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Izv. Akad. Nauk SSSR Ser. Mat., 1987, Volume 51, Issue 4, Pages 691–736 (Mi izv1316)  

This article is cited in 9 scientific papers (total in 9 papers)

Galois moduli of period $p$ group schemes over a ring of Witt vectors

V. A. Abrashkin


Abstract: Necessary and sufficient conditions (completely sufficient only when $p>2$) are obtained that are satisfied by the Galois modules of the geometric points of a finite commutative period $p$ group scheme defined over a ring of Witt vectors. As an application of these results it is proved that there are no abelian schemes over the ring of integers of the fields $\mathbf Q$, $\mathbf Q(\sqrt{-1})$, $\mathbf Q(\sqrt{\pm2})$, $\mathbf Q(\sqrt{-3})$, $\mathbf Q(\sqrt{-7})$, $\mathbf Q(\sqrt[5]{1})$. The case of the field $\mathbf Q$ answers a conjecture of Shafarevich (at the 1962 ICM in Stockholm) that there do not exist Abelian varieties or curves of genus $g\geqslant1$ defined over this field and having everywhere good reduction.
Bibliography: 15 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1988, 31:1, 1–46

Bibliographic databases:

UDC: 512.747
MSC: Primary 11S31, 14L05, 14L15; Secondary 11G10, 11S25, 14K15, 14G25
Received: 23.09.1985

Citation: V. A. Abrashkin, “Galois moduli of period $p$ group schemes over a ring of Witt vectors”, Izv. Akad. Nauk SSSR Ser. Mat., 51:4 (1987), 691–736; Math. USSR-Izv., 31:1 (1988), 1–46

Citation in format AMSBIB
\Bibitem{Abr87}
\by V.~A.~Abrashkin
\paper Galois moduli of period~$p$ group schemes over a~ring of Witt vectors
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1987
\vol 51
\issue 4
\pages 691--736
\mathnet{http://mi.mathnet.ru/izv1316}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=914857}
\zmath{https://zbmath.org/?q=an:0674.14035}
\transl
\jour Math. USSR-Izv.
\yr 1988
\vol 31
\issue 1
\pages 1--46
\crossref{https://doi.org/10.1070/IM1988v031n01ABEH001042}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. A. Abrashkin, “Modular representations of the Galois group of a local field, and a generalization of the Shafarevich conjecture”, Math. USSR-Izv., 35:3 (1990), 469–518  mathnet  crossref  mathscinet  zmath
    2. V. A. Abrashkin, “Modification of the Fontaine–Laffaille functor”, Math. USSR-Izv., 34:3 (1990), 467–516  mathnet  crossref  mathscinet  zmath
    3. Marcin Mazur, “Finite group schemes and a conjecture of Kitaoka”, crll, 2003:557 (2003), 103  crossref  mathscinet  zmath
    4. Abrashkin V., “Galois modules arising from Faltings's strict modules”, Compositio Mathematica, 142:4 (2006), 867–888  crossref  isi
    5. EKATERINA S. KHREBTOVA, DMITRY MALININ, “ON FINITE GALOIS STABLE ARITHMETIC GROUPS AND THEIR APPLICATIONS”, J. Algebra Appl, 07:06 (2008), 773  crossref
    6. H.-J. BARTELS, D. A. MALININ, “ON FINITE GALOIS STABLE SUBGROUPS OF GLnIN SOME RELATIVE EXTENSIONS OF NUMBER FIELDS”, J. Algebra Appl, 08:04 (2009), 493  crossref
    7. René Schoof, “Semistable abelian varieties with good reduction outside 15”, manuscripta math, 139:1-2 (2011), 49  crossref
    8. Pietro Ploner, “Computation of framed deformation functors”, Journal of Number Theory, 2015  crossref
    9. S. O. Gorchinskiy, Vik. S. Kulikov, A. N. Parshin, V. L. Popov, “Igor Rostislavovich Shafarevich and His Mathematical Heritage”, Proc. Steklov Inst. Math., 307 (2019), 1–21  mathnet  crossref  crossref  mathscinet  isi  elib
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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