Geodesic
Modular representations of the Galois group of a~local field, and a~generalization of the Shafarevich conjecture
Izvestiya. Mathematics , Tome 35 (1990) no. 3, pp. 469-518.

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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 $$, 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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     title = {Modular representations of the {Galois} group of a~local field, and a~generalization of the {Shafarevich} conjecture},
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     url = {http://geodesic.mathdoc.fr/item/IM2_1990_35_3_a0/}
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V. A. Abrashkin. Modular representations of the Galois group of a~local field, and a~generalization of the Shafarevich conjecture. Izvestiya. Mathematics , Tome 35 (1990) no. 3, pp. 469-518. http://geodesic.mathdoc.fr/item/IM2_1990_35_3_a0/

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