Skip to main content Accessibility help
Hostname: page-component-6c8bd87754-ncgjf Total loading time: 0.281 Render date: 2022-01-16T19:38:56.739Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

1 - A semi-stable case of the Shafarevich Conjecture

Published online by Cambridge University Press:  05 October 2014

Victor Abrashkin
University of Durham
Fred Diamond
King's College London
Payman L. Kassaei
King's College London
Minhyong Kim
University of Oxford
Get access


Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Publisher: Cambridge University Press
Print publication year: 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


[1] V. A., Abrashkin, Good reduction of two-dimensional Abelian varieties (Russian)Izv. Akad. Nauk SSSR Ser. Mat., 40 (1976), no. 2, 262–272Google Scholar
[2] V. A., Abrashkin, p-divisible groups over ℤ (Russian)Izv. Akad. Nauk SSSR Ser. Mat., 41 (1977), no. 5, 987–1007Google Scholar
[3] V. A., Abrashkin, Group schemes of period p over the ring of Witt vectors (Russian)Dokl. Akad. Nauk SSSR, 283 (1985), no. 6, 1289–1294Google Scholar
[4] V. A., Abrashkin, Honda systems of group schemes of period p (Russian)Izv. Akad. Nauk SSSR Ser. Mat., 51 (1987), no. 3, 451–484, 688; translation in Math. USSR-Izv. 30 (1988), no. 3, 419–453Google Scholar
[5] V. A., Abrashkin, Galois modules of group schemes of period p over the ring of Witt vectors (Russian)Izv. Akad. Nauk SSSR Ser. Mat., 51 (1987), no. 4, 691–736; translation in Math. USSR-Izv. 31 (1988), no. 1, 1–46Google Scholar
[6] V. A., Abrashkin, Modification of the Fontaine-Laffaille functor (Russian), Izv. Akad. Nauk SSSR Ser Mat, 53 (1989), 451–497; translation in Math. USSR-Izv., 34 (1990), 57–97Google Scholar
[7] V. A., Abrashkin, Modular representations of the Galois group of a local field and a generalization of a conjecture of Shafarevich (Russian)Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1135–1182; translation in Math. USSR-Izv. 35 (1990), no. 3, 469–518Google Scholar
[8] V. A., Abrashkin, A ramification filtration of the Galois group of a local field. III. (Russian)Izv. Ross. Akad. Nauk Ser. Mat. 62 (1998), no. 5, 3–48; translation in Izv. Math. 62 (1998), no. 5, 857–900Google Scholar
[9] V., Abrashkin, Galois modules arising from Faltings' strict modules. Compos. Math. 142 (2006), no. 4, 867–888Google Scholar
[10] V., Abrashkin, Characteristic p analogue of modules with finite crystalline height, Pure Appl. Math. Q., 5 (2009), 469–494Google Scholar
[11] V., Abrashkin, Projective varieties with bad reduction at 3 only, Doc. Math. 18 (2013), 547–619.Google Scholar
[12] Algebraic number theory. Proceedings of an instructional conference organized by the LMS. Edited by J. W. S., Cassels and A. Fröhlich Academic Press, London; Thompson Book Co., Inc., Washington, D.C. 1967
[13] A., Brumer, K., KramerNon-existence of certain semistable abelian varieties, Manuscripta Math., 106 (2001), 291–304Google Scholar
[14] Ch., Breuil, Construction de représentations p-adiques semi-stable, Ann. Sci. École Norm. Sup., 4 Ser., 31 (1998), 281–327Google Scholar
[15] Ch., Breuil, Représentations semi-stables et modules fortement divisibles, Invent. Math., 136 (1999), 89–122Google Scholar
[16] Ch., Breuil, Integral p-adic Hodge theory, Adv. Stud. Pure Math. 36 (2002), 51–80Google Scholar
[17] P., Deligne, Les corps locaux de caractristique p, limites de corps locaux de caractristique 0. (French) Representations of reductive groups over a local field, 119–157, Travaux en Cours, Hermann, Paris, 1984
[18] Dia Y, Diaz, Tables minorant la racine n-ième du discriminant d'un corps de degré n Publications Mathematiques d'Orsay, 80.06, Université de Paris-Sud, Départment de Mathématique, Bat. 425, 91405, Orsay, France
[19] J.-M., Fontaine, G., Laffaille, Construction de représentations p-adiques, Ann. Sci. École Norm. Sup., 4 Ser. 15 (1982), 547–608Google Scholar
[20] J.-M., Fontaine, Groupes finis commutatifs sur les vecteurs de Witt, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), A1423A1425Google Scholar
[21] J.-M., Fontaine, Il n'y a pas de variété abélienne sur ℤ, Invent. Math., 81 (1985), 515–538Google Scholar
[22] J.-M., Fontaine, W., Messing, p-adic periods and p-adic étale cohomology. In: Current Trends in Arithmetical Algebraic Geometry, K., Ribet, Editor, Contemporary Math. 67, Amer. Math. Soc., Providence (1987), 179–207
[23] J.-M., Fontaine, Schémas propres et lisses sur ℤ. Proceedings of the Indo-French Conference on Geometry (Bombay, 1989), 43–56, Hindustan Book Agency, Delhi, 1993
[24] Sh., Hattori, On a ramification bound of torsion semi-stable representations over a local field, J. Number theory, 129 (2009), 2474–2503Google Scholar
[25] J., Martinet, Petits discriminants des corps de nombres, London Math. Soc -Lect. note ser., 56 (1982), 151–193
[26] W., MessingThe crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture notes in Mathematics. 264, Springer-Verlag, 1972
[27] R., Schoof, Abelian varieties over ℚ with bad reduction in one prime only. Comp. Math., 141 (2005), 847–868.Google Scholar
[28] J.-P., Serre, Local FieldsBerlin, New York: Springer-Verlag, 1980
[29] I. R., Shafarevich, Algebraic number fields. Proc. Internat. Congr. Math. (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, (1963), 163–176
[30] G., Poitou, Minorations de discriminants (d'apr A. M. Odlyzko). Séminaire Bourbaki, Vol. 1975/76 28me anne, Exp. No. 479, pp. 136–153. Lecture Notes in Math., 567, Springer, Berlin, 1977 Berlin, New York: Springer-Verlag, 1980
[31] J., Tate, p-divisible groups, Proc. Conf. Local Fields (Dreibergen, 1966), Springer-Verlag, Berlin and New-York, 1967, pp. 158–183
[32] J.-P., Wintenberger, Le corps des normes de certaines extensions infinies des corps locaux; applicationAnn. Sci. Ec. Norm. Super., IV. Ser, 16 (1983), 59–89Google Scholar
[33] J.-P., Wintenberger, Extensions de Lie et groupes d'automorphismes des corps locaux de caractristique p. (French)C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 9, A477–A479Google Scholar

Send book to Kindle

To send this book to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats

Send book to Dropbox

To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox.

Available formats

Send book to Google Drive

To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive.

Available formats