Arithmetic of division fields

Brumer, Armand; Kramer, Kenneth

doi:10.1090/S0002-9939-2012-11500-X

Arithmetic of division fields

Authors: Armand Brumer and Kenneth Kramer
Journal: Proc. Amer. Math. Soc. 140 (2012), 2981-2995
MSC (2010): Primary 11F80; Secondary 11S15, 11G10, 11Y40
DOI: https://doi.org/10.1090/S0002-9939-2012-11500-X
Published electronically: January 12, 2012
MathSciNet review: 2917071
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the arithmetic of division fields of semistable abelian varieties $A_{/\mathbb{Q}}.$ The Galois group of $\mathbb{Q}(A[2])/\mathbb{Q}$ is analyzed when the conductor of is odd and squarefree. The irreducible semistable mod 2 representations of small conductor are determined under GRH. These results are used in our paper Paramodular abelian varieties of odd conductor.

1. Michael Aschbacher, Finite group theory, Cambridge Studies in Advanced Mathematics, vol. 10, Cambridge University Press, Cambridge, 1986. MR 895134
2. Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, https://doi.org/10.1006/jsco.1996.0125
3. Armand Brumer and Kenneth Kramer, Semistable abelian varieties with small division fields, Galois theory and modular forms, Dev. Math., vol. 11, Kluwer Acad. Publ., Boston, MA, 2004, pp. 13–37. MR 2059756, https://doi.org/10.1007/978-1-4613-0249-0_2
4. A. Brumer and K. Kramer, Paramodular abelian varieties of odd conductor, math arXiv:1004.4699.
5. F. Diaz y Diaz, Tables minorant la racine -ième du discriminant d'un corps de degré , Ph.D. Thesis, Publ. Math. Orsay, 1980.
6. R. H. Dye, Interrelations of symplectic and orthogonal groups in characteristic two, J. Algebra 59 (1979), no. 1, 202–221. MR 541675, https://doi.org/10.1016/0021-8693(79)90157-1
7. Jean-Marc Fontaine, Il n’y a pas de variété abélienne sur 𝑍, Invent. Math. 81 (1985), no. 3, 515–538 (French). MR 807070, https://doi.org/10.1007/BF01388584
8. Benedict H. Gross and Joe Harris, On some geometric constructions related to theta characteristics, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, pp. 279–311. MR 2058611
9. A. Grothendieck, Modèles de Néron et monodromie. Sém. de Géom. 7, Exposé IX, Lecture Notes in Math., 288, Springer-Verlag, 1973.
10. Farshid Hajir and Christian Maire, Extensions of number fields with wild ramification of bounded depth, Int. Math. Res. Not. 13 (2002), 667–696. MR 1890847, https://doi.org/10.1155/S1073792802106015
11. B. Huppert, Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703
12. John W. Jones, Wild ramification bounds and simple group Galois extensions ramified only at 2, Proc. Amer. Math. Soc. 139 (2011), no. 3, 807–821. MR 2745634, https://doi.org/10.1090/S0002-9939-2010-10628-7
13. John W. Jones and David P. Roberts, A database of local fields, J. Symbolic Comput. 41 (2006), no. 1, 80–97. MR 2194887, https://doi.org/10.1016/j.jsc.2005.09.003
14. John W. Jones and David P. Roberts, Galois number fields with small root discriminant, J. Number Theory 122 (2007), no. 2, 379–407. MR 2292261, https://doi.org/10.1016/j.jnt.2006.05.001
15. William M. Kantor, Subgroups of classical groups generated by long root elements, Trans. Amer. Math. Soc. 248 (1979), no. 2, 347–379. MR 522265, https://doi.org/10.1090/S0002-9947-1979-0522265-1
16. G. Kemper and G. Malle, The finite irreducible linear groups with polynomial ring of invariants, Transform. Groups 2 (1997), no. 1, 57–89. MR 1439246, https://doi.org/10.1007/BF01234631
17. Jack McLaughlin, Some subgroups of 𝑆𝐿_{𝑛}(𝐹₂), Illinois J. Math. 13 (1969), 108–115. MR 0237660
18. A. M. Odlyzko, Lower bounds for discriminants of number fields. II, Tôhoku Math. J. 29 (1977), no. 2, 209–216. MR 441918, https://doi.org/10.2748/tmj/1178240652
19. A. M. Odlyzko, Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results, Sém. Théor. Nombres Bordeaux (2) 2 (1990), no. 1, 119–141 (English, with French summary). MR 1061762
20. René Schoof, Abelian varieties over cyclotomic fields with good reduction everywhere, Math. Ann. 325 (2003), no. 3, 413–448. MR 1968602, https://doi.org/10.1007/s00208-002-0368-7
21. Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237
22. A. E. Zalesskiĭ and V. N. Serežkin, Finite linear groups generated by reflections, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 6, 1279–1307, 38 (Russian). MR 603578