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 Izv. Akad. Nauk SSSR Ser. Mat., 1989, Volume 53, Issue 6, Pages 1135–1182 (Mi izv1152)

Modular representations of the Galois group of a local field, and a generalization of the Shafarevich conjecture

V. A. Abrashkin

Abstract: Let $M\Gamma^{\mathrm{cris}}(\mathbf Q_p)$ be the category of crystalline representations of the Galois group of the field of fractions of the ring of Witt vectors of an algebraically closed field of characteristic $p>0$. The author describes the subfactors annihilated by multiplication by $p$ of the representations from $M\Gamma^{\mathrm{cris}}(\mathbf Q_p)$ arising from filtered modules of filtration length $<p$, and proves a generalization of the Shafarevich conjecture that there do not exist abelian schemes over $\mathbf Z$: if $X$ is a smooth proper scheme over the ring of integers of the field $\mathbf Q$ (respectively $\mathbf Q(\sqrt{-1} )$, $\mathbf Q(\sqrt{-3} )$, $\mathbf Q(\sqrt{-5})$ ), then the Hodge numbers of the complex manifold $X_{\mathbf C}$ satisfy $h^{ij}=0$ for $i\ne j$ and $i+j\leqslant3$ (respectively $i+j\leqslant2$).
Bibliography: 17 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1990, 35:3, 469–518

Bibliographic databases:

UDC: 512.7
MSC: Primary 11S25, 14F30; Secondary 11G10

Citation: V. A. Abrashkin, “Modular representations of the Galois group of a local field, and a generalization of the Shafarevich conjecture”, Izv. Akad. Nauk SSSR Ser. Mat., 53:6 (1989), 1135–1182; Math. USSR-Izv., 35:3 (1990), 469–518

Citation in format AMSBIB
\Bibitem{Abr89} \by V.~A.~Abrashkin \paper Modular representations of the Galois group of a~local field, and a~generalization of the Shafarevich conjecture \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1989 \vol 53 \issue 6 \pages 1135--1182 \mathnet{http://mi.mathnet.ru/izv1152} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1039960} \zmath{https://zbmath.org/?q=an:0733.14008} \transl \jour Math. USSR-Izv. \yr 1990 \vol 35 \issue 3 \pages 469--518 \crossref{https://doi.org/10.1070/IM1990v035n03ABEH000715} 

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This publication is cited in the following articles:
1. V. A. Abrashkin, “The image of the Galois group for some crystalline representations”, Izv. Math., 63:1 (1999), 1–36
2. Abrashkin, V, “Characteristic p Analogue of Modules with Finite Crystalline Height”, Pure and Applied Mathematics Quarterly, 5:1 (2009), 469
3. Shin Hattori, “On a ramification bound of torsion semi-stable representations over a local field”, Journal of Number Theory, 129:10 (2009), 2474
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