometrically connected components of Lubin-Tate deformation spaces with level structures”, Pure and Applied Mathematics Quarterly, 4:4 (2008), 1215–1232  crossref  mathscinet  zmath  isi
    3. Strauch M., “Deformation spaces of one-dimensional formal modules and their cohomology”, Advances in Mathematics, 217:3 (2008), 889–951  crossref  mathscinet  zmath  isi
    4. Hartl U., “A dictionary between Fontaine-Theory and its analogue in equal characteristic”, Journal of Number Theory, 129:7 (2009), 1734–1757  crossref  mathscinet  zmath  isi
    5. Abrashkin V., “Characteristic p Analogue of Modules with Finite Crystalline Height”, Pure and Applied Mathematics Quarterly, 5:1 (2009), 469–494  crossref  mathscinet  zmath  isi
    6. Abrashkin V., “Group schemes of period p > 2”, Proc London Math Soc, 101:1 (2010), 207–259  crossref  mathscinet  zmath  isi  elib
    7. Yoshida T., “On non-abelian Lubin-Tate theory via vanishing cycles”, Sapporo 2007: Algebraic and Arithmetic Structures of Moduli Spaces, Advanced Studies in Pure Mathematics, 58, 2010, 361–402  mathscinet  zmath  isi
    8. Kim W., “Galois deformation theory for norm fields and flat deformation rings”, J Number Theory, 131:7 (2011), 1258–1275  crossref  mathscinet  zmath  isi  elib
    9. Genestier A., Lafforgue V., “Fontaine's theory in equal characteristics”, Ann Sci Éc Norm Supér (4), 44:2 (2011), 263–360  crossref  mathscinet  zmath  isi
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    11. Scholze P., “The Local Langlands Correspondence for Gl(N) Over P-Adic Fields”, Invent. Math., 192:3 (2013), 663–715  crossref  mathscinet  zmath  adsnasa  isi
    12. Brasca R., “P-Adic Modular Forms of Non-Integral Weight Over Shimura Curves”, Compos. Math., 149:1 (2013), 32–62  crossref  mathscinet  zmath  isi
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    14. Fargues L., Fontaine J.-M., “Vector Bundles on Curves and P-Adic Hodge Theory”, Automorphic Forms and Galois Representations, Vol 2, London Mathematical Society Lecture Note Series, 415, eds. Diamond F., Kassaei P., Kim M., Cambridge Univ Press, 2014, 17–104  mathscinet  zmath  isi
    15. Shen X., “P-Adic Families of Automorphic Forms Over Some Unitary Shimura Varieties”, Math. Res. Lett., 23:5 (2016), 1469–1506  crossref  mathscinet  zmath  isi
    16. Ahsendorf T., Cheng Ch., Zink T., “O-Displays and Pi-Divisible Formal O-Modules”, J. Algebra, 457 (2016), 129–193  crossref  mathscinet  zmath  isi
    17. Cais B., Liu T., “on F-Crystalline Representations”, Doc. Math., 21 (2016), 223–270  mathscinet  zmath  isi
    18. Pilloni V., Stroh B., “Overconvergence, Ramification and Modularity”, Asterisque, 2016, no. 382, 195–266  mathscinet  zmath  isi
    19. Poguntke T., “Group Schemes With F-Q-Action”, Bull. Soc. Math. Fr., 145:2 (2017), 345–380  crossref  zmath  isi  scopus
    20. Lafforgue V., “Chtoucas For Reductive Groups and Parameterization of Global Langlands”, J. Am. Math. Soc., 31:3 (2018), 719–891  crossref  mathscinet  zmath  isi  scopus
    21. Liu Y., Zhang Sh., Zhang W., “A P-Adic Waldspurger Formula”, Duke Math. J., 167:4 (2018), 743–833  crossref  mathscinet  zmath  isi  scopus
    22. Fargues L., Fontaine J.-M., Colmez P., “Curves and Vector Bundles in P-Adic Hodge Theory”, Asterisque, 2018, no. 406, 51+  mathscinet  isi
    23. Cheng Ch., “Breuil O-Windows and Pi-Divisible O-Modules”, Trans. Am. Math. Soc., 370:1 (2018), 695–726  crossref  mathscinet  zmath  isi  scopus
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