Geodesic
Galois moduli of period $p$ group schemes over a ring of Witt vectors
Izvestiya. Mathematics, Tome 31 (1988) no. 1, pp. 1-46 
V. A. Abrashkin. Galois moduli of period $p$ group schemes over a ring of Witt vectors. Izvestiya. Mathematics, Tome 31 (1988) no. 1, pp. 1-46. http://geodesic.mathdoc.fr/item/IM2_1988_31_1_a0/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Shafarevich I. R., “Polya algebraicheskikh chisel”, Proc. Int. Congr. Math., Stockholm, 1962, 163–176

[2] Fontaine J.-M., “Il n'y a pas de variété abélienne sur $\mathbf{Z}$”, Invent. Math., 81:3 (1985), 515–538 | DOI | MR | Zbl

[3] Abrashkin V. A., “Gruppovye skhemy perioda $p$ nad koltsom vektorov Vitta”, Dokl. AN SSSR, 283:6 (1985), 1289–1294 | MR | Zbl

[4] Abrashkin V. A., “Sistemy Khondy gruppovykh skhem perioda $p$”, Izv. AN SSSR. Ser. matem., 51:3 (1987), 451–484 | MR | Zbl

[5] Abrashkin V. A., “Khoroshaya reduktsiya abelevykh mnogoobrazii”, Izv. AN SSSR. Ser. matem., 40:2 (1976), 262–272 | MR | Zbl

[6] Abrashkin V. A., “$p$-delimye gruppy nad $\mathbf{Z}$”, Izv. AN SSSR. Ser. matem., 41:5 (1977), 987–1007 | MR | Zbl

[7] Abrashkin V. A., “2-delimye gruppy nad $\mathbf{Z}$”, Matem. zametki, 19:5 (1976), 717–726 | MR | Zbl

[8] Raynaud M., “Schemas en groups de type $(p,\dots,p)$”, Bull. Soc. Math. France, 102 (1974), 241–280 | MR | Zbl

[9] Teit Dzh., “$p$-delimye gruppy”, Matematika, 13:2 (1969), 3–25 | MR

[10] Fontaine J.-M., “Groups finis commutatifs sur le vecteurs de Witt”, C.R. Acad. Sci., 280:21 (1975), A1423–A1425 | MR

[11] Shafarevich I. R., “Obschii zakon vzaimnosti”, Matem. sb., 26:1 (1950), 113–146 | Zbl

[12] Ivasava K., Lokalnaya teoriya polei klassov, Mir, M., 1983 | MR

[13] Martinet J., “Petits discriminants des corps de nombres”, Lond. Math. Soc. Lect. Note Ser., 56, 1982, 151–193 | MR | Zbl

[14] Steinberg R., Lektsii o gruppakh Shevalle, Mir, M., 1975 | MR | Zbl

[15] Serr Zh., Algebraicheskie gruppy i polya klassov, Mir, M., 1968