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There are no abelian varieties over \(\mathbb Z\). (Il n’y a pas de variété abélienne sur \(\mathbb Z\).) (French) Zbl 0612.14043
Over the field \(E={\mathbb Q}\), \({\mathbb Q}(\sqrt{-1})\), \({\mathbb Q}(\sqrt{-3})\), or \({\mathbb Q}(\sqrt{-5})\), there are no abelian varieties of dimension \(\geq 1\) having good reduction everywhere. This is the final result of the paper. In particular, in the case \(E={\mathbb Q}\), this assertion, which is also the title of this paper, has been known as a conjecture of Shafarevich.
Now, a little more generally, let \(J\) be a finite group scheme killed by \(p\) over the ring of integers of a number field \(E\) and \(F\) the field generated by the algebraic points of \(J\) over \(E\). A majoration of the discriminant of the field \(F\) over \(E\), which is discussed in this paper, a minoration of it given by the method of Odlyzko-Poitou-Serre, and the theory of Néron models of abelian varieties are the keys to the proof of the final result. The local version of the majoration of the discriminant, i.e., in the case of a complete valuation field \(K\) of characteristic 0, instead of the number field \(E\), was the starting point in this paper.
Reviewer: S.Koizumi

MSC:
14K15 Arithmetic ground fields for abelian varieties
11G10 Abelian varieties of dimension \(> 1\)
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
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References:
[1] [AnIV] Birch, B.J., Kuyk, W.: Modular functions of one variable IV, Antwerp. Lect. Notes Math. vol. 476. Berlin-Heidelberg-New York: Springer 1975 · Zbl 0315.14014
[2] [BBM] Berthelot, P., Breen, L., Messing, W.: Théorie de Dieudonné cristalline, II. Lect. Notes Math. vol. 930. Berlin-Heidelberg-New York: Springer 1982
[3] [De] Demazure, M.: Lecture onp-divisible groups. Lect. Notes Math. vol. 302. Berlin-Heidelberg-New York: Springer 1972
[4] [DD] Diaz y Diaz, F.: Tables minorant la racinen-ième du discriminant d’un corps de degrén. Publ. Math. d’Orsay. Orsay: Université d’Orsay 1980
[5] [Fa] Faltings, G.: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math.73, 349-366 (1983) · Zbl 0588.14026 · doi:10.1007/BF01388432
[6] [Fo] Fontaine, J.M.: Formes différentielles et modules de Tate des variétés abéliennes sur les corps locaux. Invent. Math.65, 379-409 (1982) · Zbl 0502.14015 · doi:10.1007/BF01396625
[7] [I] Illusie, L.: Déformations de groupes de Barsotti-Tate. In: Séminaire sur les pinceaux arithmétiques: La conjecture de Mordell (L. Szpiro, ed.), pp. 151-198. Astérisque, vol. 127. Paris: Soc. Math. Fr. 1985
[8] [L] Lang, S.: Algebraic number theory. Reading, MA: Addison Wesley 1970 · Zbl 0211.38404
[9] [Ma] Masley, J.M.: Where are number fields with small class number?. In: Number theory Carbondale 1979. Proceedings, pp. 221-224, Lect. Notes Math. vol. 751. Berlin-Heidelberg-New York: Springer 1979 · Zbl 0421.12005
[10] [Me] Mestre, J.-F.: Formules explicites et minorations de conducteurs de variétés algébriques. Compos. Math. (à paraître)
[11] [Pa] Par?in, A.N.: Quelques conjectures de finitude en géométrie diophantienne. Actes du Congrès Int. Math. Nice, vol. 1, pp. 467-471. Paris: Gauthier-Villars 1971
[12] [R] Raynaud, M.: Schémas en groupes de type (p, ...,p). Bull. Soc. Math. Fr.102, 241-280 (1974) · Zbl 0325.14020
[13] [Sen] Sen, S.: Ramification inp-adic Lie extensions. Invent. Math.17, 44-50 (1972) · Zbl 0242.12012 · doi:10.1007/BF01390022
[14] [Se1] Serre, J.-P.: Corps locaux. 2\(\deg\) édition, Paris: Hermann 1968
[15] [Se2] Serre, J.-P.: Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math.15, 259-331 (1972) · Zbl 0235.14012 · doi:10.1007/BF01405086
[16] [Se3] Serre, J.-P.: Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures). Seminaire Delange-Pisot-Poitou (Théorie des nombres), N\(\deg\) 19. Paris: Université de Paris 1969-1970
[17] [Sh] Shafarevich, I.: Communication au Congrès de Stockholm, 1962 (English translation) Algebraic Number Fields. Am. Math. Soc. Trans.31, 25-39 (1963)
[18] [SGA7I] Grothendieck, A.: Groupes de monodromie en géométrie algébrique. Lect. Notes Math. vol. 288. Berlin-Heidelberg-New York: Springer 1972
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