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

V. A. Abrashkin

Izvestiya. Mathematics , Tome 31 (1988) no. 1, pp. 1-46

Résumé

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.