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

V. A. Abrashkin

Izvestiya. Mathematics , Tome 35 (1990) no. 3, pp. 469-518

Résumé

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.